Repositório UFAL: Força e torque de radiação sobre uma partícula viscoelástica em um fluido ideal

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❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞❡ ❆❧❛❣♦❛s

■♥st✐t✉t♦ ❞❡ ❋ís✐❝❛

❏♦sé P❡r❡✐r❛ ▲❡ã♦ ◆❡t♦

  

❋♦rç❛ ❡ t♦rq✉❡ ❞❡ r❛❞✐❛çã♦ s♦❜r❡ ✉♠❛ ♣❛rtí❝✉❧❛

✈✐s❝♦❡❧ást✐❝❛ ❡♠ ✉♠ ✢✉✐❞♦ ✐❞❡❛❧

4 [nJ] 5

  2 4 ky

  3 2

  2 1

  4 4 2 kx 2 4

❋♦rç❛ ❞❡ r❛❞✐❛çã♦ tr❛♥s✈❡rs❛❧ ❞❡ ✉♠ ❢❡✐①❡ ❞❡ ❇❡ss❡❧ ❞❡ ♣r✐♠❡✐r❛ ♦r❞❡♠ ❡♠ ✉♠❛ ♣❛rtí❝✉❧❛ ❞❡

♣♦❧✐❡t✐❧❡♥♦✳

  

▼❛❝❡✐ó ✲ ❇r❛s✐❧

❙❡t❡♠❜r♦ ✲ ✷✵✶✺

  

❏♦sé P❡r❡✐r❛ ▲❡ã♦ ◆❡t♦

❋♦rç❛ ❡ t♦rq✉❡ ❞❡ r❛❞✐❛çã♦ s♦❜r❡ ✉♠❛ ♣❛rtí❝✉❧❛

✈✐s❝♦❡❧ást✐❝❛ ❡♠ ✉♠ ✢✉✐❞♦ ✐❞❡❛❧

  ❚❡s❡ ❛♣r❡s❡♥t❛❞❛ ♥♦ ■♥st✐t✉t♦ ❞❡ ❋í✲ s✐❝❛ ❞❛ ❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞❡ ❆❧❛❣♦❛s✱ ❝♦♠♦ r❡q✉✐s✐t♦ ♥❡❝❡ssár✐♦ ♣❛r❛ ❛ ♦❜t❡♥çã♦ ❞♦ tít✉❧♦ ❞❡ ❉♦✉t♦r ❡♠ ❋ís✐❝❛✳

  ❖r✐❡♥t❛❞♦r✿ Pr♦❢✳ ❉r✳ ●❧❛✉❜❡r ❚✳ ❙✐❧✈❛ ▼❛❝❡✐ó ✲ ❇r❛s✐❧

  ❙❡t❡♠❜r♦ ✲ ✷✵✶✺

  ❈❛t❛❧♦❣❛çã♦ ♥❛ ❢♦♥t❡ ❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞❡ ❆❧❛❣♦❛s ❇✐❜❧✐♦t❡❝❛ ❈❡♥tr❛❧ ❉✐✈✐sã♦ ❞❡ ❚r❛t❛♠❡♥t♦ ❚é❝♥✐❝♦

  ❇✐❜❧✐♦t❡❝ár✐♦ ❘❡s♣♦♥sá✈❡❧✿ ❱❛❧t❡r ❞♦s ❙❛♥t♦s ❆♥❞r❛❞❡

  ▲✹✸✼❢ ▲❡ã♦ ◆❡t♦✱ ❏♦sé P❡r❡✐r❛✳ ❋♦rç❛ ❡ t♦rq✉❡ ❞❡ r❛❞✐❛çã♦ s♦❜r❡ ✉♠❛ ♣❛rtí❝✉❧❛ ✈✐s❝♦❡❧ást✐❝❛ ❡♠ ✉♠

  ✢✉✐❞♦ ✐❞❡❛❧ ✴ ❏♦sé P❡r❡✐r❛ ▲❡ã♦ ◆❡t♦✳ ✕ ✷✵✶✺✳ ✾✽ ❢✳✿ ✐❧✳ ❖r✐❡♥t❛❞♦r✿ ●❧❛✉❜❡r ❚✳ ❙✐❧✈❛✳

  ❚❡s❡ ✭❞♦✉t♦r❛❞♦ ❡♠ ❋ís✐❝❛✮ ✕ ❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞❡ ❆❧❛❣♦❛s✳ ■♥st✐t✉t♦ ❞❡ ❋ís✐❝❛✳ Pr♦❣r❛♠❛ ❞❡ Pós✲●r❛❞✉❛çã♦ ❡♠ ❋ís✐❝❛✳ ▼❛❝❡✐ó✱ ✷✵✶✺✳

  ❇✐❜❧✐♦❣r❛✜❛✿ ❢✳ ✾✵ ✕ ✾✽✳ ✶✳ ❋♦rç❛ ❞❡ r❛❞✐❛çã♦ ❛❝úst✐❝❛✳ ✷✳ ❚♦rq✉❡ ❞❡ r❛❞✐❛çã♦✳ ✸✳ ❆❝úst✐❝❛✳ ✹✳

  ❊s♣❛❧❤❛♠❡♥t♦ ❛❝úst✐❝♦✳ ✺✳ ❱✐s❝♦❡❧❛st✐❝✐❞❛❞❡✳ ✻✳ ▼❛♥✐♣✉❧❛çã♦ ❞❡ ♣❛rtí❝✉❧❛s✳ ■✳ ❚ít✉❧♦✳

  ❈❉❯✿ ✺✸✵✳✶✹✺✳✻✿✺✸✾✳✶✷

  ❉❡❞✐❝♦ ❡st❛ t❡s❡ ❛ ♠✐♥❤❛ ❡s♣♦s❛ ▲✐❞✐❛♥❡ ❖♠❡♥❛✱ ❛♦s ♠❡✉s P❛✐s ■❧s♦♥ ▲❡ã♦ ❡ ▼❛r✐❛

  ❞❡ ❋át✐♠❛ ❡ ❛♦s ♠❡✉s ✐r♠ã♦s✱ ■❧s♦♥ ❏r✳✱ ■❡❧s♦♥✱ ■✈❡♥s ❡ ❋❡r♥❛♥❞❛✳

  

❆❣r❛❞❡❝✐♠❡♥t♦s

  ❆ ❉❡✉s ♣♦r t❡r s✐❞♦ ♠✐♥❤❛ ❢♦rt❛❧❡③❛ ❡ s✉st❡♥t♦ ♥❛s ❤♦r❛s ❞✐❢í❝❡✐s ❞✉r❛♥t❡ ❡st❛ ❡t❛♣❛ ❞❛ ♠✐♥❤❛ ✈✐❞❛✳

  ❆ ♠✐♥❤❛ ❡s♣♦s❛ ▲✐❞✐❛♥❡ ❖♠❡♥❛✱ ♣♦r ❡st❛r ❛♦ ♠❡✉ ❧❛❞♦ ❡♠ t♦❞♦s ♦s ♠♦♠❡♥t♦s ♠❡ ✐♥❝❡♥t✐✈❛♥❞♦✱ ♣❡❧♦ ♦ s❡✉ ❛♠♦r ✐♥❝♦♥❞✐❝✐♦♥❛❧ ❡ ♣♦r t♦❞❛ ❛ s✉❛ ♣❛❝✐ê♥❝✐❛ ❡ ❝♦♠♣r❡❡♥sã♦ ♥❡st❛ ❡t❛♣❛ tã♦ ❞✐❢í❝✐❧✳

  ❆♦s ♠❡✉s ♣❛✐s ■❧s♦♥ ❡ ❋át✐♠❛✱ ♣♦r t♦❞♦ ♦ s❛❝r✐❢í❝✐♦✱ ❞❡❞✐❝❛çã♦✱ ❡ ♣♦r ❛❜❞✐❝❛r❡♠ ❞❡ ♣r♦❥❡t♦s ♣❡ss♦❛✐s ♣❛r❛ q✉❡ ❡✉ t✐✈❡ss❡ ❛ ♦♣♦rt✉♥✐❞❛❞❡ ❞❡ ❡st✉❞❛r✳ ❊♥✜♠✱ ♦❜r✐❣❛❞♦ ♣♦r t❡r ♠❡ ❞❛❞♦ ❛ ❡❞✉❝❛çã♦ ❡ ❞✐r❡❝✐♦♥❛♠❡♥t♦ q✉❡ ❢♦r❛♠ ✐♠♣r❡s❝✐♥❞í✈❡✐s ♣❛r❛ ❛ ♠✐♥❤❛ ❢♦r♠❛çã♦✳

  ❆♦s ♠❡✉s ✐r♠ã♦s ■❧s♦♥ ❏r✱ ■❡❧s♦♥ ▲❡ã♦✱ ■✈❡♥s ▲❡ã♦ ❡ ■❧③❛ ❋❡r♥❛♥❞❛ ♣♦r t♦❞♦ ❝♦♠♣❛✲ ♥❤❡✐r✐s♠♦✱ ✜❞❡❧✐❞❛❞❡ ❡ ♣♦r s❡♠♣r❡ s❡r❡♠ ❡①❡♠♣❧♦s ❞❡ ❞❡t❡r♠✐♥❛çã♦ ❡ ✉♥✐ã♦ ♥❛ ♠✐♥❤❛ ✈✐❞❛✳

  ❆ ●r❛♥❞❡ ❢❛♠í❧✐❛✿ ❉✳ ❈❧❛✉❞❡t❡✱ ❉✳ ❙♦❝♦rr♦✱ ▲✐❞②❛♥❡ ■❣♥á❝✐♦✱ ■r✐s ▲✐♠❛✱ ●❛❜r✐❡❧❛ ❆③❡✈❡❞♦✱ ❘❡❣✐♥❛ ❚♦❧❡❞♦ ❡ ▼❛r❝♦s ❆♥❞ré ♣♦r t♦❞❛ ❛ ✉♥✐ã♦✱ ❛♣♦✐♦ ❡ ✐♥❝❡♥t✐✈♦✳ ❆❧é♠ ❞❛s ❞❛t❛s ✐♠♣♦rt❛♥t❡s✱ ♠♦♠❡♥t♦s ❡s♣❡❝✐❛✐s✱ q✉❡ ❡st❛♠♦s t♦❞♦s ❥✉♥t♦s ❢❛③❡♥❞♦ s❡♠♣r❡ ❜♦❛s ❝♦♥❢r❛t❡r♥✐③❛çõ❡s✱ ❝♦♠ ❜♦❛s ❝♦♥✈❡rs❛s ❡ r✐s❛❞❛s✳ ◆ã♦ ♣♦❞❡r✐❛ ❞❡✐①❛r ❞❡ ❛❣r❛❞❡❝❡r ❛♦s ♠❡✉s s♦❜r✐♥❤♦s✿ ▲❡♦♥❛r❞♦ ■❣♥á❝✐♦✱ ▼❛r✐❛ ▲ét✐❝✐❛ ❡ ■❛♥ ❘❛❢❛❡❧ ♣❡❧❛ ❛❧❡❣r✐❛ ❡ ❡♥❡r❣✐❛ ❞❡ s❡♠♣r❡✳

  ❆ t♦❞♦s ♠❡✉s ❢❛♠✐❧✐❛r❡s q✉❡ ❡st✐✈❡r❛♠ ♣r❡s❡♥t❡s ❞✉r❛♥t❡ ❛ ♠✐♥❤❛ ❢♦r♠❛çã♦✳ ❆♦ Pr♦❢✳ ●❧❛✉❜❡r ❚✳ ❙✐❧✈❛ ♣♦r ❛❝r❡❞✐t❛r ♥♦ ♠❡✉ tr❛❜❛❧❤♦✱ ♣❡❧♦ s❡✉ ❛♣♦✐♦✱ ❞❡❞✐❝❛çã♦

  ❡ ❞✐s♣♦s✐çã♦ ❡♠ ❛❥✉❞❛r✳ ❉✐s♣♦♥❞♦ ❞❡ ❤♦r❛s ❞❡ ♦r✐❡♥t❛çã♦ ♣❛r❛ ❛ ❝♦♥❝r❡t✐③❛çã♦ ❞❡st❡ tr❛❜❛❧❤♦✳ ❆♦ ❣r✉♣♦ t✉❝❛♥♦ ❛③✉❧✱ q✉❡ ❥á sã♦ ♠✐♥❤❛ ❢❛♠í❧✐❛✿ ❆♥❞ré ▼♦✉r❛ ✭●❛❜❡❤✮✱ ❊♠♠❛♥✉❡❧❧❡

  ❙❛t✐❦♦✱ ❍❡♥r✐q✉❡ ▲♦♣❡s ✭●❛❧♦ ❈❡❣♦✮ ❡ ❚❛♠✐r②s ▼❛❝❡❞♦✳ P❡❧♦s ♠♦♠❡♥t♦s ❞❡ ❞❡s❝♦♥tr❛çã♦✱ q✉❡ ♥ã♦ ❢♦r❛♠ ♣♦✉❝♦s✱ ♣❡❧❛ ❛♠✐③❛❞❡ ❞❡ ❝❛❞❛ ✉♠ ❞❡ ✈♦❝ês✳ P♦r ❡st❛r❡♠ s❡♠♣r❡ ♣r❡s❡♥t❡s✱ ✐♥❞❡♣❡♥❞❡♥t❡ ❞♦ ♠♦♠❡♥t♦ q✉❡ ❝❛❞❛ ✉♠ ❡st❛ ♣❛ss❛♥❞♦✳

  ❆♦s ❛♠✐❣♦s ❋r❡❞❡r✐❝♦ P❛ss♦s✱ ❆❧❡① ❈♦st❛ ❡ ❆♥❞❡rs♦♥ ❇❛♥❞❡✐r❛ ♣♦r ❡st❛r❡♠ s❡♠♣r❡ ❞✐s♣♦st♦s ❛ ❛❥✉❞❛r✱ ♣♦r t❡r ❛♣r❡♥❞✐❞♦ ♠✉✐t♦ ❝♦♠ ✈♦❝ês ❡ ♣❡❧♦s ✐♥❝❡♥t✐✈♦s ❞❛❞♦s ❞✉r❛♥t❡ t♦❞❛ ❛ ♠✐♥❤❛ ❢♦r♠❛çã♦✳

  ❆♦s ❛♠✐❣♦s ▲✉✐③ ❊❞✉❛r❞♦✱ ❱❛❧♠✐r♦ ❍♦r❛ ❡ ❘♦❣ér✐♦ ▼❛t✐❛s q✉❡ t✐✈❡ ❛ ♦♣♦rt✉♥✐❞❛❞❡ ❞❡ ❝♦♥✈✐✈❡r ❡ ❛♣r❡♥❞❡r ♠✉✐t♦✳

  ❛

  ❆♦s ❛♠✐❣♦s Pr♦❢✳ ❲❛♥❞❡❛r❧❡② ❉✐❛s ❡ Pr♦❢ ✳ ❙♦❝♦rr♦ ❙❡✐①❛s ♣❡❧♦ ❝♦♠♣❛♥❤❡✐r✐s♠♦✱ ♣❡❧❛s ❜♦❛s ❝♦♥✈❡rs❛s ❡ ♣❡❧♦s ✐♥❝❡♥t✐✈♦s q✉❡ ❢♦r❛♠ ❡ss❡♥❝✐❛✐s ♥❡st❛ ❡t❛♣❛✳

  ❆♦s ❛♠✐❣♦s ❞♦ ●❆❋✱ ❡♠ ❡s♣❡❝✐❛❧ ❛♦ Pr♦❢✳ ❆♥❞ré ❇❛❣❣✐♦ ♣♦r t♦❞❛ ❝♦❧❛❜♦r❛çã♦✳ ❆♦s ♣r♦❢❡ss♦r❡s ❞❛ ❜❛♥❝❛✱ ♣♦r ❛❝❡✐t❛r ♣❛rt✐❝✐♣❛r ❞❡st❡ ♠♦♠❡♥t♦ ✐♠♣♦rt❛♥t❡ ♥❛ ♠✐♥❤❛

  ❢♦r♠❛çã♦ ❡ ❝♦♥tr✐❜✉✐r ❝♦♠ ❡st❡ tr❛❜❛❧❤♦✳ ❆ t♦❞♦s ♦s ❛♠✐❣♦s ❞♦ ■♥st✐t✉t♦ ❞❡ ❋ís✐❝❛✱ ❡♠ ❡s♣❡❝✐❛❧✱ ❛♦s ❛♠✐❣♦s ❞❛ ❇❛t❝❛✈❡r♥❛✳ ❆ t♦❞♦ ❝♦r♣♦ ❞♦❝❡♥t❡ ❞♦ ■❋✱ ♣♦r ❝♦♥tr✐❜✉í❞♦ s✐❣♥✐✜❝❛t✐✈❛♠❡♥t❡ ♥❛ ♠✐♥❤❛ ❢♦r♠❛çã♦

  ❛❝❛❞ê♠✐❝❛✳ ❊♠ ❡s♣❡❝✐❛❧ ❛♦s ♣r♦❢❡ss♦r❡s✱ ❈r✐só❣♦♥♦ ❙✐❧✈❛ ❡ ❑❧é❜❡r ❙❡rr❛✳ ❆ ❈❆P❊❙ ♣❡❧♦ ❛♣♦✐♦ ✜♥❛♥❝❡✐r♦ ❞✉r❛♥t❡ ♦ ❉♦✉t♦r❛❞♦✳

  ✏❆♣❡s❛r ❞♦s ♥♦ss♦s ❞❡❢❡✐t♦s✱ ♣r❡❝✐s❛♠♦s ❡♥①❡r❣❛r q✉❡ s♦♠♦s ♣ér♦❧❛s ú♥✐❝❛s ♥♦ t❡❛tr♦ ❞❛ ✈✐❞❛ ❡ ❡♥t❡♥❞❡r q✉❡ ♥ã♦ ❡①✐st❡♠

  ♣❡ss♦❛s ❞❡ s✉❝❡ss♦ ♦✉ ♣❡ss♦❛s ❢r❛❝❛ss❛❞❛s✳ ❖ q✉❡ ❡①✐st❡ sã♦ ♣❡ss♦❛s q✉❡ ❧✉t❛♠ ♣❡❧♦s s❡✉s s♦♥❤♦s ♦✉ ❞❡s✐st❡♠ ❞❡❧❡s✳✑

  ❆✉❣✉st♦ ❈✉r②

  

❘❡s✉♠♦

  ❖ ❡st✉❞♦ ❞♦s ❢❡♥ô♠❡♥♦s ❞❡ ❢♦rç❛ ❡ t♦rq✉❡ ❞❡ r❛❞✐❛çã♦ ❛❝úst✐❝♦ t❡♠ ❛tr❛í❞♦ ✉♠ ❡♥♦r♠❡ ✐♥t❡r❡ss❡ ❞❛ ❝♦♠✉♥✐❞❛❞❡ ❝✐❡♥tí✜❝❛✱ ❞❡✈✐❞♦ ❛ ❛♣❧✐❝❛çõ❡s ❞❡ss❡s ❢❡♥ô♠❡♥♦s ❡♠ ♠❛♥✐♣✉❧❛çã♦ ❞❡ ♣❛rtí❝✉❧❛s s❡♠ ❝♦♥t❛t♦✳ ◆❡st❡ tr❛❜❛❧❤♦✱ r❡❛❧✐③❛♠♦s ✉♠❛ ❛♥á❧✐s❡ t❡ór✐❝❛ ❞❛ ❢♦rç❛ ❡ t♦rq✉❡ ❞❡ r❛❞✐❛çã♦ ❛❝úst✐❝♦ ❡①❡r❝✐❞♦ s♦❜r❡ ✉♠❛ ♣❛rtí❝✉❧❛ ✈✐s❝♦❡❧ást✐❝❛ ❤♦♠♦❣ê♥❡❛ ♥♦ ❧✐♠✐t❡ ❞♦ ❡s♣❛❧❤❛♠❡♥t♦ ❞❡ ❘❛②❧❡✐❣❤ ✭♦ r❛✐♦ ❞❛ ♣❛rtí❝✉❧❛ é ♠✉✐t♦ ♠❡♥♦r q✉❡ ♦ ❝♦♠♣r✐♠❡♥t♦ ❞❡ ♦♥❞❛ ✐♥❝✐❞❡♥t❡✮ ♣♦r ✉♠❛ ♦♥❞❛ ❝♦♠ ❣❡♦♠❡tr✐❛ ❛r❜✐trár✐❛✳ ◆♦ss♦ ❡st✉❞♦ ❜❛s❡✐❛✲s❡ ♥❛ ❡①♣❛♥sã♦ ❞❡ ♦♥❞❛s ♣❛r❝✐❛✐s ❡♠ ❝♦♦r❞❡♥❛❞❛s ❡s❢ér✐❝❛s ❞❛s ♦♥❞❛s ✐♥❝✐❞❡♥t❡ ❡ ❡s♣❛❧❤❛❞❛✳ ◆❡ss❡ ❝♦♥t❡①t♦✱ ❛ ❢♦rç❛ ❡ ♦ t♦rq✉❡ ❞❡ r❛❞✐❛çã♦ sã♦ ♦❜t✐❞♦s ❛♥❛❧✐t✐❝❛♠❡♥t❡ ❡♠ t❡r♠♦s ❞❡ ✉♠❛ sér✐❡ ✐♥✜♥✐t❛ q✉❡ ❡♥✈♦❧✈❡ ♦s ❝♦❡✜❝✐❡♥t❡s ❞❡ ❡①♣❛♥sã♦ ❞❛s ♦♥❞❛s ❡s♣❛❧❤❛❞❛ ❡ ✐♥❝✐❞❡♥t❡✳ ❆ss✉♠✐♠♦s q✉❡ ❛ ♣❛rtí❝✉❧❛ s❡ ❝♦♠♣♦rt❛ ❝♦♠♦ ✉♠ só❧✐❞♦ ✈✐s❝♦❡❧ást✐❝♦ ❧✐♥❡❛r✱ q✉❡ ♦❜❡❞❡❝❡ ♦ ♠♦❞❡❧♦ ❞❡ ❑❡❧✈✐♥✲❱♦✐❣t ❢r❛❝✐♦♥ár✐♦✳ ❋ór♠✉❧❛s ❛♥❛❧ít✐❝❛s ♣❛r❛ ❛ ❢♦rç❛ ❡ t♦rq✉❡ ❞❡ r❛❞✐❛çã♦ sã♦ ♦❜t✐❞❛s ❝♦♥s✐❞❡r❛♥❞♦ ✉♠❛ ❛♣r♦①✐♠❛çã♦ ❞❡ ❜❛✐①❛ ❡ ❞❡ ❛❧t❛ ❢r❡q✉ê♥❝✐❛ ♥♦ ♠♦❞❡❧♦ ✈✐s❝♦❡❧ást✐❝♦✳ ❆ t❡♦r✐❛ ❞❡s❡♥✈♦❧✈✐❞❛ é ✉s❛❞❛ ♣❛r❛ ❞❡s❝r❡✈❡r ❛ ✐♥t❡r❛çã♦ ❞❡ ♦♥❞❛s ❛❝úst✐❝❛s ✭♦♥❞❛ ♣❧❛♥❛ ♣r♦❣r❡ss✐✈❛✱ ♦♥❞❛ ♣❧❛♥❛ ❡st❛❝✐♦♥ár✐❛✱ ❢❡✐①❡s ❞❡ ❇❡ss❡❧ ❞❡ ♦r❞❡♠ ③❡r♦ ❡ ❞❡ ♣r✐♠❡✐r❛ ♦r❞❡♠✮ ❝♦♠ ♣❛rtí❝✉❧❛s ❞❡ ♣♦❧✐❡t✐❧❡♥♦ ❞❡ ❜❛✐①❛ ❡ ❞❡ ❛❧t❛ ❞❡♥s✐❞❛❞❡✳ ❖s ♥♦ss♦s r❡s✉❧t❛❞♦s ♠♦str❛♠ q✉❡ ❛ ❢♦rç❛ ❞❡ r❛❞✐❛çã♦ ❛❝úst✐❝❛ ❛①✐❛❧ ♣♦❞❡ s❡r ♥❡❣❛t✐✈❛ ✭✐st♦ é✱ ❡♠ ♦♣♦s✐çã♦ à ❞✐r❡çã♦ ❞❡ ♣r♦♣❛❣❛çã♦ ❞❛ ♦♥❞❛✮ q✉❛♥❞♦ ✉♠❛ ❞❡t❡r♠✐♥❛❞❛ ❝♦♥❞✐çã♦ ❡♥✈♦❧✈❡♥❞♦ ♦s ♣❛râ♠❡tr♦s ❢ís✐❝♦s ❞❛ ♣❛rtí❝✉❧❛ é s❛t✐s❢❡✐t❛✳ ❚♦rq✉❡ ❞❡ r❛❞✐❛çã♦ ❛❝úst✐❝♦ ♥❡❣❛t✐✈♦ ❞❡✈✐❞♦ ❛ ✉♠ ❢❡✐①❡ ❞❡ ❇❡ss❡❧ ❞❡ ♣r✐♠❡✐r❛ ♦r❞❡♠ t❛♠❜é♠ ♣♦❞❡ ♦❝♦rr❡r q✉❛♥❞♦ ❛ ♠❡s♠❛ ❝♦♥❞✐çã♦ ❞❛ ❢♦rç❛ ❞❡ r❛❞✐❛çã♦ ♥❡❣❛t✐✈❛ ❢♦r ❛t❡♥❞✐❞❛✳ ◆♦t❛✈❡❧♠❡♥t❡✱ ❡st❛ é ❛ ♣r✐♠❡✐r❛ ✈❡③ q✉❡ ❛ ❢♦rç❛ ❞❡ r❛❞✐❛çã♦ ♥❡❣❛t✐✈❛ é ♣r❡✈✐st❛ s♦❜r❡ ✉♠❛ ♣❛rtí❝✉❧❛ ❤♦♠♦❣ê♥❡❛ ♥♦ r❡❣✐♠❡ ❞❡ ❡s♣❛❧❤❛♠❡♥t♦ ❘❛②❧❡✐❣❤✳ ❆❧é♠ ❞✐ss♦✱ ❛ ❡st❛❜✐❧✐❞❛❞❡ tr❛♥s✈❡rs❛❧ ❞❛ ❢♦rç❛ ❞❡ r❛❞✐❛çã♦ ❛❝úst✐❝❛ ❣❡r❛❞❛ ♣❡❧♦ ❢❡✐①❡ ❞❡ ❇❡ss❡❧ t❛♠❜é♠ é ✐♥✈❡st✐❣❛❞❛✳ ▼♦str❛♠♦s q✉❡ ✉♠ ❢❡✐①❡ ❞❡ ❇❡ss❡❧ tr❛t♦r ✸❉ ❝♦♠♣❧❡t♦ ❛t✉❛ s♦❜r❡ ✉♠❛ ♣❛rtí❝✉❧❛ ❞❡ ♣♦❧✐❡t✐❧❡♥♦ ❞❡ ❛❧t❛ ❞❡♥s✐❞❛❞❡ ✭P❊❆❉✮✳ ◆❛ ❛♥á❧✐s❡ ❞❛ ❢♦rç❛ ❞❡ r❛❞✐❛çã♦ ❛❝úst✐❝❛ ❣❡r❛❞❛ s♦❜r❡ ✉♠❛ ♣❛rtí❝✉❧❛ ✈✐s❝♦❡❧ást✐❝❛ ♣♦r ✉♠❛ ♦♥❞❛ ♣❧❛♥❛ ❡st❛❝✐♦♥ár✐❛✱ ❞❡s✈✐♦s r❡❧❡✈❛♥t❡s s✉r❣✐r❛♠ ❡♠ ❝♦♠♣❛r❛çã♦ ❝♦♠ ♣❛rtí❝✉❧❛ só❧✐❞❛ ❡❧ást✐❝❛✳ ❆ ❛♠♣❧✐t✉❞❡ ❞❛ ❢♦rç❛ ❡ t♦rq✉❡ ❞❡ r❛❞✐❛çã♦ s♦❜r❡ ✉♠❛ P❊❆❉ ❞❡s❝r✐t❛ ♣❡❧♦ ♠♦❞❡❧♦ ✈✐s❝♦❡❧ást✐❝♦ ❛♣r❡s❡♥t❛ ✉♠ ❝♦♠♣♦rt❛♠❡♥t♦ ❞✐❢❡r❡♥t❡ ✭❢♦rç❛ ❞❡ r❛❞✐❛çã♦ ♥❡❣❛t✐✈❛✮ ❝♦♠♣❛r❛❞♦s ❝♦♠ ♦s ♦✉tr♦s ♠❛t❡r✐❛✐s ✭só❧✐❞❛ ❡❧ást✐❝❛ ❡ ✢✉✐❞❛ ❝♦♠ ❛❜s♦rçã♦ ❧♦♥❣✐t✉❞✐♥❛❧✮ ❞❡✈✐❞♦ ❛ ♦♥❞❛ ♣❧❛♥❛ ♣r♦❣r❡ss✐✈❛ ❡ ❢❡✐①❡s ❞❡ ❇❡ss❡❧✳ P♦r ✜♠✱ ❛❝r❡❞✐t❛♠♦s q✉❡ ❡st❡ ❡st✉❞♦ ♣♦❞❡ ❛❥✉❞❛r ❛ ♠❡❧❤♦r❛r ❛✐♥❞❛ ♠❛✐s ♦ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❞❡ ❞✐s♣♦s✐t✐✈♦s ❞❡ ❧❡✈✐t❛çã♦ ❛❝úst✐❝❛✱ ♠❛♥✐♣✉❧❛çã♦ ❞❡ ♣❛rtí❝✉❧❛s ❡♠ ❛❝✉st♦✢✉í❞✐❝❛ ❡ ♣✐♥ç❛s ❛❝úst✐❝❛s✳

  P❛❧❛✈r❛s✲❝❤❛✈❡✿ ❋♦rç❛ ❞❡ r❛❞✐❛çã♦ ❛❝úst✐❝❛✱ ❚♦rq✉❡ ❞❡ r❛❞✐❛çã♦ ❛❝úst✐❝♦✱ ❊s♣❛❧❤❛✲ ♠❡♥t♦ ❛❝úst✐❝♦✱ ❱✐s❝♦❡❧❛st✐❝✐❞❛❞❡✱ ▼❛♥✐♣✉❧❛çã♦ ❞❡ ♣❛rtí❝✉❧❛s✳

  

❆❜str❛❝t

  ❚❤❡ st✉❞② ♦❢ ❛❝♦✉st✐❝ r❛❞✐❛t✐♦♥ ❢♦r❝❡ ❛♥❞ t♦rq✉❡ ♣❤❡♥♦♠❡♥❛ ❤❛s ❛ttr❛❝t❡❞ ❛♥ ❡♥♦r♠♦✉s ✐♥t❡r❡st ♦❢ t❤❡ s❝✐❡♥t✐✜❝ ❝♦♠♠✉♥✐t②✱ ❞✉❡ t♦ ❛♣♣❧✐❝❛t✐♦♥s ♦❢ t❤❡s❡ ♣❤❡♥♦♠❡♥❛ ✐♥ ♥♦♥❝♦♥✲ t❛❝t ♣❛rt✐❝❧❡s ♠❛♥✐♣✉❧❛t✐♦♥✳ ■♥ t❤✐s ✇♦r❦✱ ✇❡ ♣❡r❢♦r♠ ❛ t❤❡♦r❡t✐❝❛❧ ❛♥❛❧②s✐s ♦❢ ❛❝♦✉st✐❝ r❛❞✐❛t✐♦♥ ❢♦r❝❡ ❛♥❞ t♦rq✉❡ ❡①❡rt❡❞ ♦♥ ❛ ❤♦♠♦❣❡♥❡♦✉s ✈✐s❝♦❡❧❛st✐❝ ♣❛rt✐❝❧❡ ✐♥ t❤❡ ❘❛②❧❡✐❣❤ s❝❛tt❡r✐♥❣ ❧✐♠✐t ✭t❤❡ ♣❛rt✐❝❧❡ r❛❞✐✉s ✐s ♠✉❝❤ s♠❛❧❧❡r t❤❛♥ t❤❡ ✐♥❝✐❞❡♥t ✇❛✈❡❧❡♥❣t❤✮ ❜② ❛ ✇❛✈❡ ✇✐t❤ ❛r❜✐tr❛r② ❣❡♦♠❡tr②✳ ❖✉r st✉❞② ✐s ❜❛s❡❞ ♦♥ t❤❡ ♣❛rt✐❛❧✲✇❛✈❡ ❡①♣❛♥s✐♦♥ ✐♥ s♣❤❡✲ r✐❝❛❧ ❝♦♦r❞✐♥❛t❡s ♦❢ t❤❡ ✐♥❝✐❞❡♥t ❛♥❞ s❝❛tt❡r❡❞ ✇❛✈❡s✳ ■♥ t❤✐s ❝♦♥t❡①t✱ t❤❡ r❛❞✐❛t✐♦♥ ❢♦r❝❡ ❛♥❞ t♦rq✉❡ ❛r❡ ♦❜t❛✐♥❡❞ ❛♥❛❧②t✐❝❛❧❧② ✐♥ t❡r♠s ♦❢ ❛♥ ✐♥✜♥✐t❡ s❡r✐❡s ✇❤✐❝❤ ✐♥✈♦❧✈❡s t❤❡ s❝❛t✲ t❡r✐♥❣ ❛♥❞ ✐♥❝✐❞❡♥t ❡①♣❛♥s✐♦♥ ❝♦❡✣❝✐❡♥ts✳ ❲❡ ❛ss✉♠❡ t❤❛t t❤❡ ♣❛rt✐❝❧❡ ❜❡❤❛✈❡s ❛s ❛ ❧✐♥❡❛r ✈✐s❝♦❡❧❛st✐❝ s♦❧✐❞✱ ✇❤✐❝❤ ♦❜❡②s t❤❡ ❢r❛❝t✐♦♥❛❧ ❑❡❧✈✐♥✲❱♦✐❣t ♠♦❞❡❧✳ ❆♥❛❧②t✐❝❛❧ ❡①♣r❡ss✐♦♥s ❢♦r t❤❡ r❛❞✐❛t✐♦♥ ❢♦r❝❡ ❛♥❞ t♦rq✉❡ ❛r❡ ♦❜t❛✐♥❡❞ ❝♦♥s✐❞❡r✐♥❣ t❤❡ ❧♦✇✲ ❛♥❞ ❤✐❣❤✲❢r❡q✉❡♥❝② ❛♣♣r♦①✐♠❛t✐♦♥ ✐♥ t❤❡ ✈✐s❝♦❡❧❛st✐❝ ♠♦❞❡❧✳ ❚❤❡ ❞❡✈❡❧♦♣❡❞ t❤❡♦r② ✐s ✉s❡❞ t♦ ❞❡s❝r✐❜❡ t❤❡ ✐♥t❡r❛❝t✐♦♥ ♦❢ ❛❝♦✉st✐❝ ✇❛✈❡s ✭tr❛✈❡❧✐♥❣ ❛♥❞ st❛♥❞✐♥❣ ♣❧❛♥❡ ✇❛✈❡s✱ ❛♥❞ ③❡r♦ ❛♥❞ ✜rst✲♦r❞❡r ❇❡ss❡❧ ❜❡❛♠s✮ ✇✐t❤ ❛ ❧♦✇✲❛♥❞ ❤✐❣❤✲❞❡♥s✐t② ♣♦❧②❡t❤②❧❡♥❡ ♣❛rt✐❝❧❡✳ ❖✉r r❡s✉❧ts s❤♦✇ t❤❛t t❤❡ ❛①✐❛❧ ❛❝♦✉st✐❝ r❛❞✐❛t✐♦♥ ❢♦r❝❡ ♠✐❣❤t ❜❡❝♦♠❡ ♥❡❣❛t✐✈❡ ✭✐✳❡✳ ✐♥ ♦♣♣♦s✐t✐♦♥ t♦ t❤❡ ✇❛✈❡ ♣r♦♣❛❣❛t✐♦♥ ❞✐r❡❝t✐♦♥✮ ✇❤❡♥ ❛ ❝❡rt❛✐♥ ❝♦♥❞✐t✐♦♥ ✐♥✈♦❧✈✐♥❣ t❤❡ ♣❤②s✐❝❛❧ ♣❛r❛♠❡t❡rs ♦❢ t❤❡ ♣❛rt✐❝❧❡ ✐s s❛t✐s✜❡❞✳ ◆❡❣❛t✐✈❡ ❛❝♦✉st✐❝ r❛❞✐❛t✐♦♥ t♦rq✉❡ ❞✉❡ ❛ ❜❡❛♠ ♦❢ ✜rst✲♦r❞❡r ❇❡ss❡❧ ♠❛② ❛❧s♦ ♦❝❝✉r ✇❤❡♥ t❤❡ s❛♠❡ ❝♦♥❞✐t✐♦♥ ♦❢ ♥❡❣❛t✐✈❡ r❛❞✐❛t✐♦♥ ❢♦r❝❡ ✐s ♠❡t✳ ❘❡♠❛r❦❛❜❧②✱ t❤✐s ✐s t❤❡ ✜rst t✐♠❡ t❤❛t ♥❡❣❛t✐✈❡ r❛❞✐❛t✐♦♥ ❢♦r❝❡ ✐s ♣r❡❞✐❝t❡❞ ♦♥ ❛ ❤♦♠♦❣❡♥❡♦✉s ♣❛rt✐❝❧❡ ✐♥ t❤❡ ❘❛②❧❡✐❣❤ s❝❛tt❡r✐♥❣ r❡❣✐♠❡✳ ❋✉rt❤❡r♠♦r❡✱ t❤❡ st❛❜✐❧✐t② ♦❢ t❤❡ tr❛♥s✈❡rs❡ ❛❝♦✉st✐❝ r❛❞✐❛t✐♦♥ ❢♦r❝❡ ❣❡♥❡r❛t❡❞ ❜② ❛ ❇❡ss❡❧ ❜❡❛♠ ✐s ❛❧s♦ ✐♥✈❡st✐❣❛t❡❞✳ ❲❡ s❤♦✇ ❛ ❢✉❧❧ ✸❉ tr❛❝t♦r ❇❡ss❡❧ ✈♦rt❡① ❜❡❛♠ ❛❝t✐♥❣ ♦♥ t❤❡ ❤✐❣❤✲❞❡♥s✐t② ♣♦❧②❡t❤②❧❡♥❡ ✭❍❉P❊✮✳ ■♥ t❤❡ ❛♥❛❧②s✐s ♦❢ ❛❝♦✉st✐❝ r❛❞✐❛t✐♦♥ ❢♦r❝❡ ❣❡♥❡r❛t❡❞ ♦♥ ❛ ✈✐s❝♦❡❧❛st✐❝ ♣❛rt✐❝❧❡ ❜② ❛ st❛♥❞✐♥❣ ♣❧❛♥❡ ✇❛✈❡✱ r❡❧❡✈❛♥t ❞❡✈✐❛t✐♦♥s ❛r♦s❡ ✐♥ ❝♦♠♣❛r✐s♦♥ ✇✐t❤ t❤❡ s♦❧✐❞ ❡❧❛st✐❝ ♠♦❞❡❧ ❢♦r t❤❡ ♣❛rt✐❝❧❡s✳ ❚❤❡ ♠❛❣♥✐t✉❞❡ ♦❢ t❤❡ r❛❞✐❛t✐♦♥ ❢♦r❝❡ ❛♥❞ t♦rq✉❡ ♦♥ ❛ ❍❉P❊ ❞❡s❝r✐❜❡❞ ❜② t❤❡ ✈✐s❝♦❡❧❛st✐❝ ♠♦❞❡❧ ❜❡❤❛✈❡s ❞✐✛❡r❡♥t❧② ✭♥❡❣❛t✐✈❡ r❛❞✐❛t✐♦♥ ❢♦r❝❡✮ ❝♦♠♣❛r❡❞ ✇✐t❤ ♦t❤❡r ♠❛t❡r✐❛❧s ✭s♦❧✐❞ ❡❧❛st✐❝ ❛♥❞ ❝♦♠♣r❡ss✐❜❧❡ ✢✉✐❞ ♣❛rt✐❝❧❡✮ ❞✉❡ t♦ tr❛✈❡❧✐♥❣ ♣❧❛♥❡ ✇❛✈❡ ❛♥❞ ❇❡ss❡❧ ❜❡❛♠s✳ ❋✐♥❛❧❧②✱ ✇❡ ❜❡❧✐❡✈❡ t❤❛t t❤✐s st✉❞② ♠❛② ❤❡❧♣ ❢✉rt❤❡r ❡♥❤❛♥❝❡ t❤❡ ❞❡✈❡❧♦♣♠❡♥t ♦❢ ❛❝♦✉st✐❝ ❧❡✈✐t❛t✐♦♥✱ ♣❛rt✐❝❧❡ ❤❛♥❞❧✐♥❣ ✐♥ ❛❝♦✉st♦✢✉✐❞s✱ ❛♥❞ ❛❝♦✉st✐❝❛❧ t✇❡❡③❡rs ❞❡✈✐❝❡s✳

  ❑❡②✇♦r❞s✿ ❆❝♦✉st✐❝ r❛❞✐❛t✐♦♥ ❢♦r❝❡✱ ❆❝♦✉st✐❝ r❛❞✐❛t✐♦♥ t♦rq✉❡✱ ❆❝♦✉st✐❝ s❝❛tt❡r✐♥❣✱ ❱✐s❝♦❡❧❛st✐❝✐t②✱ P❛rt✐❝❧❡ ♠❛♥✐♣✉❧❛t✐♦♥✳

  

▲✐st❛ ❞❡ sí♠❜♦❧♦s

  ❙í♠❜♦❧♦ ◆♦♠❡ ❯♥✐❞❛❞❡s✭❙✳■✳✮ 3

  ρ ❞❡♥s✐❞❛❞❡ ❞♦ ✢✉✐❞♦ ✐❞❡❛❧ kg · m 3

  ρ ❞❡♥s✐❞❛❞❡ kg · m m

  ♠❛ss❛ kg c ✈❡❧♦❝✐❞❛❞❡ ❞♦ s♦♠ ♥♦ ✢✉✐❞♦ ✐❞❡❛❧ m/s c

  ✈❡❧♦❝✐❞❛❞❡ ❞♦ s♦♠ ❧♦♥❣✐t✉❞✐♥❛❧ m/s c

  s

  ✈❡❧♦❝✐❞❛❞❡ ❞♦ s♦♠ ❞❡ ❝✐s❛❧❤❛♠❡♥t♦ m/s τ t❡♠♣♦ ❞❡ r❡❧❛①❛çã♦ ❧♦♥❣✐t✉❞✐♥❛❧ s τ

  s

  t❡♠♣♦ ❞❡ r❡❧❛①❛çã♦ ❞❡ ❝✐s❛❧❤❛♠❡♥t♦ s a r❛✐♦ ❞❛ ♣❛rtí❝✉❧❛ m w

  ✈❡t♦r ❞❡s❧♦❝❛♠❡♥t♦ m v ✈❡❧♦❝✐❞❛❞❡ ❞❛ ♣❛rtí❝✉❧❛ m/s

  ǫ t❡♥s♦r ❞❡❢♦r♠❛çã♦

  σ t❡♥s♦r ❞❡ t❡♥sõ❡s ❡♠ só❧✐❞♦s

  ♣ ♣r❡ssã♦ ❛❝úst✐❝❛ Pa 3 L ❞❡♥s✐❞❛❞❡ ▲❛❣r❛♥❣✐❛♥❛ ❛❝úst✐❝❛ J/m 2

  φ /s

  ♣♦t❡♥❝✐❛❧ ❞❡ ✈❡❧♦❝✐❞❛❞❡ m 2 ψ 1, s 2, s

  /s ✱ ψ ♣♦t❡♥❝✐❛✐s ❡s❝❛❧❛r❡s ❞❡ ❉❡❜②❡ m 1

  ω ❢r❡q✉ê♥❝✐❛ ❛♥❣✉❧❛r s 1

  λ ❝♦♠♣r✐♠❡♥t♦ ❞❡ ♦♥❞❛ m 1

  ❦ ♥ú♠❡r♦ ❞❡ ♦♥❞❛ m 1

  ❦ ♥ú♠❡r♦ ❞❡ ♦♥❞❛ ❧♦♥❣✐t✉❞✐♥❛❧ m 1

  ❦ s ♥ú♠❡r♦ ❞❡ ♦♥❞❛ ❞❡ ❝✐s❛❧❤❛♠❡♥t♦ m a nm ❝♦❡✜❝✐❡♥t❡ ❢♦r♠❛ ❞♦ ❢❡✐①❡ s n

  ❝♦❡✜❝✐❡♥t❡ ❞❡ ❡s♣❛❧❤❛♠❡♥t♦ ❡s❝❛❧❛r b n n n ✱c ✱d ❝♦❡✜❝✐❡♥t❡s ❞❡ ❡①♣❛♥sã♦

  ε ♣❛râ♠❡tr♦ ❞❡ ❡s❝❛❧❛ ❡♠ t❡r♠♦s ❞♦ t❛♠❛♥❤♦ ❞❛ ♣❛rtí❝✉❧❛

  ε j ♣❛râ♠❡tr♦ ❞❡ ❡s❝❛❧❛ ❡♠ t❡r♠♦s t❡♠♣♦ ❞❡ r❡❧❛①❛çã♦ ϕ

  â♥❣✉❧♦ ❛③✐♠✉t❛❧ θ

  â♥❣✉❧♦ r❛❞✐❛❧ j n (1) ❢✉♥çã♦ ❞❡ ❇❡ss❡❧ ❡s❢ér✐❝❛ ❞❡ ♦r❞❡♠ ♥ h n (2) ❢✉♥çã♦ ❞❡ ❍❛♥❦❡❧ ❡s❢ér✐❝❛ ❞❡ ♣r✐♠❡✐r❛ ♦r❞❡♠ ♥ h n m ❢✉♥çã♦ ❞❡ ❍❛♥❦❡❧ ❡s❢ér✐❝❛ ❞❡ ♣r✐♠❡✐r❛ ♦r❞❡♠ ♥ Y n m

  ❤❛r♠ô♥✐❝♦s ❡s❢ér✐❝♦s P n ❢✉♥çõ❡s ❛ss♦❝✐❛❞❛s ❞❡ ▲❡❣❡♥❞r❡ ❞❡ ♦r❞❡♠ ♥

  r❛❞

  F ❢♦rç❛ ❞❡ r❛❞✐❛çã♦ ❛❝úst✐❝❛ N

  Y , Y , Y x y z ❝♦♠♣♦♥❡♥t❡s ❛❞✐♠❡♥s✐♦♥❛✐s ❞❛ ❢♦rç❛ ❞❡ r❛❞✐❛çã♦

  r❛❞

  N t♦rq✉❡ ❞❡ r❛❞✐❛çã♦ ❛❝úst✐❝♦ N · m

  N x , N y , N z ❝♦♠♣♦♥❡♥t❡s ❛❞✐♠❡♥s✐♦♥❛✐s ❞♦ t♦rq✉❡ ❞❡ r❛❞✐❛çã♦ 1

  ♦♣❡r❛❞♦r ❣r❛❞✐❡♥t❡ m ∇ 1

  ∇· ♦♣❡r❛❞♦r ❞✐✈❡r❣❡♥t❡ m

  − 1

  ∇× 2 ♦♣❡r❛❞♦r r♦t❛❝✐♦♥❛❧ m 2

  ∇ ♦♣❡r❛❞♦r ▲❛♣❧❛❝✐❛♥♦ m r

  ✈❡t♦r ♣♦s✐çã♦ ♠ (x, y, z)

  ❝♦♦r❞❡♥❛❞❛s ❝❛rt❡s✐❛♥❛s (r, θ, φ)

  ❝♦♦r❞❡♥❛❞❛s ❡s❢ér✐❝❛s (e , e , e ) x y z ❝♦♠♣♦♥❡♥t❡s ❝❛rt❡s✐❛♥❛s ✉♥✐trát✐❛s n

  ✈❡t♦r ♥♦r♠❛❧ à s✉♣❡r❢í❝✐❡ n

  ❝

  í♥❞✐❝❡ ❞❡ r❡❢r❛çã♦ ❝♦♠♣❧❡①♦ n R í♥❞✐❝❡ ❞❡ r❡❢r❛çã♦ ❞♦ ♠❛t❡r✐❛❧

  α ˜ ❢✉♥çã♦ ❞❡ ❛❜s♦rçã♦ ❛❞✐♠❡♥s✐♦♥❛❧

  ❘❡ ♣❛rt❡ r❡❛❧

  ■♠ ♣❛rt❡ ✐♠❛❣✐♥ár✐❛

  √ ✐ = −1

  ✉♥✐❞❛❞❡ ✐♠❛❣✐♥ár✐❛

  ▲✐st❛ ❞❡ ❚❛❜❡❧❛s

   ✳ ✳ ✳ ✳ ✳ ✺✶

   ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✺

   ✳ ✳ ✳ ✼✽

  ▲✐st❛ ❞❡ ❋✐❣✉r❛s

  

  

  ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✵

  

  

  ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✼

  

  

  

  

  ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✶

  

  

  

  ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✺

  

  

  ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✺

  

  

  

  

  ✳ ✳ ✳ ✳ ✹✻

  

  

  

  ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✹

  

  

  

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✼

  

  

  

  ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✶

  

  

  

  ✼✻

  

  

  

  ✼✼

  

  

  

  ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽✵

  

  

  

  

  ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽✶

  

  

  

  ✳ ✳ ✳ ✳ ✳ ✽✷

  

  

  

  

  ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽✸

  

  

  

  

  ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽✺

  

  

  

  

  ✳ ✳ ✳ ✳ ✽✻

  ❙✉♠ár✐♦

   ✶✻

  ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✻ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✻ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✽

  ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✾ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✾ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✵ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✶

  ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✶ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✷ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✷

   ✷✹

  ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✹ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✺ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✻ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✼

  ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✽ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✾ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✵ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✶

  ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✸ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✹ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✹ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✻ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✻ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✼

  ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✽ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✶

   ✹✹ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✹ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✵

   ✺✸ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✸ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✹

  ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✹ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✻

  ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✶ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✶ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✷

  ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✹ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✺ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✻

   ✻✽

  ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✽ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✾ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✵ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✵

  ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✹ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✹ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✽ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✾ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✾ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽✷ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽✹ ✳ ✳ ✳ ✳ ✳ ✽✹

   ✽✽

  ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽✽ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽✾

  1 ■♥tr♦❞✉çã♦

  ❖♥❞❛s ❛❝úst✐❝❛s ❝♦♥s✐st❡♠ ❡♠ ♣❡rt✉r❜❛çõ❡s q✉❡ tr❛♥s♣♦rt❛♠ ❡♥❡r❣✐❛ ❡ ♠♦♠❡♥t♦✳ ❖ ♠♦♠❡♥t♦ ❞❛ ♦♥❞❛ ✐♥❝✐❞❡♥t❡ ♣♦❞❡ s❡r tr❛♥s❢❡r✐❞♦ ♣❛r❛ ♦❜❥❡t♦s s✉s♣❡♥s♦s✳ ❊st❛ tr❛♥s❢❡rê♥✲ ❝✐❛ ❞♦ ♠♦♠❡♥t♦ ❧✐♥❡❛r ❡ ❛♥❣✉❧❛r ❞❛ ♦♥❞❛ ❛❝úst✐❝❛ ❤❛r♠ô♥✐❝❛ ♣❛r❛ ✉♠ ♦❜❥❡t♦ s✉s♣❡♥s♦ ♣♦❞❡ ❞❛r ♦r✐❣❡♠ ❛♦s ❢❡♥ô♠❡♥♦s ❞❡ ❢♦rç❛ ❡ t♦rq✉❡ ❞❡ r❛❞✐❛çã♦

  ◗✉❛♥❞♦ ✉♠ ❝❛♠♣♦ ❛❝úst✐❝♦ é ❛♣❧✐❝❛❞♦ ❡♠ ✉♠ ✢✉✐❞♦ ❝♦♥t❡♥❞♦ ♣❛rtí❝✉❧❛s s✉s♣❡♥s❛s✱ ❡ss❛s ♣❛rtí❝✉❧❛s s❡rã♦ ❛❢❡t❛❞❛s ❞❡✈✐❞♦ ❛ ❢♦rç❛ ❞❡ r❛❞✐❛çã♦ ❛❝úst✐❝❛✳ ❆ ♠❛♥✐♣✉❧❛çã♦ ❞❡ ♣❛rtí❝✉❧❛s ♣♦r ✉❧tr❛ss♦♠ ✉t✐❧✐③❛ ❛ ❢♦rç❛ ❞❡ r❛❞✐❛çã♦ ♣❛r❛ ❢♦r♥❡❝❡r ✉♠ ♠ét♦❞♦ ❞❡ ♠❛♥✐♣✉✲ ❧❛çã♦ ❞❡ ♣❛rtí❝✉❧❛s s❡♠ ❝♦♥t❛t♦✳ ❊st❡ ♠ét♦❞♦ t♦r♥♦✉✲s❡ ♣r♦♠✐ss♦r ♥❛ ár❡❛ ❞❡ ❜✐♦t❡❝♥♦❧♦✲ ❣✐❛ ❡ ♣❡r♠✐t❡ ❛ ♠❛♥✐♣✉❧❛çã♦ ❞❡ ❞✐✈❡rs❛s ♣❛rtí❝✉❧❛s✱ ✐♥❝❧✉✐♥❞♦ ❝é❧✉❧❛s ❜✐♦❧ó❣✐❝❛s ❡ ♦✉tr♦s ♠✐❝r♦r❣❛♥✐s♠♦s ❊st❡s ♠ét♦❞♦s ❞❡♣❡♥❞❡♠ s♦♠❡♥t❡ ❞❛s ♣r♦♣r✐❡❞❛❞❡s ♠❡❝â♥✐❝❛s ❞❛ ♣❛rtí❝✉❧❛ ❡ ❞♦ ✢✉✐❞♦✳ ❆❧é♠ ❞❡ s❡r tr❛♥s❧❛❞❛❞❛ ♦✉ ❛♣r✐s✐♦♥❛❞❛✱ ✉♠❛ ♣❛rtí❝✉❧❛ ♣♦❞❡ s❡r ❛❥✉st❛❞❛ ♣❛r❛ ❣✐r❛r ❞❡✈✐❞♦ ❛♦ t♦rq✉❡ ❞❡ r❛❞✐❛✲ çã♦ ❛❝úst✐❝♦ ❈♦♥s❡q✉❡♥t❡♠❡♥t❡✱ ✉♠ ❣r❛✉ ❞❡ ❧✐❜❡r❞❛❞❡ ❞❡ r♦t❛çã♦ t❛♠❜é♠ é ❛✈❛❧✐❛❞♦ ♥❛ ♠❛♥✐♣✉❧❛çã♦ ❞❡ ♣❛rtí❝✉❧❛s ❝♦♠ ❜❛s❡ ❡♠ ♠ét♦❞♦s ❛❝úst✐❝♦s✳

  ◆❡st❡ ❝❛♣ít✉❧♦ ❢❛r❡♠♦s ✉♠❛ r❡✈✐sã♦ ❤✐stór✐❝❛ ❞♦s ❢❡♥ô♠❡♥♦s ❞❡ ❢♦rç❛ ❡ t♦rq✉❡ ❞❡ r❛❞✐❛çã♦ ❛❝úst✐❝♦ ❣❡r❛❞♦s ♣♦r ♠❡✐♦ ❞❛ ✐♥t❡r❛çã♦ ❞❡ ✉♠❛ ♦♥❞❛ ❛❝úst✐❝❛ ❝♦♠ ✉♠ ♦❜❥❡t♦✳ ❆❧é♠ ❞✐ss♦✱ ✐r❡♠♦s ❛❜♦r❞❛r ❛❧❣✉♠❛s ❛♣❧✐❝❛çõ❡s t❡❝♥♦❧ó❣✐❝❛s ❡ ❛♣r❡s❡♥t❛r ❛ ♠♦t✐✈❛çã♦ ❞❡st❡ tr❛❜❛❧❤♦✳

  ✶✳✶ ❘❡✈✐sã♦ ❍✐stór✐❝❛ ✶✳✶✳✶ ❋♦rç❛ ❞❡ r❛❞✐❛çã♦ ❛❝úst✐❝❛

  ❖ ❡st✉❞♦ ❞❡ ❢♦rç❛ ❞❡ r❛❞✐❛çã♦ ❛❝úst✐❝❛ ❡♠ ♣❛rtí❝✉❧❛s ❡s❢ér✐❝❛s t❡✈❡ ✐♥í❝✐♦ ❡♠ ✶✾✸✹ ◆❡ss❡ tr❛❜❛❧❤♦ ♣✐♦♥❡✐r♦✱ ❑✐♥❣ ♦❜t❡✈❡ ✉♠❛ ❡①♣r❡ssã♦ ❣❡r❛❧ ♣❛r❛ ❛ ❢♦rç❛ ❞❡ r❛❞✐❛çã♦ ❛❝úst✐❝❛ ❛♣❧✐❝❛❞❛ ❛ ✉♠❛ ❡s❢❡r❛ rí❣✐❞❛ ✭t❛♠❛♥❤♦ ❛r❜✐trár✐♦✮ s✉s♣❡♥s❛ ❡♠ ✉♠ ✢✉✐❞♦ ✐❞❡❛❧ ❞❡✈✐❞♦ ❛ ✉♠❛ ♦♥❞❛ ♣❧❛♥❛ ♣r♦❣r❡ss✐✈❛ ❡ ✉♠❛ ♦♥❞❛ ♣❧❛♥❛ ❡st❛❝✐♦♥ár✐❛✳ ❆ ❢♦rç❛ ❞❡ r❛❞✐❛çã♦ ❢♦✐ ❡①♣r❡ss❛ ❡♠ t❡r♠♦s ❞❡ ✉♠❛ sér✐❡ ✐♥✜♥✐t❛✱ ❝♦♥❤❡❝✐❞❛ ❝♦♠♦ ❡①♣❛♥sã♦ ❞❡ ♦♥❞❛s ♣❛r❝✐❛✐s✳ ❯♠❛ ❡①t❡♥sã♦ ❞♦ tr❛❜❛❧❤♦ ❞❡ ❑✐♥❣ ❝♦♥s✐❞❡r❛♥❞♦ ✉♠❛ ♦♥❞❛ ❡s❢ér✐❝❛ ✐♥❝✐❞❡♥t❡ ❢♦✐ ♣r♦♣♦st♦ ♣♦r ❊♠❜❧❡t♦♥ P❡r❝❡❜❡✉✲s❡ q✉❡ ❛ ♣❛rtí❝✉❧❛ ♣♦❞❡ s❡r t❛♥t♦ ❛tr❛í❞❛ ❝♦♠♦ r❡♣❡❧✐❞❛

  ♣❡❧❛ ❢♦♥t❡ ❡♠ ❢✉♥çã♦ ❞❛ ❞✐stâ♥❝✐❛ r❡❧❛t✐✈❛✳ P♦st❡r✐♦r♠❡♥t❡✱ ❨♦s✐♦❦❛ ❡ ❑❛✇❛s✐♠❛ ♦❜t✐✈❡r❛♠ ✉♠❛ ❢ór♠✉❧❛ ♣❛r❛ ❢♦rç❛ ❞❡ r❛❞✐❛çã♦ ❛❝úst✐❝❛ ❛❣✐♥❞♦ s♦❜r❡ ✉♠❛ ♣❛rtí❝✉❧❛ ❡s❢ér✐❝❛ ❝♦♠♣r❡ssí✈❡❧ s✉s♣❡♥s❛ ❡♠ ✉♠ ✢✉✐❞♦ ✐❞❡❛❧✳ ❊❧❡s ❛♥❛❧✐s❛r❛♠ ❛ ✐♥t❡r❛çã♦ ❞❡ ✉♠❛ ♦♥❞❛ ♣❧❛♥❛ ❡st❛❝✐♦♥ár✐❛ ✐♥❝✐❞❡♥t❡ ❡♠ ✉♠❛ ♣❛rtí❝✉❧❛ ❝♦♠♣r❡ssí✈❡❧ ❡ ♥♦t❛r❛♠ q✉❡ ♣♦❞❡✲s❡ ♠♦✈❡r ❛ ♣❛rtí❝✉❧❛ ♣❛r❛ ✉♠ ♥ó ♦✉ ❛♥t✐✲♥ó ❞❡ ♣r❡ssã♦ ❞❡♣❡♥❞♦ ❞❛s ♣r♦♣r✐❡❞❛❞❡s ❛❝úst✐❝❛s ❞♦ ✢✉✐❞♦ ❡ ❞❛ ♣❛rtí❝✉❧❛✳ ❊♠ ✶✾✺✼✱ ❲❡st❡r✈❡❧t ❡♥❝♦♥tr♦✉ ✉♠❛ ❡①♣r❡ssã♦ ♣❛r❛ ❢♦rç❛ ❞❡ r❛❞✐❛çã♦ ❡♠ t❡r♠♦s ❞❛ ❢✉♥çã♦ ❞❡ ❡s♣❛❧❤❛♠❡♥t♦ ❛ss✐♥tót✐❝♦ s♦❜r❡ ✉♠ ♦❜stá❝✉❧♦ ❛r❜✐trár✐♦ ❞❡✈✐❞♦ ❛ ✉♠❛ ♦♥❞❛ ♣❧❛♥❛ ✐♥❝✐❞❡♥t❡✳ ❙✉❜s❡q✉❡♥t❡♠❡♥t❡✱ ●♦r❦♦✈ ♣r♦♣ôs ✉♠ ♠♦❞❡❧♦ ❞❡ ❢♦rç❛ ❞❡ r❛❞✐❛çã♦ ❡①❡r❝✐❞❛ s♦❜r❡ ✉♠❛ ♣❛rtí❝✉❧❛ ❝♦♠♣r❡ssí✈❡❧ ❞❡✈✐❞♦ ❛ ✉♠ ❢❡✐①❡ ✐♥❝✐❞❡♥t❡ ❞❡ ❣❡♦♠❡tr✐❛ ❛r❜✐trár✐❛✳ ❊❧❡ ♠♦str♦✉ q✉❡ ♣❛r❛ ✉♠❛ ♦♥❞❛ ♣❧❛♥❛ ❡st❛❝✐♦♥ár✐❛ ❛ ❢♦rç❛ t❡♠ ❝♦♥tr✐❜✉✐çõ❡s ✐♠♣♦rt❛♥t❡s ❞♦s t❡r♠♦s ❞❡ ✐♥t❡r❢❡rê♥❝✐❛ ❡♥tr❡ ♦♥❞❛s ✐♥❝✐❞❡♥t❡ ❡ ❡s♣❛❧❤❛❞❛✱ ❡♥q✉❛♥t♦ q✉❡ ♣❛r❛ ♦♥❞❛ ♣❧❛♥❛ ♣r♦❣r❡ss✐✈❛ ❛ ♠❛❣♥✐t✉❞❡ ❞♦ ♠♦♠❡♥t♦ ❧✐♥❡❛r tr❛♥s♠✐t✐❞♦ à ♣❛rtí❝✉❧❛ é ❞❡t❡r♠✐♥❛❞♦ s♦♠❡♥t❡ ♣❡❧❛s ♦♥❞❛s ❡s♣❛❧❤❛❞❛s✳ ❆ss✐♠✱ ❛ ♠❛❣♥✐t✉❞❡ ❞❛ ❢♦rç❛ ❞❡ r❛❞✐❛çã♦ ♠é❞✐❛ ❡♠ ✉♠❛ ♦♥❞❛ ❡st❛❝✐♦♥ár✐❛ é ♠✉✐t♦ ♠❛✐♦r q✉❡ ❡♠ ✉♠❛ ♦♥❞❛ ♣❧❛♥❛ ♣r♦❣r❡ss✐✈❛✳ ◆♦ ❡♥t❛♥t♦✱ ♦ ♠ét♦❞♦ ✉t✐❧✐③❛❞♦ ♣♦r ●♦r❦♦✈ t❡♠ ✉♠❛ ❧✐♠✐t❛çã♦✱ ❡❧❡ ❝♦♥s✐❞❡r♦✉ q✉❡ ❛ ♣❛rtí❝✉❧❛ ❡s❢ér✐❝❛ ❝♦♠♣r❡ssí✈❡❧ é ♠✉✐t♦ ♠❡♥♦r q✉❡ ♦ ❝♦♠♣r✐♠❡♥t♦ ❞❡ ♦♥❞❛ ✐♥❝✐❞❡♥t❡✳ ◆❡st❛ ❛♣r♦①✐♠❛çã♦✱ s♦♠❡♥t❡ ♦s ❝♦❡✜❝✐❡♥t❡s ❞❡ ❡s♣❛❧❤❛♠❡♥t♦ ❞❡ ♠♦♥♦♣♦❧♦ ❡ ❞✐♣♦❧♦ sã♦ ❝♦♥s✐❞❡r❛❞♦s✳ ◆②❜♦r❣ ❢❡③ ✉s♦ ❞♦s ♠ét♦❞♦s ❞❡s❡♥✈♦❧✈✐❞♦s ♣♦r ❑✐♥❣ ❡ ❊♠❜❧❡t♦♥ ♣❛r❛ ❡st✉❞❛r ❛ ❢♦rç❛ ❞❡ r❛❞✐❛çã♦ s♦❜r❡ ✉♠❛ ♣❡q✉❡♥❛ ❡s❢❡r❛ rí❣✐❞❛ ❞❡✈✐❞♦ ✉♠❛ ♦♥❞❛ ❡s❢ér✐❝❛✳ ❊❧❡ ❡①♣r❡ss♦✉ ❛ ❢♦rç❛ ❞❡ r❛❞✐❛çã♦ ❝♦♠♦ ❣r❛❞✐❡♥t❡ ❞❛ ❞❡♥s✐❞❛❞❡ ❞❡ ❡♥❡r❣✐❛ ♣♦t❡♥❝✐❛❧ ❡ ❝✐♥ét✐❝❛ ❞♦ ❢❡✐①❡ ✐♥❝✐❞❡♥t❡ ❡ ♠♦str♦✉ q✉❡ ❛ ♣❛rt❡ ♣r♦❣r❡ss✐✈❛ ❞❛ ♦♥❞❛ ❡s❢ér✐❝❛ ♣♦❞❡ s❡r ❞❡s❝r✐t❛ ♣❡❧❛ t❡♦r✐❛ ❞❡ ●♦r❦♦✈ ❝♦♠ ✉♠ t❡r♠♦ ❞❡ ❝♦rr❡çã♦✳ ❆ t❡♦r✐❛ ❞❡ ❢♦rç❛ ❞❡ r❛❞✐❛çã♦ ❛❝úst✐❝❛ s♦❜r❡ ✉♠❛ ❡s❢❡r❛ ❡❧ást✐❝❛ s✉s♣❡♥s❛ ❡♠ ✉♠ ✢✉✐❞♦ ✐❞❡❛❧ ❞❡✈✐❞♦ ❛ ✉♠❛ ♦♥❞❛ ♣❧❛♥❛ ♣r♦❣r❡ss✐✈❛ é ❞❡s❡♥✈♦❧✈✐❞❛ ♣♦r ❍❛s❡❣❛✇❛ ❖ ❝á❧❝✉❧♦ ❞❛ ❢♦rç❛ ❞❡ r❛❞✐❛çã♦ ❡♠ ✉♠❛ ♣❡q✉❡♥❛ ❡s❢❡r❛ ❝♦♠♣r❡ssí✈❡❧ ❞❡✈✐❞♦ ❛ ✉♠ ❢❡✐①❡ ❡s❢ér✐❝♦ ❢♦❝❛❧✐③❛❞♦ ❢♦✐ ♣r♦♣♦st♦ ♣♦r ❲✉ ❡ ❉✉ ❖s r❡s✉❧t❛❞♦s ❢♦r❛♠ ❡①♣r❡ss♦s ❡♠ t❡r♠♦s ❞❛s ❞❡♥s✐❞❛❞❡s ❞❡ ❡♥❡r❣✐❛ ♣♦t❡♥❝✐❛❧ ❡ ❝✐♥ét✐❝❛ ❞♦ ❢❡✐①❡ ✐♥❝✐❞❡♥t❡✳ ◆❡ss❡ ❡st✉❞♦✱ ❡❧❡s ♠♦str❛r❛♠ q✉❡ ♦ ❢❡✐①❡ ❢♦❝❛❧✐③❛❞♦ ♣♦❞❡ s❡r ✉♠❛ ❛❧t❡r♥❛t✐✈❛ ♣❛r❛ ❧❡✈✐t❛çã♦ ❞❡ ♣❡q✉❡♥❛s ❡s❢❡r❛s ❞❡ ❛❧t❛ ❞❡♥s✐❞❛❞❡✳ ❯♠ ❡st✉❞♦ ♠❛✐s ❛❜r❛♥❣❡♥t❡ s♦❜r❡ ❢❡✐①❡s ❞❡ ❇❡ss❡❧ t❡♠ s✐❞♦ r❡❛❧✐③❛❞♦ ❞❡✈✐❞♦ ❛ ❝❛r❛❝t❡ríst✐❝❛ ❞❡ss❡ ❢❡✐①❡ ❡♠ ❣❡r❛r ❢♦rç❛ ❞❡ r❛❞✐❛çã♦ ♥❡❣❛t✐✈❛✳ ▼❛rst♦♥ ♦❜t❡✈❡ ✉♠❛ ❡①♣r❡ssã♦ ❞❛ ❢♦rç❛ ❞❡ r❛❞✐❛çã♦ ❣❡r❛❞❛ ♣❡❧❛ ✐♥t❡r❛çã♦ ❞❡ ✉♠ ❢❡✐①❡ ❞❡ ❇❡ss❡❧ ❞❡ ♦r❞❡♠ ③❡r♦ ❝♦♠ ✉♠❛ ♣❛rtí❝✉❧❛ ✢✉✐❞❛ ♥♦ ❡✐①♦ ❞♦ ❢❡✐①❡✳ ❊❧❡ ❛♣r❡s❡♥t♦✉ ❛s ❝♦♥❞✐çõ❡s ♣❛r❛ ♣r♦❞✉③✐r ✉♠❛ ❢♦rç❛ ♥❡❣❛t✐✈❛ ✭✐st♦ é✱ ♦♣♦st❛ à ❞✐r❡çã♦ ❞❡ ♣r♦♣❛❣❛çã♦ ❞♦ ❢❡✐①❡✮ s♦❜r❡ ❛ ♣❛rtí❝✉❧❛✳ ❆❧é♠ ❞✐ss♦✱ ❡❧❡ ♣r♦♣ôs ✉♠ ❡st✉❞♦ ❞❛ ❢♦rç❛ ❞❡ r❛❞✐❛çã♦ ❞❡✈✐❞♦ ❛ ✉♠ ❢❡✐①❡ ❞❡ ❇❡ss❡❧ ❞❡ ♣r✐♠❡✐r❛ ♦r❞❡♠ s♦❜r❡ ✉♠❛ ♣❛rtí❝✉❧❛ ❡s❢ér✐❝❛ ✐♠❡rs❛ ❡♠ á❣✉❛✱ ❡ ♦❜t❡✈❡ ❢♦rç❛ ❞❡ r❛❞✐❛çã♦ ♥❡❣❛t✐✈❛ ❝♦♠♦ ✐♥✈❡st✐❣❛❞♦ ❛♥t❡r✐♦r♠❡♥t❡ ♣❛r❛ ❢❡✐①❡ ❞❡ ❇❡ss❡❧ ❞❡ ♦r❞❡♠ ③❡r♦✳ ▼✐tr✐ ❞❡♠♦str♦✉ q✉❡ ❛ ❢♦rç❛ ❞❡ r❛❞✐❛çã♦ ❞❡ ✉♠ ❢❡✐①❡ ❞❡ ❇❡ss❡❧ ❞❡ ❛❧t❛ ♦r❞❡♠ ❛❣✐♥❞♦ s♦❜r❡ ✉♠❛ ❡s❢❡r❛ é ♦♣♦st❛ ❛ ❞✐r❡çã♦ ❞❡ ♣r♦♣❛❣❛çã♦ ❞♦ ❢❡✐①❡✳ ❖ ❝♦♠♣♦rt❛♠❡♥t♦ ❞❡ ✉♠ ❢❡✐①❡ ❞❡ ❇❡ss❡❧ tr❛t♦r ❞❡ ♦r❞❡♠ ③❡r♦ ❛t✉❛♥❞♦ ❡♠ ✉♠❛ ❡s❢❡r❛ ✢✉✐❞❛ ❢♦✐ ❛♣r❡s❡♥t❛❞♦ ❊❧❡ t❛♠❜é♠ ❛♣r❡s❡♥t♦✉ ❛❧❣✉♥s tr❛❜❛❧❤♦s s♦❜r❡ ❢♦rç❛ ❞❡ r❛❞✐❛çã♦ ❡♠ ✉♠ ❢❡✐①❡ ❞❡ ❇❡ss❡❧ ❞❡ ♦♥❞❛ ❡st❛❝✐♦♥ár✐❛ ❝♦♠ ✐♥t✉✐t♦ ❞❡ ❛♣❧✐❝❛r ❡ss❡ ❡st✉❞♦ ❡♠ ♠❛♥✐♣✉❧❛çã♦ ❞❡ ♣❛rtí❝✉❧❛s✳ ❙✐❧✈❛ ♦❜t❡✈❡ ✉♠❛ s♦❧✉çã♦ ❞❛ ❢♦rç❛ ❞❡ r❛❞✐❛çã♦ ❡①❡r❝✐❞❛ ♣❛r❛ ✉♠ ❢❡✐①❡ ❛❝úst✐❝♦ ❞❡ ❢♦r♠❛ ❛r❜✐trár✐❛ s♦❜r❡ ✉♠ ♦❜❥❡t♦ ❞❡ ❣❡♦♠❡tr✐❛ ❛r❜✐trár✐❛✳ ❊st❛

  ❡①♣r❡ssã♦ ❞❛ ❢♦rç❛ ❞❡ r❛❞✐❛çã♦ é ♦❜t✐❞❛ ❝♦♠♦ ✉♠❛ ❢✉♥çã♦ ❞♦s ❝♦❡✜❝✐❡♥t❡s ❞❡ ❢♦r♠❛ ❞♦ ❢❡✐①❡ ❞❡ ✉♠❛ ♦♥❞❛ ✐♥❝✐❞❡♥t❡ ❡ ❞♦s ❝♦❡✜❝✐❡♥t❡s ❞❡ ❡s♣❛❧❤❛♠❡♥t♦✳ P♦st❡r✐♦r♠❡♥t❡✱ ❩❤❛♥❣ ❡ ▼❛rst♦♥ ❞❡s❡♥✈♦❧✈❡r❛♠ ✉♠❛ ✐♥t❡r♣r❡t❛çã♦ ❣❡♦♠étr✐❝❛ ❞❛ ❢♦rç❛ ❞❡ r❛❞✐❛çã♦ ❡①❡r❝✐❞❛ ♣❡❧♦ ❢❡✐①❡ ❞❡ ❇❡ss❡❧ ❞❡ ♦r❞❡♠ ③❡r♦ ❡♠ ✉♠❛ ❡s❢❡r❛ ♥♦ ❡✐①♦ ❡♠ t❡r♠♦s ❞❛s s❡çõ❡s tr❛♥s✈❡rs❛✐s ❞❡ ❛❜s♦rçã♦✱ ❡s♣❛❧❤❛♠❡♥t♦ ❡ ❞❛ ♣♦tê♥❝✐❛ ❡①tr❛í❞❛ ❞♦ ❢❡✐①❡✳ ❯♠ ❡st✉❞♦ s❡♠❡❧❤❛♥t❡ s♦❜r❡ ✉♠❛ ❝❧❛ss❡ ♠❛✐s ❛♠♣❧❛ ❞❡ ❢❡✐①❡s ❛❝úst✐❝♦s ❢♦✐ t❛♠❜é♠ r❡❛❧✐③❛❞♦ ♣♦r ❡st❡ ❣r✉♣♦ ❆③❛r♣❡②✈❛♥❞ r❡❛❧✐③♦✉ ✉♠ ❡st✉❞♦ t❡ór✐❝♦ s♦❜r❡ ❛ ❢♦rç❛ ❞❡ r❛❞✐❛çã♦ ❡①❡r❝✐❞❛ s♦❜r❡ ✉♠❛ ❡s❢❡r❛ ♣♦r♦s❛ ❞❡ ❛❧✉♠í♥✐♦ ♥♦ ❡✐①♦✳ ◆❡st❡ ❝❛s♦✱ ♦ ❛✉♠❡♥t♦ ❞❛ ♣♦r♦s✐❞❛❞❡ ❞❡❣r❛❞❛ ❛ ❢♦rç❛ ❞❡ r❛❞✐❛çã♦ s♦❜r❡ ✉♠❛ ♣❛rtí❝✉❧❛ ♣❡q✉❡♥❛✳ ❊❧❡ t❛♠❜é♠ ❛♣r❡s❡♥t♦✉ ✉♠ ❡st✉❞♦ s♦❜r❡ ❛ ♠❛♥✐♣✉❧❛çã♦ ❛❝úst✐❝❛ ❞❡ ❝❛s❝❛s ❡s❢ér✐❝❛s ♣♦r♦s❛s ❡ ♦s r❡s✉❧t❛❞♦s ♦❜t✐❞♦s ✐♥❞✐❝❛♠ q✉❡ ❛ ♠❛♥✐♣✉❧❛çã♦ ❞❡ ❝❛s❝❛s ❞❡ ❜❛✐①❛ ♣♦r♦s✐❞❛❞❡ é ♣♦ssí✈❡❧ ✉s❛♥❞♦ ❢❡✐①❡s ❞❡ ❇❡ss❡❧ ❝♦♠ ❣r❛♥❞❡s â♥❣✉❧♦s ❝ô♥✐❝♦s ❆ ❢♦rç❛ ❞❡ r❛❞✐❛çã♦ ❞❡ ✉♠ ❢❡✐①❡ ❞❡ ❇❡ss❡❧ ❡①❡r❝✐❞♦ s♦❜r❡ ✉♠❛ ♣❡q✉❡♥❛ ♣❛rtí❝✉❧❛ ❢♦✐ ✐♥✈❡st✐❣❛❞♦ ♣♦r ❇❛r❡s❝❤ ❡ ❝♦❧❛❜♦r❛❞♦r❡s ❆❧é♠ ❞✐ss♦✱ ❙❛♣♦③❤♥✐❦♦✈ ❞❡s❡♥✈♦❧✈❡✉ ✉♠❛ ❛❜♦r❞❛❣❡♠ t❡ór✐❝❛ ♣❛r❛ ❝❛❧❝✉❧❛r ❢♦rç❛ ❞❡ r❛❞✐❛çã♦ ❞❡ ✉♠ ❢❡✐①❡ ❛❝úst✐❝♦ ❛r❜✐trár✐♦ s♦❜r❡ ✉♠❛ ❡s❢❡r❛ ❡❧ást✐❝❛ ❞❡♥tr♦ ❞❡ ✉♠ ❧íq✉✐❞♦ ♦✉ ❣ás✳ ❋ór♠✉❧❛s ❡①❛t❛s ❞❛ ❢♦rç❛ ❞❡ r❛❞✐❛çã♦ ❛❝úst✐❝❛ ❡①❡r❝✐❞❛ ♣♦r ✉♠❛ ♦♥❞❛ ❤❛r♠ô♥✐❝❛ ❛r❜✐trár✐❛ s♦❜r❡ ✉♠❛ ♣❡q✉❡♥❛ ♣❛rtí❝✉❧❛ ❝♦♠♣r❡ssí✈❡❧ ❛❜s♦r✈❡❞♦r❛✱ s✉s♣❡♥s❛ ♥✉♠ ✢✉✐❞♦ ✐❞❡❛❧✱ ❢♦✐ ♣r♦♣♦st❛ ♣♦r ❙✐❧✈❛

  ❆ ❢♦rç❛ ❞❡ r❛❞✐❛çã♦ ❛❝úst✐❝❛ ❡♥tr❡ ❞✉❛s ♦✉ ♠❛✐s ♣❛rtí❝✉❧❛s ♥ã♦ ❛❜s♦r✈❡❞♦r❛s t❛♠❜é♠ tê♠ s✐❞❛ ❛♥❛❧✐s❛❞❛ ❆❧❣✉♥s ❡s❢♦rç♦s tê♠ s✐❞♦ ❞❡❞✐❝❛❞♦ ❛ ❝♦♠♣r❡❡♥❞❡r ❛ ❢♦rç❛ ❞❡ r❛❞✐❛çã♦ ❛❣✐♥❞♦ s♦❜r❡ ✉♠❛ ♣❛rtí❝✉❧❛ r❡✈❡st✐❞❛ ♣♦r ✉♠❛ ❝❛♠❛❞❛ ✈✐s❝♦❡❧ást✐❝❛ ❱❡r✐✜❝♦✉✲s❡ q✉❡✱ ♥❛ ♣r❡s❡♥ç❛ ❞❡ ✉♠❛ ❝❛♠❛❞❛ ✈✐s❝♦❡❧ást✐❝❛✱ ❛ ❢♦rç❛ ❞❡ r❛❞✐❛çã♦ ❛①✐❛❧ ♣♦r ♦♥❞❛s ♣r♦❣r❡ss✐✈❛s ♣♦❞❡ t♦r♥❛r✲s❡ ♥❡❣❛t✐✈❛✱ ♦ q✉❡ s✐❣♥✐✜❝❛ q✉❡ ❛ ❢♦rç❛ ❡ ❛ ❞✐r❡çã♦ ❞❡ ♣r♦♣❛❣❛çã♦ ❞❛ ♦♥❞❛ sã♦ ❝♦♥trár✐❛s

  ✶✳✶✳✷ ❚♦rq✉❡ ❞❡ r❛❞✐❛çã♦ ❛❝úst✐❝♦

  ❯♠❛ ♦♥❞❛ ❛❝úst✐❝❛ ♣♦❞❡ tr❛♥s♣♦rt❛r ♠♦♠❡♥t♦ ❛♥❣✉❧❛r✱ ❡ ❡st❡ ♣♦❞❡ s❡r tr❛♥s❢❡r✐❞♦ ♣❛r❛ ✉♠ ♦❜❥❡t♦ s✉s♣❡♥s♦✳ ❊ss❛ ✐♥t❡r❛çã♦ ❞❛ ♦♥❞❛ ❛❝úst✐❝❛ ❝♦♠ ✉♠ ♦❜❥❡t♦ s✉s♣❡♥s♦ ♥✉♠ ✢✉✐❞♦ ♣♦❞❡ ❣❡r❛r ✉♠ t♦rq✉❡ ♠é❞✐♦✱ ❝♦♥❤❡❝✐❞♦ ❝♦♠♦ t♦rq✉❡ ❞❡ r❛❞✐❛çã♦ ❛❝úst✐❝♦✳ ❯♠❛ ❞❛s ♣r✐♠❡✐r❛s ♦❜s❡r✈❛çõ❡s ❛ r❡s♣❡✐t♦ ❞❡s❞❡ ❢❡♥ô♠❡♥♦ ❢♦✐ r❡❧❛t❛❞❛ ♣♦r ❘❛②❧❡✐❣❤ ❖ t♦rq✉❡ ❞❡ r❛❞✐❛çã♦ ❘❛②❧❡✐❣❤ é ♣r♦✈♦❝❛❞♦ ♣❡❧♦ ♠♦♠❡♥t♦ ❞♦ ❡str❡ss❡ ❞❡ r❛❞✐❛çã♦ ❛❣✐♥❞♦ s♦❜r❡ ✉♠ ♦❜❥❡t♦ ❛ss✐♠étr✐❝♦✳ ❯♠ tr❛❜❛❧❤♦ ♣✐♦♥❡✐r♦ ❛ r❡s♣❡✐t♦ ❞❡st❡ ❢❡♥ô♠❡♥♦ ❢♦✐ ♣r♦♣♦st♦ ♣♦r ▼❛✐❞❛♥✐❦ ◆❡st❡ tr❛❜❛❧❤♦ ❡❧❡ ❞❡r✐✈♦✉ ✉♠❛ ❡①♣r❡ssã♦ ♣❛r❛ ♦ t♦rq✉❡ ❞❡ r❛❞✐❛çã♦ s♦❜r❡ ✉♠ ♦❜❥❡t♦ ❞❡ ❢♦r♠❛ ❛r❜✐trár✐❛ ❞❡✈✐❞♦ ❛ ♣r❡ssã♦ ❞❡ r❛❞✐❛çã♦ ❛❝úst✐❝❛ ❢❛③❡♥❞♦ ♦ ✉s♦ ❞♦ t❡♦r❡♠❛ ❞♦ ♠♦♠❡♥t♦ ❛♥❣✉❧❛r✳ ❯♠❛ ❢ór♠✉❧❛ ❛❧t❡r♥❛t✐✈❛ ♣❛r❛ ♦ t♦rq✉❡ ❞❡ r❛❞✐❛çã♦ q✉❡ ❞❡♣❡♥❞❡ ❞❡ ✈❛r✐❛çõ❡s ♥❛ ❛♠♣❧✐t✉❞❡ ❞❡ ❡s♣❛❧❤❛♠❡♥t♦ ♣❛r❛ ♠✉❞❛♥ç❛s ♥❛ ♦r✐❡♥t❛çã♦ ❞❛ ♦♥❞❛ ♣❧❛♥❛ ✐♥❝✐❞❡♥t❡ ❢♦✐ ♦❜t✐❞♦ ♣♦r ❙♠✐t❤ ♣♦r ♠❡✐♦ ❞❡ ✉♠❛ ❣❡♥❡r❛❧✐③❛çã♦ ❞♦ t❡♦r❡♠❛ ót✐❝♦✳ ❍❡❢♥❡r ❡ ▼❛rst♦♥ ♠♦str❛r❛♠ q✉❡ ✉♠ t♦rq✉❡ ❞❡ r❛❞✐❛çã♦ ❛①✐❛❧ ♣♦❞❡ s❡r ❞❡s❡♥✈♦❧✈✐❞♦ ❡♠ ✉♠ ♦❜❥❡t♦ ❛❜s♦r✈❡❞♦r ♣♦r ✉♠ ❢❡✐①❡ ❞❡ ✈♦rt✐❝✐❞❛❞❡ ♣❛r❛①✐❛❧✳ ❋❛♥ ❡ ❝♦❧❛❜♦r❛❞♦r❡s ❡st❛❜❡❧❡❝❡r❛♠ ✉♠❛ t❡♦r✐❛ ♣❛r❛ ❝❛❧❝✉❧❛r ♦ t♦rq✉❡ ❞❡ r❛❞✐❛çã♦ s♦❜r❡ ✉♠ ❡s♣❛❧❤❛❞♦r ❞❡ ❢♦r♠❛ ❛r❜✐trár✐❛ ♣❛r❛ ✉♠❛ ♦♥❞❛ ❛❝úst✐❝❛ ❛r❜✐trár✐❛✳ ❊ss❡ ❡st✉❞♦ é ❧✐♠✐t❛❞♦✱ ♣♦✐s ♦ ❡s♣❛❧❤❛❞♦r t❡♠ q✉❡ s❡r ♣❡q✉❡♥♦ q✉❛♥❞♦ ❝♦♠♣❛r❛❞♦ ❛♦ ❝♦♠♣r✐♠❡♥t♦ ❞❡ ♦♥❞❛✳ ❩❤❛♥❣ ❡ ▼❛rst♦♥

  ❡st✉❞❛r❛♠ t❡♦r✐❝❛♠❡♥t❡ ♦ t♦rq✉❡ ❞❡ r❛❞✐❛çã♦ ❛①✐❛❧ s♦❜r❡ ✉♠ ♦❜❥❡t♦ ❝♦♠ s✐♠❡tr✐❛ ❛①✐❛❧ s✉s♣❡♥s♦ ♥✉♠ ✢✉✐❞♦ ✐❞❡❛❧ ✳ ❊❧❡s t❛♠❜é♠ ❡①♣r❡ss❛r❛♠ ♦ t♦rq✉❡ ❞❡ r❛❞✐❛çã♦ ❝♦♠♦ ❛ ✐♥t❡❣r❛❧ ❞♦ ✢✉①♦ ❞♦ ♠♦♠❡♥t♦ ❛♥❣✉❧❛r ♠é❞✐♦ s♦❜r❡ ✉♠❛ s✉♣❡r❢í❝✐❡ ❡s❢ér✐❝❛ ❞✐st❛♥t❡ ❞♦ ♦❜❥❡t♦ ❡s♣❛❧❤❛❞♦r ❢♦r♥❡❝❡r❛♠ ✉♠❛ ❡①♣r❡ssã♦ ❣❡r❛❧ ♣❛r❛ ♦ t♦rq✉❡ ❞❡ r❛❞✐❛çã♦ tr✐❞✐♠❡♥s✐♦♥❛❧ ❡①❡r❝✐❞♦ ♣♦r ✉♠ ❢❡✐①❡ ❛❝úst✐❝♦ ❞❡ ❢r❡♥t❡ ❞❡ ♦♥❞❛ ❛r❜✐trár✐❛ s♦❜r❡ ✉♠ ♦❜❥❡t♦ ❞❡ q✉❛❧q✉❡r ❢♦r♠❛ ❣❡♦♠étr✐❝❛ ❡♠ t❡r♠♦s ❞♦s ❝♦❡✜❝✐❡♥t❡s ❞❡ ❢♦r♠❛ ❞♦ ❢❡✐①❡ ❡ ❞❡ ❡s♣❛❧❤❛♠❡♥t♦✳ ❆❧é♠ ❞✐ss♦✱ ❙✐❧✈❛ ❞❡s❡♥✈♦❧✈❡✉ ✉♠❛ ❡①♣r❡ssã♦ ♣❛r❛ t♦rq✉❡ ❞❡ r❛❞✐❛çã♦ ❛❝úst✐❝♦ s♦❜r❡ ✉♠❛ ♣❡q✉❡♥❛ ♣❛rtí❝✉❧❛ ✢✉✐❞❛ ♣❛r❛ ❝❛♠♣♦ ❛❝úst✐❝♦ ❛r❜✐trár✐♦ ❯♠ ❡st✉❞♦ s♦❜r❡ t♦rq✉❡ ❞❡ r❛❞✐❛çã♦ ❛①✐❛❧ ❞❡ ✉♠ ❢❡✐①❡ ❞❡ ❇❡ss❡❧ ♣r♦❣r❡ss✐✈♦ ❡ ❡st❛❝✐♦♥ár✐♦ s♦❜r❡ ✉♠❛ ❝❛s❝❛ ❡s❢ér✐❝❛ ❢♦✐ ❞❡s❡♥✈♦❧✈✐❞♦ ♣♦r ▼✐tr✐ ❛♣r❡s❡♥t❛r❛♠ ♦ ♣r✐♠❡✐r♦ t❡st❡ q✉❛♥t✐t❛t✐✈♦ ❞❡ tr❛♥s❢❡rê♥❝✐❛ ❞♦ ♠♦♠❡♥t♦ ❛♥❣✉❧❛r ♣❛r❛ ✉♠ ♦❜❥❡t♦ ❛❜s♦r✈❡❞♦r ✐♠❡rs♦ ❡♠ ✉♠ ❧✐q✉✐❞♦ ✈✐s❝♦s♦✳ ◆❛ ❡①♣❡r✐ê♥❝✐❛ r❡❛❧✐③❛❞❛✱ ❡❧❡ ♦❜s❡r✈♦✉ q✉❡ é ♣♦ssí✈❡❧ ❣✐r❛r ✉♠ ❞✐s❝♦ ❞❡ t❛♠❛♥❤♦ ♠✐❧✐♠étr✐❝♦ ✉t✐❧✐③❛♥❞♦ ❢❡✐①❡ ❣❡r❛❞♦ ♣♦r ✉♠ tr❛♥s❞✉t♦r ❡s❢ér✐❝♦✳ ❙❝❤✇❛r③ ❡ ❝♦❧❛❜♦r❛❞♦r❡s ✜③❡r❛♠ ✉♠❛ ❛♥á❧✐s❡ t❡ór✐❝❛ ❞♦ t♦rq✉❡ ❞❡ r❛❞✐❛çã♦ ❛❝úst✐❝♦ ❡ r❡❛❧✐③❛r❛♠ ✉♠ ❡①♣❡r✐♠❡♥t♦ q✉❡ é ❝❛♣❛③ ❞❡ r♦t❛❝✐♦♥❛r ♣❛rtí❝✉❧❛s ♥ã♦ ❡s❢ér✐❝❛s ❞❡ ❢♦r♠❛ ❝♦♥tr♦❧❛❞❛ ♣♦r ♠❡✐♦ ❞❡ ♦♥❞❛s ❛❝úst✐❝❛s ❡st❛❝✐♦♥ár✐❛s✳

  ✶✳✷ ❆♣❧✐❝❛çõ❡s

  ❖ ✐♥t❡r❡ss❡ ❡♠ ❢♦rç❛ ❞❡ r❛❞✐❛çã♦ ❛❝úst✐❝❛ t❡♠ ❝r❡s❝✐❞♦ ♥❛s ú❧t✐♠❛s ❞é❝❛❞❛s ❞❡✈✐❞♦ ❛ ❛♣❧✐❝❛çõ❡s ❡♠ ♠❛♥✐♣✉❧❛çã♦ ❞❡ ♣❛rtí❝✉❧❛s s❡♠ ❝♦♥t❛t♦s s✉s♣❡♥s❛ ❡♠ ✉♠ ✢✉✐❞♦✳ ❉✐❢❡r❡♥✲ t❡s ❛♣❧✐❝❛çõ❡s ❜✐♦t❡❝♥♦❧ó❣✐❝❛s s✉r❣✐r❛♠✱ ✐♥❝❧✉✐♥❞♦ ❧❡✈✐t❛çã♦ ❛❝úst✐❝❛✱ ♣✐♥ç❛s ❛❝úst✐❝❛s ❡ ♠✐❝r♦✢✉í❞✐❝❛ ✭❧❛❜✲♦♥✲❛✲❝❤✐♣✮✳

  ✶✳✷✳✶ ▲❡✈✐t❛çã♦ ❛❝úst✐❝❛

  ❆ ❧❡✈✐t❛çã♦ é ✉♠ ♣r♦❝❡ss♦ ♥♦ q✉❛❧ ✉♠ ♦❜❥❡t♦ é s✉s♣❡♥s♦ ❡♠ ✉♠❛ ♣♦s✐çã♦ ❡stá✈❡❧ s❡♠ ❝♦♥t❛t♦✳ ❯♠ s✐st❡♠❛ ❞❡ ❧❡✈✐t❛çã♦ ❛❝úst✐❝❛ é ❝♦♠♣♦st♦ ♣♦r ✉♠ tr❛♥s❞✉t♦r✱ s✉♣❡r❢í❝✐❡ ❝♦♠ ✈✐❜r❛çã♦ ❤❛r♠ô♥✐❝❛ ❝❛♣❛③ ❞❡ ♣r♦❞✉③✐r ✉♠❛ ♦♥❞❛ s♦♥♦r❛ ❞❡ ❢r❡q✉ê♥❝✐❛ ❞❡✜♥✐❞❛ ❡ ✉♠ r❡✢❡t♦r✳ ❉❡st❛ ❢♦r♠❛✱ ✉♠❛ ♦♥❞❛ ❡st❛❝✐♦♥ár✐❛ é ❣❡r❛❞❛ ❞❡✈✐❞♦ ❛ r❡✢❡①õ❡s ❡♥tr❡ ♦ tr❛♥s❞✉t♦r ❡ ✉♠ r❡✢❡t♦r✳ ❆ ❧❡✈✐t❛çã♦ ❛❝úst✐❝❛ é ✉♠❛ ❞❛s ❛♣❧✐❝❛çõ❡s q✉❡ ✉t✐❧✐③❛ ♦♥❞❛ ❡st❛❝✐♦♥ár✐❛✳

  ❊♠ ✶✾✽✺✱ ❚r✐♥❤ ✉t✐❧✐③♦✉ ♦ ♠ét♦❞♦ ❞❡ ♦♥❞❛ ❡st❛❝✐♦♥ár✐❛ ♣❛r❛ ❧❡✈✐t❛r ❛♠♦str❛s ❝♦♠ ◦ ◦ C C

  ❞✐♠❡♥sõ❡s tí♣✐❝❛s q✉❡ ✈❛r✐❛♠ ❞❡ 100 µm ❛ 5 mm ❡ t❡♠♣❡r❛t✉r❛s ❡♥tr❡ 40 ❡ 500 ✳ ●❛♠♠❡❧ ❡ ❝♦❧❛❜♦r❛❞♦r❡s ✉t✐❧✐③❛r❛♠ ✉♠❛ s✐r❡♥❡ ♦♣❡r❛♥❞♦ ❛ 3260 Hz ❡ ✉♠ r❡✢❡t♦r ❛♣r♦♣r✐❛❞♦ ♣❛r❛ ♠♦str❛r q✉❡ é ♣♦ssí✈❡❧ ❧❡✈✐t❛r ❧✐✈r❡♠❡♥t❡ ✉♠❛ ❜♦❧❛ ❞❡ ❛ç♦ ❞❡ 1 cm ❞❡ ❞✐â♠❡tr♦✳ ❳✐❡ ❡ ❲❡✐ ♥♦t❛r❛♠ ✉♠ ❛✉♠❡♥t♦ ♥♦tá✈❡❧ ❞❛ ❢♦rç❛ ❞❡ ❧❡✈✐t❛çã♦ ♣r♦❥❡t❛♥❞♦ ❛❞❡q✉❛❞❛♠❡♥t❡ ❛ ❢♦r♠❛ ❞♦ r❡✢❡t♦r✳ ■st♦ ♣❡r♠✐t✐✉ ❧❡✈✐t❛r ❜♦❧❛s ❞❡ t✉♥❣stê♥✐♦ ❞❡ ❛❧t❛ 3

  ❞❡♥s✐❞❛❞❡ 18, 9 g/cm ✳ ❇r❛♥❞t ♠♦str♦✉ q✉❡ ♦♥❞❛s ❞❡ ✉❧tr❛ss♦♠ ♣♦❞❡ ❧❡✈✐t❛r ❜♦❧❛s ♣❡s❛❞❛s ❞❡ t✉♥❣stê♥✐♦✳ ❊st❡ ♠ét♦❞♦ ❞❡ ♠❛♥t❡r ♦s ♦❜❥❡t♦s s✉s♣❡♥s♦s s❡♠ ❝♦♥t❛t♦ ♥♦ ❛r ♣♦❞❡♠ s❡r ❛♣❧✐❝❛❞♦s ♣❛r❛ ❛ ✐♥✈❡st✐❣❛çã♦ ❡ ♦ ♣r♦❝❡ss❛♠❡♥t♦ ❞♦s ♥♦✈♦s ♠❛t❡r✐❛✐s✳ ❋♦✲ r❡st✐ ❡ ❝♦❧❛❜♦r❛❞♦r❡s ❛♣r❡s❡♥t❛r❛♠ ✉♠❛ ❞❡♠♦♥str❛çã♦ ❡①♣❡r✐♠❡♥t❛❧ ❡ t❡ór✐❝❛ ❞❡ ✉♠ ❝♦♥❝❡✐t♦ q✉❡ ♣❡r♠✐t❡ ❧❡✈✐t❛r ❡ ♠♦✈✐♠❡♥t❛r ❝♦♥tr♦❧❛❞❛♠❡♥t❡ ❛ ♠❛tér✐❛ s❡♠ ❝♦♥t❛t♦✱

  ✐♥❝❧✉✐♥❞♦ ♠♦✈✐♠❡♥t♦ ♦r❜✐t❛❧ ♦✉ r♦t❛çã♦ ❞❛ ❣♦tí❝✉❧❛s ❡ ♣❛rtí❝✉❧❛s ♥♦ ❛r✳ ❆♥❞r❛❞❡ ❡ ❝♦❧❛❜♦✲ r❛❞♦r❡s ✜③❡r❛♠ ✉♠❛ ❛♥á❧✐s❡ ❞❡ ✉♠ ❧❡✈✐t❛❞♦r ❛❝úst✐❝♦ ❢♦r♠❛❞♦ ♣♦r ✉♠ tr❛♥s❞✉t♦r ❞❡ ✉❧tr❛ss♦♠ ❡ ✉♠ r❡✢❡t♦r ❝ô♥❝❛✈♦✳ ❊♠ ❝♦♥tr❛st❡ ❝♦♠ ❧❡✈✐t❛❞♦r❡s tr❛❞✐❝✐♦♥❛✐s✱ ❛ ❣❡♦♠❡tr✐❛ ❛♣r❡s❡♥t❛❞❛ ♥❡st❡ tr❛❜❛❧❤♦ ♥ã♦ ♥❡❝❡ss✐t❛ q✉❡ ❛ ❞✐stâ♥❝✐❛ ❞❡ s❡♣❛r❛çã♦ ❡♥tr❡ ♦ tr❛♥s❞✉t♦r ❡ ♦ r❡✢❡t♦r s❡❥❛ ♠ú❧t✐♣❧❛ ❞❡ ♠❡✐♦ ❝♦♠♣r✐♠❡♥t♦ ❞❡ ♦♥❞❛✳ ❆❧é♠ ❞✐ss♦✱ ❡❧❡s ❞❡♠♦♥str❛r❛♠ q✉❡ ❛s ♣❛rtí❝✉❧❛s ❞❡ ❧❡✈✐t❛çã♦ ♣♦❞❡♠ s❡r❡♠ ♠❛♥✐♣✉❧❛❞❛s ❛tr❛✈és ❞♦ ❝♦♥tr♦❧❡ ❞❛ ♣♦s✐çã♦ ❞♦ r❡✢❡t♦r✱ ♠❛♥t❡♥❞♦ ♦ tr❛♥s❞✉t♦r ♥✉♠❛ ♣♦s✐çã♦ ✜①❛✱ ❋✐❣✳

  ▲❡✈✐t❛çã♦ ❛❝úst✐❝❛ ❞❡ ♣❛rtí❝✉❧❛s ❞❡ ♣♦❧✐❡st✐r❡♥♦ ♣❛r❛ três ❝♦♥✜❣✉r❛çõ❡s ❞✐❢❡r❡♥t❡s

  ❋✐❣✉r❛ ✶✳✶✿

  ❞♦ ❧❡✈✐t❛❞♦r✳ ❋♦♥t❡✿ ▼❛r❝♦ ❆♥❞r❛❞❡ ❡ ❝♦❧❛❜♦r❛❞♦r❡s✱ ✷✵✶✺ ✶✳✷✳✷ P✐♥ç❛ ❛❝úst✐❝❛

  ❯♠❛ ❞❛s ♣r✐♠❡✐r❛s ❛♣❧✐❝❛çõ❡s ❞❛ ❢♦rç❛ ❞❡ r❛❞✐❛çã♦ ❢♦✐ ❞❡♥♦♠✐♥❛❞❛ ♣✐♥ç❛ ❛❝úst✐❝❛✱ q✉❡ é ✉♠❛ té❝♥✐❝❛ ♣❛r❛ ❛♣r✐s✐♦♥❛♠❡♥t♦ ❞❡ ♣❛rtí❝✉❧❛s q✉❡ ✉t✐❧✐③❛ ❢❡✐①❡s ❛❝úst✐❝♦s✳ ❉✉❛s ❛❜♦r❞❛❣❡♥s ❞✐❢❡r❡♥t❡s tê♠ s✐❞♦ ✉t✐❧✐③❛❞❛s ♣❛r❛ ❛♣r✐s✐♦♥❛♠❡♥t♦ ❞❡ ♠✐❝r♦♣❛rtí❝✉❧❛s ❞❡♥♦♠✐✲ ♥❛❞❛s ❞❡ ♦♥❞❛ ❡st❛❝✐♦♥ár✐❛ ❡ ♠ét♦❞♦ ❞❡ ❢❡✐①❡ ú♥✐❝♦✳ ❖ ♣r✐♠❡✐r♦ s✐st❡♠❛ ❞❡ ♣✐♥ç❛ ❛❝úst✐❝♦ ❢♦✐ ♣r♦♣♦st♦ ♣♦r ❲✉ ❝♦♠ ✉♠ ❞✐s♣♦s✐t✐✈♦ ❝❛♣❛③ ❞❡ ❛♣r✐s✐♦♥❛r ❡st❛✈❡❧♠❡♥t❡ ♣❛rtí❝✉❧❛s ❞❡ ❧❛t❡① ❡ ♦✈♦s ❞❡ rã ❞❡ 270 µm ❞❡ ❞✐â♠❡tr♦ ❡♠ á❣✉❛ ❊ss❡ ❞✐s♣♦s✐t✐✈♦ é ❝♦♠♣♦st♦ ♣♦r ❞♦✐s ❢❡✐①❡s ❢♦❝❛❧✐③❛❞♦s ❝♦♥tr❛♣r♦♣❛❣❛♥t❡s ♣❛r❛ ❢♦r♠❛r ✉♠❛ ♦♥❞❛ ❡st❛❝✐♦♥ár✐❛ ❞❡ 3.5 MHz ❝❛♣❛③ ❞❡ ❛♣r✐s✐♦♥❛r ♠✐❝r♦♣❛rtí❝✉❧❛s✳ ❑♦③✉❦❛ ❡ ❝♦❧❛❜♦r❛❞♦r❡s ❛♣r❡s❡♥t❛r❛♠ ✉♠ ♠ét♦❞♦ ❞❡ ♣✐♥ç❛ ❛❝úst✐❝❛ ✉t✐❧✐③❛♥❞♦ ✉♠❛ ♦♥❞❛ ❡st❛❝✐♦♥ár✐❛ ❞❡ 2.1 MHz ❣❡r❛❞❛ ♣♦r ✉♠ tr❛♥s❞✉t♦r ❢♦❝❛❧✐③❛❞♦ ❡ ✉♠ r❡✢❡t♦r ❝❛♣❛③ ❞❡ ❛♣r✐s✐♦♥❛r ♠✐❝r♦♣❛rtí❝✉❧❛s ❞❡ ❛❧✉♠í♥✐♦ ❝♦♠ ❞✐â♠❡tr♦ ♠é❞✐♦ ❞❡ 16 µm✳ ❖♥❞❛s ❡st❛❝✐♦♥ár✐❛s ✐♥❝❧✐♥❛❞❛s sã♦ ♣r♦❞✉③✐❞❛s ♣♦r ✉♠❛ ❝♦♥✜❣✉r❛çã♦ ❞❡ três tr❛♥s❞✉t♦r❡s ♦♣❡r❛♥❞♦ ❛ 1.67 MHz ❢♦r❛♠ ✉s❛❞❛s ♣❛r❛ ❛♣r✐s✐♦♥❛r ❡ tr❛♥s♣♦rt❛r ❣♦t❛s ❞❡ sí❧✐❝❛ ❞❡ 100 µm ❞❡ ❞✐â♠❡tr♦

  P♦r ♦✉tr♦ ❧❛❞♦✱ ❛ ❛❜♦r❞❛❣❡♠ ❞❡ ♣✐♥ç❛s ❛❝úst✐❝❛s ❛tr❛✈és ❞♦ ♠ét♦❞♦ ❞❡ ❢❡✐①❡ ú♥✐❝♦ ✉t✐✲ ❧✐③❛ tr❛♥s❞✉t♦r❡s ❜❡♠ ❢♦❝❛❧✐③❛❞♦s ♣❛r❛ ❛♣r✐s✐♦♥❛r ♣❛rtí❝✉❧❛s ♥♦ ♣♦♥t♦ ❢♦❝❛❧ ❞♦ ❞✐s♣♦s✐t✐✈♦✳ ▲❡❡ ❡ ❝♦❧❛❜♦r❛❞♦r❡s ❞❡s❡♥✈♦❧✈❡r❛♠ ✉♠ ❞✐s♣♦s✐t✐✈♦ ❛❝úst✐❝♦ ❞❡ ❢❡✐①❡ ú♥✐❝♦ ✉t✐❧✐③❛♥❞♦ ✉♠ tr❛♥s❞✉t♦r ❞❡ 30 MHz ❝❛♣❛③ ❞❡ ❛♣r✐s✐♦♥❛r ❣♦tí❝✉❧❛s ❞❡ ❧✐♣í❞✐♦s ❞❡ 126 µm ❞❡ ❞✐â♠❡tr♦s✳ ❚r❛♥s❞✉t♦r❡s ❝♦♠ ❢r❡q✉ê♥❝✐❛s ♠❛✐s ❛❧t❛s q✉❡ ♦♣❡r❛♠ ❛ 200 MHz ❢♦r❛♠ ♣r♦❥❡t❛❞♦s ♣❛r❛ ✐♠♦❜✐❧✐③❛r ❡ tr❛♥s❧❛❞❛r ✉♠❛ ❝é❧✉❧❛ ❞❡ ❧❡✉❝❡♠✐❛ ❞❡ 10 µm ❈❤♦❡ ❡ ❝♦❧❛❜♦r❛❞♦r❡s ❞❡s❝r❡✈❡r❛♠ ✉♠ ❛♣r✐s✐♦♥❛♠❡♥t♦ ❛❝úst✐❝♦ ❞❡ ♠✐❝r♦❡s❢❡r❛s ✈❛r✐❛♥❞♦ ❞❡ ❞✐â♠❡tr♦ ❡♥tr❡ 70 ❛ 90 µm ✉t✐❧✐③❛♥❞♦ ✉♠ tr❛♥s❞✉t♦r ❡q✉✐♣❛❞♦ ❝♦♠ ✉♠❛ ❧❡♥t❡ ❞❡ ❋r❡s♥❡❧

  ♠✉❧t✐❢♦❝❛❧ ❣❡r❛♥❞♦ ✉♠ ❢❡✐①❡ ❞❡ ❇❡ss❡❧ ❞❡ ♦r❞❡♠ ③❡r♦ ❞❡ 17.9 MHz✳ ❯♠ tr❛♥s❞✉t♦r ❢♦✐ ❢❛❜r✐❝❛❞♦ ♣♦r ❍s✉ ❡ ❝♦❧❛❜♦r❛❞♦r❡s ❝♦♠ ❢r❡q✉ê♥❝✐❛ ❝❡♥tr❛❧ ❞❡ 57.5 MHz ❛♣r✐s✐♦♥❛♥❞♦ ♠✐❝r♦♣❛rtí❝✉❧❛s ❞❡ ♣♦❧✐❡st✐r❡♥♦ ❞❡ 15 µm ❞❡ ❞✐â♠❡tr♦✳ ❖♥❞❛s ❞❡ ❇❡ss❡❧ ❡st❛❝✐♦♥ár✐❛ ❢♦r❛♠ ❣❡r❛❞♦s ♣♦r ✉♠ ❝♦♥❥✉♥t♦ ❞❡ ✻✹ ❡❧❡♠❡♥t♦s ♦♣❡r❛♥❞♦ ❛ 2.35 MHz ♦r❣❛♥✐③❛❞♦s ❡♠ ❝ír❝✉❧♦s ♣❛r❛ ♠❛♥✐♣✉❧❛r ♠✐❝r♦♣❛rtí❝✉❧❛s ❞❡ ♣♦❧✐❡st✐r❡♥♦ ❞❡ 90 µm ❞❡ ❞✐â♠❡tr♦ ❡♠ ✷❉

  ✶✳✷✳✸ ❆❝✉st♦✢✉í❞✐❝❛

  ❆ ♠✐❝r♦✢✉í❞✐❝❛ ♣♦❞❡ s❡r ❞❡✜♥✐❞❛ ❝♦♠♦ ❛ ❝✐ê♥❝✐❛ q✉❡ ♦♣❡r❛ ❡♠ ♣❡q✉❡♥♦s ✢✉✐❞♦s s✉s✲ ♣❡♥s♦s ❡♠ ❝❛♥❛✐s ❞❡ ❝♦rt❡s tr❛♥s✈❡rs❛✐s ❝♦♠ ❞✐♠❡♥sõ❡s ♠✐❝r♦♠étr✐❝❛s✳ ❆❝✉st♦✢✉í❞✐❝❛✱ ♦✉ s❡❥❛✱ ✉♠❛ té❝♥✐❝❛ q✉❡ ✉t✐❧✐③❛ ❝❛♠♣♦s ❛❝úst✐❝♦s ❡♠ ♠✐❝r♦♣❛rtí❝✉❧❛s s✉s♣❡♥s❛s ❡♠ ♠✐❝r♦✲ ✢✉í❞✐❝❛✱ t❡♠ ❛tr❛í❞♦ ✉♠❛ ❛t❡♥çã♦ ❡s♣❡❝✐❛❧ ♣♦rq✉❡ ♣❡r♠✐t❡ ✉♠❛ s❡♣❛r❛çã♦ ❞❡ ♣❛rtí❝✉❧❛s ❜❛s❡❛❞❛s ♥❛s ♣r♦♣r✐❡❞❛❞❡s ♠❡❝â♥✐❝❛s✱ t❛✐s ❝♦♠♦✱ t❛♠❛♥❤♦✱ ❢♦r♠❛✱ ❞❡♥s✐❞❛❞❡ ❡ ❝♦♠♣r❡s✲ s✐❜✐❧✐❞❛❞❡✳ ❉✐s♣♦s✐t✐✈♦s ♥♦ ❞♦♠í♥✐♦ ❞❡ ❛❝✉st♦✢✉í❞✐❝❛ sã♦✱ ❡♠ ❣❡r❛❧✱ ❜❛s❡❛❞♦s ❡♠ ♦♥❞❛s ❡st❛❝✐♦♥ár✐❛s ❖♥❞❛s ❛❝úst✐❝❛s ❡st❛❝✐♦♥ár✐❛s t♦r♥♦✉✲s❡ ✉♠ ♠ét♦❞♦ ♣❛❞rã♦ ♣❛r❛ ♠❛♥✐♣✉❧❛çã♦ ❞❡ ❝é❧✉❧❛s ❡ ♣❛rtí❝✉❧❛s ❡♠ ❝❤✐♣s ♠✐❝r♦✢✉í❞✐❝♦s

  ❖♥❞❛s ❛❝úst✐❝❛s ❡st❛❝✐♦♥ár✐❛s ❞❡ s✉♣❡r❢í❝✐❡ tê♠ s✐❞♦ ✉t✐❧✐③❛❞❛s ♣❛r❛ ❛♣r✐s✐♦♥❛r ♣❛rtí❝✉✲ ❧❛s ❝♦♠ ❞✐â♠❡tr♦ ♠❡♥♦r q✉❡ 16 µm s✉s♣❡♥s❛s ❡♠ ❝❛♥❛✐s ♠✐❝r♦✢✉í❞✐❝♦s ❈♦♠ ♦ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❞❡ s✐st❡♠❛s ♠✐❝r♦✢✉í❞✐❝♦s ❡ ♦ ❝♦♥❝❡✐t♦ ❞❛ t❡❝♥♦❧♦❣✐❛ ❞❡ ❧❛❜♦r❛tór✐♦ ❞❡♥tr♦ ❞❡ ✉♠ ❝❤✐♣ ✭❧❛❜✲♦♥✲❛❝❤✐♣✮✱ ♦ ♠♦✈✐♠❡♥t♦ ❞❡ ♣❛rtí❝✉❧❛s r❡s✉❧t❛♥t❡ ❞❛ ❢♦rç❛ ❞❡ r❛❞✐✲ ❛çã♦ ❛❝úst✐❝❛ tê♠ ♣r♦♠♦✈✐❞♦ ✉♠❛ r❡✈♦❧✉çã♦ ❡♠ ❛♣❧✐❝❛çõ❡s ❜✐♦♠é❞✐❝❛s ◆❡ss❛s ❛♣❧✐❝❛çõ❡s é ❝♦♠✉♠ q✉❡ ❛s ♣❛rtí❝✉❧❛s tr❛t❛❞❛s ♣♦ss✉❛♠ ❞✐♠❡♥sõ❡s ♠✉✐t♦ ♠❡♥♦r❡s ❞♦ q✉❡ ♦ ❝♦♠♣r✐♠❡♥t♦ ❞❡ ♦♥❞❛ ❛❝úst✐❝❛✳ ■st♦ ❝♦rr❡s♣♦♥❞❡ ❛♦ ❝❤❛♠❛❞♦ r❡❣✐♠❡ ❞❡ ❡s♣❛❧❤❛♠❡♥t♦ ❘❛②❧❡✐❣❤✳ ❊st❡ ❧✐♠✐t❡ é ❢❛❝✐❧♠❡♥t❡ ❡♥❝♦♥tr❛❞♦ ❡♠ ❞✐s♣♦s✐t✐✈♦s ❞❡ ❛❝✉st♦✢✉í❞✐❝❛ ♦♣❡r❛♥❞♦ ❡♠ 2 MHz ❡ ♠❛♥✐♣✉❧❛çã♦ ❞❡ ♣❛rtí❝✉❧❛s tã♦ ♣❡q✉❡♥❛s q✉❛♥t♦ 1 µm ❡♠ ✉♠ ♠❡✐♦ ❝♦♠♦ ❛ á❣✉❛✳

  ✶✳✸ ▼♦t✐✈❛çã♦

  ❈é❧✉❧❛s ❜✐♦❧ó❣✐❝❛s ♦✉ ♣♦❧í♠❡r♦s s❡ ❝♦♠♣♦rt❛♠ ❝♦♠♦ ♣❛rtí❝✉❧❛s só❧✐❞❛s ✈✐s❝♦❡❧ást✐✲ ❝❛s P♦rt❛♥t♦✱ ✉♠❛ ✐♥✈❡st✐❣❛çã♦ ♠❛✐s ❛♠♣❧❛ s♦❜r❡ ❝♦♠♦ ❛ ✈✐s❝♦❡❧❛st✐❝✐❞❛❞❡ ❞❛ ♣❛rtí✲ ❝✉❧❛ ❛❢❡t❛ ❛ ❢♦rç❛ ❡ ♦ t♦rq✉❡ ❞❡ r❛❞✐❛çã♦ ❛❝úst✐❝♦ ♥❛ ❛♣r♦①✐♠❛çã♦ ❞❡ ❘❛②❧❡✐❣❤ é ❞❡s❡❥❛❞❛✳ ■ss♦ ♥♦s ❞❡✉ ❛ ♠♦t✐✈❛çã♦ ♣❛r❛ ❛♥❛❧✐s❛r t❡♦r✐❝❛♠❡♥t❡ ❡ss❡s ❢❡♥ô♠❡♥♦s ❛t✉❛♥t❡s ❡♠ ♣❡✲ q✉❡♥❛s ♣❛rtí❝✉❧❛s ✈✐s❝♦❡❧ást✐❝❛s ❝♦♥s✐❞❡r❛♥❞♦ ♦♥❞❛s ♣❧❛♥❛s ♣r♦❣r❡ss✐✈❛s ❡ ❡st❛❝✐♦♥ár✐❛s ❡ ❢❡✐①❡s ❞❡ ❇❡ss❡❧ ❞❡ ♦r❞❡♠ ③❡r♦ ❡ ♣r✐♠❡✐r❛ ♦r❞❡♠✳ ❆ss✉♠✐♠♦s q✉❡ ♦s ❡❢❡✐t♦s t❡r♠♦✲✈✐s❝♦s♦s sã♦ ♥❡❣❧✐❣❡♥❝✐❛❞♦s ♥♦s ♣r♦❜❧❡♠❛s ❞❡ ❢♦rç❛ ❡ t♦rq✉❡ ❞❡ r❛❞✐❛çã♦✳ ❆ss✐♠✱ ♦ ✢✉✐❞♦ é ❝♦♥s✐❞❡✲ r❛❞♦ ❝♦♠♦ ♥ã♦✲✈✐s❝♦s♦✳ ❊st❛ s✉♣♦s✐çã♦ r❡q✉❡r q✉❡ ♦ r❛✐♦ ❞❛ ♣❛rtí❝✉❧❛ ❡ ♦ ❝♦♠♣r✐♠❡♥t♦ ❞❡

  = p2D /ω

  t t

  ♦♥❞❛ ✐♥❝✐❞❡♥t❡ ❞❡✈❡♠ s❡r ♠✉✐t♦ ♠❛✐♦r❡s ❞♦ q✉❡ ❛s ❝❛♠❛❞❛s ❧✐♠✐t❡ tér♠✐❝❛ δ = p2ν /ω

  ✈ t

  ❡ ✈✐s❝♦s❛ δ ♥❛ s✉♣❡r❢í❝✐❡ ❞❛ ♣❛rtí❝✉❧❛ ♦♥❞❡ D é ❛ ❞✐❢✉s✐✈✐❞❛❞❡ tér♠✐❝❛✱ ω

  é ❛ ❢r❡q✉ê♥❝✐❛ ❛♥❣✉❧❛r ❡ ν é ❛ ✈✐s❝♦s✐❞❛❞❡ ❝✐♥ét✐❝❛✳ ❆❧é♠ ❞✐ss♦✱ ♥♦ss♦ ❡st✉❞♦ ♣♦❞❡ ♣❡r♠✐t✐r ♥♦✈♦s ❛✈❛♥ç♦s ♥❛s té❝♥✐❝❛s ❞❡ ❛❝✉st♦❢♦rét✐❝❛✳

  ✶✳✹ ❆rt✐❣♦s ❡ ♣✉❜❧✐❝❛çõ❡s

  ❏✳ P✳ ▲❡ã♦✲◆❡t♦ ❛♥❞ ●✳ ❚✳ ❙✐❧✈❛✱ ❆❝♦✉st✐❝ r❛❞✐❛t✐♦♥ ❢♦r❝❡ ❛♥❞ t♦rq✉❡ ❡①❡rt❡❞ ♦♥ ❛ s♠❛❧❧ ✈✐s❝♦❡❧❛st✐❝ ♣❛rt✐❝❧❡ ✐♥ ❛♥ ✐❞❡❛❧ ✢✉✐❞✱ ❛r❳✐✈✿♣❤②s✐❝s✳❝❧❛ss✲♣❤✴✶✺✵✽✳✵✶✾✵✽✱ ✷✺ ♣á❣✐♥❛s✱ ✺ ✜✲ ❣✉r❛s✳ ❙✉❜♠❡t✐❞♦ ❛♦ ❏♦✉r♥❛❧ ♦❢ ❙♦✉♥❞ ❛♥❞ ❱✐❜r❛t✐♦♥✱ ✷✵✶✺✳ ❚r❛❜❛❧❤♦s ❡♠ ❈♦♥❣r❡ss♦s

  ✶✳ ❏✳ P✳ ▲✳ ◆❡t♦✱ ●✳ ❚✳ ❙✐❧✈❛✱ ❆❝♦✉st✐❝ r❛❞✐❛t✐♦♥ ❢♦r❝❡ ❛♥❞ t♦rq✉❡ ♦♥ ❛ ❢r❛❝t✐♦♥❛❧ ✈✐s❝♦❡❧❛st✐❝ ♣❛rt✐❝❧❡✳ ✷✵✶✺✳ ✭❳❳❳❱■■■ ❊◆❋▼❈ ❇r❛③✐❧✐❛♥ P❤②s✐❝❛❧ ❙♦❝✐❡t② ▼❡❡t✐♥❣✮✳

  ✷✳ ❏✳ P✳ ▲✳ ◆❡t♦✱ ●✳ ❚✳ ❙✐❧✈❛✱ ■♥❞✉❝❡❞ r❛❞✐❛t✐♦♥ ❢♦r❝❡ ❛♥❞ t♦rq✉❡ ♦♥ ❛ ✈✐s❝♦❡❧❛st✐❝ ♣❛rt✐❝❧❡ ✐♥ ❛♥ ✐❞❡❛❧ ✢✉✐❞✳ ✷✵✶✺✳ ✭✶✼✵t❤ ▼❡❡t✐♥❣ ♦❢ t❤❡ ❆❝♦✉st✐❝❛❧ ❙♦❝✐❡t② ♦❢ ❆♠❡r✐❝❛✮✳

  ✶✳✺ ❆♣r❡s❡♥t❛çã♦ ❣❡r❛❧ ❞♦ tr❛❜❛❧❤♦

  ❊st❡ tr❛❜❛❧❤♦ é ✈♦❧t❛❞♦ ❛♦ ❡st✉❞♦ t❡ór✐❝♦ ❞❛ ❢♦rç❛ ❡ ❞♦ t♦rq✉❡ ❞❡ r❛❞✐❛çã♦ ❛❝úst✐❝♦ s♦✲ ❜r❡ ✉♠❛ ♣❛rtí❝✉❧❛ ✈✐s❝♦❡❧ást✐❝❛ s✉s♣❡♥s❛ ❡♠ ✉♠ ✢✉✐❞♦ ✐❞❡❛❧✳ ❊♠ ♣❛rt✐❝✉❧❛r✱ ♦ ♠♦❞❡❧♦ ❞❡ ❑❡❧✈✐♥✲❱♦✐❣t ❢r❛❝✐♦♥ár✐♦ é ✉t✐❧✐③❛❞♦ ♣❛r❛ ❞❡s❝r❡✈❡r ❛ ♣r♦♣❛❣❛çã♦ ❞❡ ♦♥❞❛s ❡♠ ♣❛rtí❝✉❧❛s ✈✐s❝♦❡❧ást✐❝❛s ❊st❡ ♠♦❞❡❧♦ ❢♦✐ ❡s❝♦❧❤✐❞♦ ♣♦rq✉❡ ♣♦❞❡ ❞❡s❝r❡✈❡r ❛ ❞❡♣❡♥❞ê♥❝✐❛ ❞❛ ❧❡✐ ❞❡ ♣♦tê♥❝✐❛ ❞❛ ❢r❡q✉ê♥❝✐❛ ♦❜s❡r✈❛❞♦s ❡①♣❡r✐♠❡♥t❛❧♠❡♥t❡ ❡♠ ❞✐✈❡rs♦s ♠❛t❡r✐❛✐s ✈✐s❝♦❡❧ást✐✲ ❝♦s ❞❡r✐✈❛♠♦s ❛s ❡q✉❛çõ❡s ❞❡ ♦♥❞❛ ❞❡ ❝✐s❛❧❤❛♠❡♥t♦ ❡ ❧♦♥❣✐t✉❞✐♥❛❧ ❞❡❝♦rr❡♥t❡s ❞❛ r❡❧❛çã♦ ❞❡ ❡str❡ss❡✲❞❡❢♦r♠❛çã♦ ❝♦♠ ❜❛s❡ ♥♦ ♠♦❞❡❧♦ ❞❡ ❑❡❧✈✐♥✲❱♦✐❣t ❢r❛✲ ❝✐♦♥ár✐♦ ♦✉ s❡❥❛✱ ❛ ❧❡✐ ❞❡ ❍♦♦❦❡ ❣❡♥❡r❛❧✐③❛❞❛✱ ❛ss✉♠✐♥❞♦ q✉❡ ❛s ❝♦♠♣♦♥❡♥t❡s ❞♦ ❡str❡ss❡ sã♦ ❧✐♥❡❛r♠❡♥t❡ r❡❧❛❝✐♦♥❛❞❛s ❝♦♠ ❛s ❝♦♠♣♦♥❡♥t❡s ❞❛ ❞❡❢♦r♠❛çã♦✱ ❝♦♠ t❡r♠♦s ❞❡ ❞❡r✐✈❛❞❛ ❢r❛❝✐♦♥ár✐❛ ♥♦ t❡♠♣♦✳

  ◆♦ ❝❛♣ít✉❧♦ ❛♣r❡s❡♥t❛♠♦s ❛ t❡♦r✐❛ ❞❡ ❡s♣❛❧❤❛♠❡♥t♦ ❞❛ ♦♥❞❛ ❛❝úst✐❝❛ ♣♦r ✉♠❛ ♣❛rtí✲ ❝✉❧❛ ❡s❢ér✐❝❛✳ ➱ ✐♠♣♦rt❛♥t❡ r❡ss❛❧t❛r q✉❡ ❛ t❡♦r✐❛ ❞❡ ❡s♣❛❧❤❛♠❡♥t♦ ❡stá ❞✐r❡t❛♠❡♥t❡ ❧✐❣❛❞❛ ❛♦s ❢❡♥ô♠❡♥♦s ❞❡ ❢♦rç❛ ❡ t♦rq✉❡ ❞❡ r❛❞✐❛çã♦ ❛❝úst✐❝♦ ❞❡r✐✈❛♠♦s ♦s ❝♦❡✜❝✐❡♥t❡s ❞❡ ❡s♣❛❧❤❛♠❡♥t♦ ❛❞✐♠❡♥s✐♦♥❛❧ ♣❛r❛ ❛ ❛♣r♦①✐♠❛çã♦ ❞❡ ♠♦♥♦♣♦❧♦ ❡ ❞✐♣♦❧♦ q✉❡ s❡rá ✉t✐❧✐③❛❞♦ ♥❛s ❢ór♠✉❧❛s ❞❡ ❡①♣❛♥sã♦ ❞❡ ♦♥❞❛ ♣❛r❝✐❛❧ ♣❛r❛ ❢♦rç❛ ❡ t♦rq✉❡ ❞❡ r❛❞✐❛çã♦ ❛❝úst✐❝♦✳

  ◆♦ ❝❛♣ít✉❧♦ ♥ós ❞❡s❝r❡✈❡♠♦s ❝♦♠♦ ♦❜t❡r ❛s ❢ór♠✉❧❛s ❞❡ ❢♦rç❛ ❡ t♦rq✉❡ ❞❡ r❛❞✐❛çã♦ ❛❝úst✐❝♦ ♣❛r❛ ❛ ❛♣r♦①✐♠❛çã♦ ♠♦♥♦♣♦❧♦✲❞✐♣♦❧♦✳ P❛r❛ s✐♠♣❧✐✜❝❛r ❛✐♥❞❛ ♠❛✐s ❡ss❛s ❢ór♠✉❧❛s✱ ❡①♣❛♥❞✐♠♦s ♥❛ s❡çã♦ ❛ r❡❧❛çã♦ ❞❡ ❞✐s♣❡rsã♦ ✈✐s❝♦❡❧ást✐❝❛ ❡♠ sér✐❡ ❞❡ ❚❛②❧♦r ♥❛s ❛♣r♦①✐♠❛çõ❡s ❞❡ ❜❛✐①❛ ❡ ❛❧t❛ ❢r❡q✉ê♥❝✐❛✳

  ❆♣❧✐❝❛♠♦s ♦ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❞❛ t❡♦r✐❛ ♥♦ ❝❛♣ít✉❧♦ ♣❛r❛ ❡st✉❞❛r ♦ ❝♦♠♣♦rt❛♠❡♥t♦ ❞❛s ♣❛rtí❝✉❧❛s ❞❡ ♣♦❧✐❡t✐❧❡♥♦ ❞❡ ❜❛✐①❛ ❡ ❛❧t❛ ❞❡♥s✐❞❛❞❡ ✐♥t❡r❛❣✐♥❞♦ ❝♦♠ ♦♥❞❛s ❛❝úst✐❝❛s✳ ◆❛ s❡çã♦ ❛♣r❡s❡♥t❛♠♦s ❡①♣r❡ssõ❡s ❛♥❛❧ít✐❝❛s ♣❛r❛ ❢♦rç❛ ❡ t♦rq✉❡ ❞❡ r❛❞✐❛çã♦ ❛❝úst✐❝♦ ❡①❡r❝✐❞♦ s♦❜r❡ ✉♠❛ ♣❛rtí❝✉❧❛ ✈✐s❝♦❡❧ást✐❝❛ ♣♦r ♦♥❞❛s ♣❧❛♥❛s ♣r♦♣❛❣❛♥t❡s ❡ ❡st❛❝✐♦♥ár✐❛s✱ ❢❡✐①❡s ❞❡ ❇❡ss❡❧ ❞❡ ♦r❞❡♠ ③❡r♦ ❡ ♣r✐♠❡✐r❛ ♦r❞❡♠✳ ❘❡s✉❧t❛❞♦s t❡ór✐❝♦s ❛♥t❡r✐♦r❡s ♣❛r❛ ❛s ♣❛rtí❝✉❧❛s ❢❡✐t❛s ❞❡ ♦✉tr♦s ♠❛t❡r✐❛✐s ✭♣❛rtí❝✉❧❛ ✢✉✐❞❛ ❡ só❧✐❞❛✮ sã♦ r❡❝✉♣❡r❛❞♦s✳ ◆ós ❡st❛❜❡❧❡❝❡♠♦s ❛ ❝♦♥❞✐çã♦ ❞❡ ♦❜t❡r ❢♦rç❛ ❡ t♦rq✉❡ ❞❡ r❛❞✐❛çã♦ ❛❝úst✐❝♦ ♥❡❣❛t✐✈♦ ❡♠ t❡r♠♦s ❞♦s t❡♠♣♦s ❞❡ r❡❧❛①❛çã♦ ❧♦♥❣✐t✉❞✐♥❛❧ ❡ ❞❡ ❝✐s❛❧❤❛♠❡♥t♦ ❞❛ ♣❛rtí❝✉❧❛✳

  P♦r ✜♠✱ ❛♣r❡s❡♥t❛♠♦s ♥♦ ❝❛♣ít✉❧♦ ❛s ♣r✐♥❝✐♣❛✐s ❝♦♥❝❧✉sõ❡s ❞❡st❡ tr❛❜❛❧❤♦ ❛ ♣❛rt✐r ❞♦s r❡s✉❧t❛❞♦s ♦❜t✐❞♦s✱ ❜❡♠ ❝♦♠♦ ❛s ♥♦ss❛s ♣❡rs♣❡❝t✐✈❛s ♣❛r❛ ♦ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❞❡ ♥♦✈♦s tr❛❜❛❧❤♦s✳

  2 ❊q✉❛çõ❡s ❞♦ ♠♦❞❡❧♦

  ❊♠ ❣❡r❛❧✱ ❛ ♠❛tér✐❛ ❛♣r❡s❡♥t❛✲s❡ ♥♦ ❡st❛❞♦ só❧✐❞♦ ♦✉ ✢✉✐❞♦✳ ❖ t❡r♠♦ ✏✢✉✐❞♦✑ ♣♦❞❡ s❡r ✉s❛❞♦ ♣❛r❛ ✉♠ ❣ás ♦✉ ✉♠ ❧íq✉✐❞♦✳ ❆ ❝♦♠♣r❡❡♥sã♦ ❞♦s ♣r✐♥❝í♣✐♦s ❜ás✐❝♦s ❞❛ ❢ís✐❝❛ ❞♦s ✢✉✐❞♦s ❡ só❧✐❞♦s sã♦ ❞❡ s✉♠❛ ✐♠♣♦rtâ♥❝✐❛ ♥❛ ❛♥á❧✐s❡ ❞❡ s✐st❡♠❛s ❡♠ q✉❡ ♦s ♠❡s♠♦s sã♦ ♠❡✐♦s ❛t✉❛♥t❡s✳ ❚❛♥t♦ ✢✉✐❞♦s ❝♦♠♦ só❧✐❞♦s ❝❧áss✐❝♦s ♦❜❡❞❡❝❡♠ ❛ ❝♦♥s❡r✈❛çã♦ ❞❡ ♠❛ss❛✱ ❞♦ ♠♦♠❡♥t♦ ❡ ❞❡ ❡♥❡r❣✐❛✳ ◆❡st❡ ❝❛♣ít✉❧♦✱ ❛s ❡q✉❛çõ❡s ❞❡ ❞✐♥â♠✐❝❛ ❞♦ ✢✉✐❞♦ sã♦ ❞❡r✐✈❛❞❛s ❞❛s r❡❧❛çõ❡s ❢✉♥❞❛♠❡♥t❛✐s q✉❡ ❞❡s❝r❡✈❡♠ ❛ t❛①❛ ❞❡ ✈❛r✐❛çã♦ ❞❛s ❞❡♥s✐❞❛❞❡s ❞❡ ✢✉①♦ ❞❡ ♠❛ss❛✱ ♠♦♠❡♥t♦ ❡ ❡♥❡r❣✐❛✳ ❆♣r❡s❡♥t❛r❡♠♦s t❛♠❜é♠ ❝♦♠♦ ✉♠❛ ♦♥❞❛ s❡ ♣r♦♣❛❣❛ ❡♠ ✉♠ ✢✉✐❞♦ ✐❞❡❛❧✱ ❝♦♥s✐❞❡r❛♥❞♦ q✉❡ ❛ ❡q✉❛çã♦ ❞❡ ♦♥❞❛ ❝♦rr❡s♣♦♥❞❡♥t❡ é ❝♦♥❤❡❝✐❞❛ ❝♦♠♦ ❡q✉❛çã♦ ❞❡ ❍❡❧♠❤♦❧t③✳ ❆ s♦❧✉çã♦ ❞❡st❛ ❡q✉❛çã♦ ❡♠ ❝♦♦r❞❡♥❛❞❛s ❡s❢ér✐❝❛s é ❛♣r❡s❡♥t❛❞❛ ❡♠ t❡r♠♦s ❞♦ ♠ét♦❞♦ ❞❡ ♦♥❞❛ ♣❛r❝✐❛❧ ♣❛r❛ t♦❞❛s ❛s ♦♥❞❛s ❡♥✈♦❧✈✐❞❛s ♥♦ ♣r♦❜❧❡♠❛✳ ❆❧é♠ ❞✐ss♦✱ ❛♣r❡s❡♥t❛r❡♠♦s t❛♠❜é♠ ❛s ❡q✉❛çõ❡s ❞❡ ❝♦♥s❡r✈❛çã♦ ♣❛r❛ só❧✐❞♦s✳ ❊ ♣♦r ✜♠✱ ❡①♣❛♥❞✐♠♦s ♥♦ss❛ ❛♥á❧✐s❡ ❝♦♥s✐❞❡r❛♥❞♦ ❛ ♣r♦♣❛❣❛çã♦ ❞❡ ♦♥❞❛ ❡♠ ✉♠ só❧✐❞♦ ✈✐s❝♦❡❧ást✐❝♦ ❛tr❛✈és ❞♦ ♠♦❞❡❧♦ ❑❡❧✈✐♥✲❱♦✐❣t✳ ❊ss❡s ❡st✉❞♦s ❞❛ ♣r♦♣❛❣❛çã♦ ❞❡ ♦♥❞❛s ❡♠ ✉♠ ❞❡t❡r♠✐♥❛❞♦ ♠❡✐♦ ✭✢✉✐❞♦ ♦✉ só❧✐❞♦✮ s❡rã♦ ✐♠♣♦rt❛♥t❡s ♣❛r❛ ❝♦♠♣r❡❡♥❞❡r ❛ t❡♦r✐❛ ❞❡ ❡s♣❛❧❤❛♠❡♥t♦ ❛❝úst✐❝♦ q✉❡ s❡rá ❛♣r❡s❡♥t❛❞❛ ♥♦ ♣ró①✐♠♦ ❝❛♣ít✉❧♦✳

  ✷✳✶ ❊q✉❛çõ❡s ❞❡ ❝♦♥s❡r✈❛çã♦ ♣❛r❛ ✉♠ ✢✉✐❞♦ ✐❞❡❛❧

  ❆♣❡s❛r ❞❡ ❣❛s❡s ❡ ❧íq✉✐❞♦s s❡r❡♠ ❝♦♠♣♦st♦s ❞❡ át♦♠♦s ❡ ♠♦❧é❝✉❧❛s✱ é út✐❧ ❝♦♥s✐❞❡rá✲ ❧♦s ❝♦♠♦ ♠❡✐♦s ❝♦♥tí♥✉♦s✳ ❈♦♥s✐❞❡r❡ ✉♠ ✢✉✐❞♦ ✐❞❡❛❧✱ ❝✉❥❛s ♣r♦♣r✐❡❞❛❞❡s ♠❛❝r♦s❝ó♣✐❝❛s sã♦ ❝♦♥st❛♥t❡s✳ ❯♠❛ ♠❛♥❡✐r❛ ❞❡ ❞❡s❝r❡✈❡r ♦ ♠♦✈✐♠❡♥t♦ ❞❡ ✉♠ ✢✉✐❞♦ é ♣♦r ♠❡✐♦ ❞❡ ✉♠ ❡❧❡♠❡♥t♦ ❞❡ ✢✉✐❞♦ q✉❡ ❡stá ✜①♦ ♥♦ ❡s♣❛ç♦ ❝♦♠ r❡❧❛çã♦ ❛ ✉♠ s✐st❡♠❛ ❞❡ ❝♦♦r❞❡♥❛❞❛s ❡♠ q✉❛❧q✉❡r ✐♥st❛♥t❡ ❞❡ t❡♠♣♦ t✳ ■r❡♠♦s ❝♦♥s✐❞❡r❛r ✉♠ ❡❧❡♠❡♥t♦ ❞❡ ✢✉✐❞♦ ❛r❜✐trár✐♦ ❞❡ ✈♦❧✉♠❡ V ✜①♦ ♥♦ ❡s♣❛ç♦ ❡ ❧♦❝❛❧✐③❛❞♦ ❡♠ ✉♠❛ ♣♦s✐çã♦ r ❝♦♥t❡♥❞♦ ✉♠ ♥ú♠❡r♦ s✉✜❝✐❡♥✲ t❡♠❡♥t❡ ❣r❛♥❞❡ ❞❡ át♦♠♦s ♦✉ ♠♦❧é❝✉❧❛s q✉❡ ❡♥❝♦♥tr❛♠✲s❡ ❡♠ ❡q✉✐❧í❜r✐♦ t❡r♠♦❞✐♥â♠✐❝♦ ❧♦❝❛❧✳ ■st♦ ♥♦s ♣❡r♠✐t❡ ❞❡✜♥✐r ❣r❛♥❞❡③❛s ❢ís✐❝❛s ♣❛r❛ ♦ ❡❧❡♠❡♥t♦ ❞❡ ✢✉✐❞♦✱ t❛✐s ❝♦♠♦ ❞❡♥s✐❞❛❞❡✱ ✈❡❧♦❝✐❞❛❞❡✱ ♣r❡ssã♦✱ ❡♥❡r❣✐❛ ✐♥t❡r♥❛ ❡ ❡♥tr♦♣✐❛✳

  ◆❡st❡ ❝❛♣ít✉❧♦✱ ❢❛r❡♠♦s ♦ ✉s♦ r❡❝♦rr❡♥t❡ ❞♦ t❡♦r❡♠❛ ❞❛ ❞✐✈❡r❣ê♥❝✐❛ ❞❡ ●❛✉ss ♣❛r❛ ✉♠ ❞❡t❡r♠✐♥❛❞♦ ❝❛♠♣♦ ✈❡t♦r✐❛❧ B(r) q✉❡ é ❞❛❞♦ ♣♦r Z Z 3 2 r = B r , V S ∇ · B(r)❞ (r) · n❞ ✭✷✳✶✮

  2 3

  r r ♦♥❞❡ n é ♦ ✈❡t♦r ✉♥✐tár✐♦ ♥♦r♠❛❧ q✉❡ ❛♣♦♥t❛ ♣❛r❛ ❢♦r❛ ❞❛ s✉♣❡r❢í❝✐❡ S ✱ ❞ ❡ ❞ ✱ sã♦ ♦s ❡❧❡♠❡♥t♦s ❞❡ ✐♥t❡❣r❛çã♦ ❞❡ ár❡❛ ❡ ❞❡ ✈♦❧✉♠❡✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✳

  ✷✳✶✳✶ ❊q✉❛çã♦ ❞❡ ❝♦♥s❡r✈❛çã♦ ❞❛ ♠❛ss❛

  ❈♦♥s✐❞❡r❡ ✉♠ ✈♦❧✉♠❡ ✜①♦ V ❞❡ ✉♠ ❡❧❡♠❡♥t♦ ❞❡ ✢✉✐❞♦ ❧♦❝❛❧✐③❛❞♦ ♥❛ ♣♦s✐çã♦ r✳ ❆ss✉✲ ♠✐♠♦s q✉❡ ♦ ❡❧❡♠❡♥t♦ ❞❡ ✢✉✐❞♦ ❝♦♥té♠ N át♦♠♦s ♦✉ ♠♦❧é❝✉❧❛s ❡♠ ✉♠ ✐♥st❛♥t❡ ❞❡ t❡♠♣♦ t

  ✳ ❊♥tã♦✱ ♣♦❞❡♠♦s ❞❡✜♥✐r ❛ ❞❡♥s✐❞❛❞❡ ρ(r, t) ❞❛ s❡❣✉✐♥t❡ ❢♦r♠❛✿ X N (t)

  1 ρ(r, t) = m , i

  ✭✷✳✷✮

  V i=1 ♦♥❞❡ m i é ❛ ♠❛ss❛ ❞❡ ✉♠ át♦♠♦ ♦✉ ♠♦❧é❝✉❧❛ ❞❡♥tr♦ ❞♦ ❡❧❡♠❡♥t♦ ❞❡ ✈♦❧✉♠❡ V ✳ ❆ ✈❡❧♦❝✐❞❛❞❡ ❞♦ ❝❡♥tr♦ ❞❡ ♠❛ss❛ ❞♦ ❡❧❡♠❡♥t♦ ❞❡ ✈♦❧✉♠❡ V ❞♦ ✢✉✐❞♦ ♣❛r❛ q✉❛❧q✉❡r ✐♥st❛♥t❡ ❞❡ t❡♠♣♦ t é ❞❡✜♥✐❞❛ ❞❛ s❡❣✉✐♥t❡ ❢♦r♠❛✿ X N (t)

  1 v (r, t) = m v . i i ✭✷✳✸✮

  ρ(r, t)V i=1 i ♦♥❞❡ v é ❝♦♥❤❡❝✐❞♦ ❝♦♠♦ ❛ ✈❡❧♦❝✐❞❛❞❡ ❞♦ ❡❧❡♠❡♥t♦ ❞❡ ✢✉✐❞♦✱ ❡ v é ❛ ✈❡❧♦❝✐❞❛❞❡ ❞♦ i✲és✐♠♦ át♦♠♦ ♦✉ ♠♦❧é❝✉❧❛ ❞❡♥tr♦ ❞♦ ❡❧❡♠❡♥t♦ ❞❡ ✈♦❧✉♠❡ V ✳

  ❉❡♥tr♦ ❞❡ ✉♠ ✢✉✐❞♦✱ ❛ ♠❛ss❛ ♥ã♦ ♣♦❞❡ s❡r ❝r✐❛❞❛ ♦✉ ❞❡str✉í❞❛✱ ❡♥tã♦ ❛ ♠❛ss❛ t♦t❛❧ ♣♦❞❡ ✈❛r✐❛r ❛♣❡♥❛s ❞❡✈✐❞♦ ❛♦ ✢✉①♦ ❞❡ ♠❛ss❛✳ ❖ ✢✉①♦ ❞❡ ♠❛ss❛ ♣♦r ✉♥✐❞❛❞❡ ❞❡ ár❡❛ 2 − 1

  ♣♦r ✉♥✐❞❛❞❡ ❞❡ t❡♠♣♦ ✭❝♦♠ ✉♥✐❞❛❞❡ ❦❣ ♠ s ✮ é ❞❡✜♥✐❞♦ ❝♦♠♦ ❞❡♥s✐❞❛❞❡ ρ(r, t) ✈❡③❡s ❛ ✈❡❧♦❝✐❞❛❞❡ ❞❡ ❝♦♥✈❡❝çã♦ v(r, t)✳ ❆ss✐♠✱ ❛ ✈❛r✐❛çã♦ ❞❛ ♠❛ss❛ t♦t❛❧ ❞♦ ✈♦❧✉♠❡ V ♣♦r ✉♥✐❞❛❞❡ ❞❡ t❡♠♣♦ ❞❡♣❡♥❞❡ ❞♦ ✢✉①♦ ❞❡ ♠❛ss❛ ρv ❛tr❛✈és ❞❛ s✉♣❡r❢í❝✐❡ S ❡ ❞♦ ✈♦❧✉♠❡ V ✱ ✜❣✉r❛ P♦rt❛♥t♦✱ ❛ ❝♦♥s❡r✈❛çã♦ ❞❡ ♠❛ss❛ ❡①✐❣❡ q✉❡ Z Z 3 2 t + ∂ ρ r r = 0, V S ❞ ρv · n❞ ✭✷✳✹✮ t = ∂/∂t t

  ♦♥❞❡ ∂ ✳ ❯♠❛ ✈❡③ q✉❡ ♦ ✈♦❧✉♠❡ V é ✜①♦✱ ❛ ❞❡r✐✈❛❞❛ t❡♠♣♦r❛❧ ∂ ♣♦❞❡ s❡r ❝❛❧❝✉❧❛❞❛ ❞✐r❡t❛♠❡♥t❡ ♥❛ ❢✉♥çã♦ ❞❡♥s✐❞❛❞❡✳ P♦r ♦✉tr♦ ❧❛❞♦✱ ♣♦❞❡♠♦s ✉t✐❧✐③❛r ♦ t❡♦r❡♠❛ ❞❛ ❞✐✈❡r❣ê♥❝✐❛ ❞❡ ●❛✉ss✱ ❊q✳ ♣❛r❛ tr❛♥s❢♦r♠❛r ❡ss❛ ✐♥t❡❣r❛❧ ❡♠ ✉♠❛ ✐♥t❡❣r❛❧ ❞❡ ✈♦❧✉♠❡✱ Z Z 2 3 r r

  = .

  ✭✷✳✺✮ S ρv · n ❞ ∇ · (ρv)❞ V ❙✉❜st✐t✉✐♥❞♦ ❛ ❡q✉❛çã♦ ♦❜t❡♠♦s

  ∂ t ✭✷✳✻✮

  ρ + ∇ · ρv = 0, q✉❡ ❝♦rr❡s♣♦♥❞❡ ❛ ❡q✉❛çã♦ ❞❡ ❝♦♥s❡r✈❛çã♦ ❞❛ ♠❛ss❛✳

  ✷✳✶✳✷ ❊q✉❛çã♦ ❞❡ ❝♦♥s❡r✈❛çã♦ ❞♦ ♠♦♠❡♥t♦

  ❖ ♠♦♠❡♥t♦ ❧✐♥❡❛r ❞❡♥tr♦ ❞❡ ✉♠ ✈♦❧✉♠❡ ✜①♦ V ♣❛r❛ q✉❛❧q✉❡r ✐♥st❛♥t❡ ❞❡ t❡♠♣♦ t é ρv✳ ❉❛ ♠❡s♠❛ ❢♦r♠❛ q✉❡ ❛ ✈❛r✐❛çã♦ ❞❛ ♠❛ss❛✱ ♦ ♠♦♠❡♥t♦ ❧✐♥❡❛r ♣♦❞❡ ✈❛r✐❛r ❞❡✈✐❞♦ à ❝♦♥✈❡❝çã♦✳ ◆❡ss❡ ❝❛s♦✱ ♦ ✢✉①♦ ❞❡ ♠♦♠❡♥t♦ é ❞❡✜♥✐❞♦ ❝♦♠♦ ρvv✳ ❆ ✈❛r✐❛çã♦ ❞❛ t❛①❛ ❞♦ ♠♦♠❡♥t♦ ❧✐♥❡❛r ❡♠ ✉♠ ✈♦❧✉♠❡ V ❞❡✈❡rá s❡r ✐❣✉❛❧ ❛ s♦♠❛ ❞❛s ❝♦♥tr✐❜✉✐çõ❡s ❞♦ ✢✉①♦ ❞❡ ♠♦♠❡♥t♦✱ ❞❛ ❢♦rç❛ ❞❡✈✐❞♦ ❛ ♣r❡ssã♦ ♥♦r♠❛❧ ❛ s✉♣❡r❢í❝✐❡ S ❡ ❞❛s ❢♦rç❛s ✈♦❧✉♠étr✐❝❛s✱ q✉❡ sã♦ ❢♦rç❛s ❡①t❡r♥❛s q✉❡ ❛t✉❛♠ ❛♦ ❧♦♥❣♦ ❞❡ t♦❞♦ ♦ ✈♦❧✉♠❡ ❞♦ ✢✉✐❞♦✱ ♣♦❞❡♠♦s ❝✐t❛r ❝♦♠♦ ❡①❡♠♣❧♦ ❞❡ss❛s ❢♦rç❛s ✈♦❧✉♠étr✐❝❛s ❛ ❢♦rç❛ ❣r❛✈✐t❛❝✐♦♥❛❧ ❡ ❛ ❢♦rç❛ ❡❧étr✐❝❛✳ ❆ss✐♠✱ ❛ ✈❛r✐❛çã♦ ❞❛ t❛①❛ ❞♦ ♠♦♠❡♥t♦ ❧✐♥❡❛r ♥♦ ✈♦❧✉♠❡ V ♣❛r❛ ✉♠ ✢✉✐❞♦ ✐❞❡❛❧ é ❞❛❞♦ ♣♦r Z Z Z Z 3 2 2 3

  ❱

  ∂ ρv p n t + + + r r r f r = 0, V S S ❞ ❞ ρvv · n❞ ❞ ✭✷✳✼✮ V ♦♥❞❡ ❛ q✉❛♥t✐❞❛❞❡ vv é ✉♠ ❞✐❛❞❡ ✭t❡♥s♦r ❞❡ s❡❣✉♥❞❛ ♦r❞❡♠✮✱ p(r, t) é ❛ ♣r❡ssã♦ ❡♠

  ❱

  r ❡ f é ❛ ❞❡♥s✐❞❛❞❡ ❞❡ ❢♦rç❛ ✈♦❧✉♠étr✐❝❛✳ P♦❞❡♠♦s ✉t✐❧✐③❛r ♦ t❡♦r❡♠❛ ❞❡ ●❛✉ss ♣❛r❛ t❡♥s♦r❡s ❡♠ ✐♥t❡❣r❛✐s

  ✈♦❧✉♠étr✐❝❛s✳ ❈♦♥s❡q✉❡♥t❡♠❡♥t❡✱ ❛ ❡q✉❛çã♦ ❞❡ ❝♦♥s❡r✈❛çã♦ ❞♦ ♠♦♠❡♥t♦ ♥❛ ❢♦r♠❛ ✐♥t❡❣r❛❧ é ❞❛❞❛ ♣♦r Z Z Z Z 3 3 3 3

  ❱ ∂ ρv r r r f r = 0. t + + + V ❞ ∇p n❞ V V ∇ · (ρvv) ❞ ❞ ✭✷✳✽✮ V

  ❈♦♠♦ ♦ ✈♦❧✉♠❡ V é ❛r❜✐trár✐♦✱ ❡♥❝♦♥tr❛♠♦s ❛ ❡q✉❛çã♦ ❞✐❢❡r❡♥❝✐❛❧ ♣❛r❝✐❛❧ ❞❛ ❝♦♥s❡r✈❛çã♦ ❞♦ ♠♦♠❡♥t♦

  ❱ ∂ = 0. t

  (ρv) + ∇p + ∇ · (ρvv) + f ✭✷✳✾✮

  ❱

  ◆♦ ❝❛s♦ ♣❛rt✐❝✉❧❛r✱ ❡♠ q✉❡ ❛s ❢♦rç❛s ✈♦❧✉♠étr✐❝❛s f sã♦ ♠✉✐t♦ ♠❡♥♦r❡s q✉❡ ❛s ♦✉tr❛s

  ❱

  = 0 ❢♦rç❛s ❞❛ ❊q✳ ♣♦❞❡♠♦s ❝♦♥s✐❞❡r❛r q✉❡ f ✳ ❆❧é♠ ❞✐ss♦✱ ❞❡✜♥✐♠♦s ♦ t❡♥s♦r ❞❡ t❡♥sõ❡s ♣❛r❛ ✉♠ ✢✉✐❞♦ ♥ã♦✲✈✐s❝♦s♦ ❝♦♠♦ S

  = pI + ρvv, ✭✷✳✶✵✮

  ♦♥❞❡ ♦ ❣r❛❞✐❡♥t❡ ❞❛ ♣r❡ssã♦ ♣♦❞❡ s❡r ❡①♣r❡ss♦ ❡♠ t❡r♠♦s ❞❡ ✉♠ t❡♥s♦r ❞❡ s❡❣✉♥❞❛ ♦r❞❡♠ ❝♦♠♦ ∇p = ∇ · (pI)✱ ❝♦♠ I s❡♥❞♦ ♦ t❡♥s♦r ✉♥✐tár✐♦✳ P♦rt❛♥t♦✱ ♣♦❞❡♠♦s r❡❡s❝r❡✈❡r ❛ ❊q✳ ❞❛ s❡❣✉✐♥t❡ ❢♦r♠❛✿

  ∂ t (ρv) + ∇ · S = 0. ✭✷✳✶✶✮

  ❊ss❛ é ❛ ❡q✉❛çã♦ ❞❡ ❝♦♥s❡r✈❛çã♦ ❞♦ ♠♦♠❡♥t♦ ❧✐♥❡❛r q✉❡ ✐r❡♠♦s ✉t✐❧✐③❛r ♣❛r❛ ❞❡r✐✈❛çã♦ ❞❛ ❡q✉❛çã♦ ❞❡ ♦♥❞❛ ❧✐♥❡❛r ♥❛ ♣ró①✐♠❛ s❡çã♦✳ P♦ré♠✱ é út✐❧ r❡❡s❝r❡✈❡r ❡ss❛ ❡q✉❛çã♦ ❡♠ ✉♠ ♥♦✈♦ ❢♦r♠❛t♦ q✉❡ s❡rá ✉t✐❧✐③❛❞❛ ♥❛ ❞❡♠♦str❛çã♦ ❞❛ ❢ór♠✉❧❛ ❞❡ ❢♦rç❛ ♥♦ ❝❛♣ít✉❧♦ ✹✳ ❈♦♥s✐❞❡r❡ ❛ r❡❧❛çã♦ ♠❛t❡♠át✐❝❛✱

  ✭✷✳✶✷✮ ∇ · ρvv = (∇ · ρv)v + ρv · ∇v.

  ❋♦rç❛s ❛t✉❛♥❞♦ s♦❜r❡ ❛s ♣❛rtí❝✉❧❛s ❞❡ ♠❛ss❛ m i ❡ ✈❡❧♦❝✐❞❛❞❡ v i q✉❡ ♦❝✉♣❛♠ ✉♠

  ❋✐❣✉r❛ ✷✳✶✿

  (t) ❡❧❡♠❡♥t♦ ❞❡ ✢✉✐❞♦ ❞❡ ✈♦❧✉♠❡ V ✳ mi mi mi mi mi mi mi mi mi mi mi mi mi mi o) mi mi mi mi v mi mi

i mi

p

mi mi mi mi mi mi mi ssã p mi (pre mi mi mi mi

mi

mi mi mi mi mi mi mi p mi ρv V (volume) mi

p

mi mi mi ρvv (fluxo do momento) Ev (fluxo de energia) (fluxo de massa) mi Fluido mi mi mi Elemento de fluido mi mi

mi

mi ❋♦♥t❡✿ ❆✉t♦r✱ ✷✵✶✺✳

  ❯s❛♥❞♦ ❛ ❡q✉❛çã♦ ❞❡ ❝♦♥s❡r✈❛çã♦ ❞❛ ♠❛ss❛✱ ❊q✳ ❞❡♥tr♦ ❞❡ss❛ r❡❧❛çã♦ ❛❝✐♠❛✱ t❡♠♦s t ∇ · ρvv = −(∂ ρ)v + ρv · ∇v. ✭✷✳✶✸✮

  ❙✉❜st✐t✉✐♥❞♦ ❡ss❛ r❡❧❛çã♦ ♥❛ ❡q✉❛çã♦ ❞❡ ❝♦♥s❡r✈❛çã♦ ❞♦ ♠♦♠❡♥t♦✱ ❊q✳ ♦❜t❡♠♦s ρD v t t t = −∇p, ✭✷✳✶✹✮

  ♦♥❞❡ ♦ ♦♣❡r❛❞♦r D ≡ (∂ + v · ∇) é ❛ ❞❡r✐✈❛❞❛ t❡♠♣♦r❛❧ ♠❛t❡r✐❛❧✳ ❊ss❛ é ✉♠❛ ♦✉tr❛ ❢♦r♠❛ ❞❡ ❡s❝r❡✈❡r ❛ ❡q✉❛çã♦ ❞♦ ♠♦✈✐♠❡♥t♦✳

  ✷✳✶✳✸ ❈♦♥s❡r✈❛çã♦ ❞❛ ❡♥❡r❣✐❛

  ❆ ú❧t✐♠❛ ❡q✉❛çã♦ ❞❡ ❝♦♥s❡r✈❛çã♦ ♣❛r❛ ✉♠ ✢✉✐❞♦ ✐❞❡❛❧ ❛ s❡r ❛♣r❡s❡♥t❛❞❛ é ❣♦✈❡r♥❛❞❛ ♣♦r ❧❡✐s ❞❛ t❡r♠♦❞✐♥â♠✐❝❛ ❡ ♣♦r ❝♦♥✈❡❝çã♦✳ ❆♦ tr❛❜❛❧❤❛r ❝♦♠ t❡r♠♦❞✐♥â♠✐❝❛ ❞♦s ✢✉✐❞♦s é ❝♦♥✈❡♥✐❡♥t❡ ✉t✐❧✐③❛r q✉❛♥t✐❞❛❞❡s t❡r♠♦❞✐♥â♠✐❝❛s ♣♦r ✉♥✐❞❛❞❡ ❞❡ ♠❛ss❛✳ ❉❡st❛ ❢♦r♠❛✱ ✉s❛r❡♠♦s ❛ ❧❡tr❛ u ♣❛r❛ ❞❡♥♦♠✐♥❛r ❛ ❡♥❡r❣✐❛ ✐♥t❡r♥❛ ♣♦r ✉♥✐❞❛❞❡ ❞❡ ♠❛ss❛ ❞❡ ✉♠ ✈♦❧✉♠❡ ✜①♦ V ✱ ❡ ❛ ❧❡tr❛ s ♣❛r❛ ❞❡♥♦♠✐♥❛r ❛ ❡♥tr♦♣✐❛ ♣♦r ✉♥✐❞❛❞❡ ❞❡ ♠❛ss❛✳ ❆ ♣r✐♠❡✐r❛ ❧❡✐ ❞❛ t❡r♠♦❞✐♥â♠✐❝❛ r❡❧❛❝✐♦♥❛ ♦ ❛✉♠❡♥t♦ ❞❛ ❡♥❡r❣✐❛ ✐♥t❡r♥❛ ❝♦♠ ♦ tr❛❜❛❧❤♦ r❡❛❧✐③❛❞♦ s♦❜r❡ ♦ ❡❧❡♠❡♥t♦ ❞❡ ✈♦❧✉♠❡✱ ❡ ♦ ❝❛❧♦r T ds ❞❛ s❡❣✉✐♥t❡ ❢♦r♠❛✿ 1 p

  ) = T ❞u = T ❞s − p❞(ρ ❞s + ❞ρ, ✭✷✳✶✺✮ 2

  ρ ♦♥❞❡ T é ❛ t❡♠♣❡r❛t✉r❛✳ ❆ss✉♠✐r❡♠♦s q✉❡ ♦ tr❛♥s♣♦rt❡ ❞❡ ❝❛❧♦r ♥♦ ✢✉✐❞♦ é ❧❡♥t♦ ❝♦♠✲ ♣❛r❛❞♦ ❝♦♠ ❛ ♣r♦♣❛❣❛çã♦ ❛❝úst✐❝❛✳ ❉❡st❛ ❢♦r♠❛✱ ❡♠ ✉♠ ♣r♦❝❡ss♦ ❛❞✐❛❜át✐❝♦ ✭❞s = 0✮✱

  − 2

  ❛ ♣r✐♠❡✐r❛ ❧❡✐ ❞❛ t❡r♠♦❞✐♥â♠✐❝❛ t♦r♥❛✲s❡ ❞u = pρ ❞ρ✳ ❆ss✐♠✱ t❛♥t♦ ❛ ❡♥❡r❣✐❛ ✐♥t❡r♥❛ ❝♦♠♦ ❛ ♣r❡ssã♦ ♣♦❞❡ s❡r ❡①♣r❡ss❛ ❝♦♠♦ ♣❡❧❛ ❡q✉❛çã♦ ❞❡ ❡st❛❞♦ ❡♠ ❢✉♥çã♦ ❞❛ ❞❡♥s✐❞❛❞❡✱ p = p(ρ),

  ✭✷✳✶✻✮ u = u(ρ). ✭✷✳✶✼✮

  ❆ ❞❡♥s✐❞❛❞❡ ❞❡ ❡♥❡r❣✐❛ ❡♠ ✉♠ ✈♦❧✉♠❡ V é ❞❛❞❛ ♣❡❧❛ s♦♠❛ ❞❛ ❞❡♥s✐❞❛❞❡ ❞❡ ❡♥❡r❣✐❛ ❝✐♥ét✐❝❛ ❡ ❛ ❞❡♥s✐❞❛❞❡ ❞❡ ❡♥❡r❣✐❛ ✐♥t❡r♥❛✱ 2

  ρv (r, t) E(r, t) = + ρu(r, t).

  ✭✷✳✶✽✮

  2 ❉❡ ♠❛♥❡✐r❛ ❛♥á❧♦❣❛ ❛ ✈❛r✐❛çã♦ ❞♦ ♠♦♠❡♥t♦ ❧✐♥❡❛r✱ ❊q✳ ❛ ❞❡♥s✐❞❛❞❡ ❞❡ ❡♥❡r❣✐❛

  ❡♠ ✉♠ ✢✉✐❞♦ ❞❡ ✈♦❧✉♠❡ V ♣♦❞❡ ✈❛r✐❛r ♣♦r ❝♦♥✈❡❝çã♦ ❛tr❛✈és ❞❛ s✉♣❡r❢í❝✐❡ S ❡ ❞❡✈✐❞♦ ❛ ✈❛r✐❛çã♦ ❞♦ tr❛❜❛❧❤♦ r❡❛❧✐③❛❞♦ ♣♦r ✉♠❛ ❢♦rç❛ ❣❡r❛❞❛ ♣❡❧❛ ♣r❡ssã♦ ♥❛ s✉♣❡r❢í❝✐❡ S ✳ ❆ 2

  /2 + ρu)v ❞❡♥s✐❞❛❞❡ ❞❡ ❡♥❡r❣✐❛ ❞❡ ❝♦♥✈❡❝çã♦ é ❞❛❞❛ ❡♠ t❡r♠♦s ❞♦ ✢✉①♦ ❞❡ ❡♥❡r❣✐❛ (ρv ✱ ❡♥q✉❛♥t♦ ❛ ✈❛r✐❛çã♦ ❞♦ tr❛❜❛❧❤♦ ❞❡✈✐❞♦ ❛ ♣r❡ssã♦ é pv ✭✈❡❥❛ ✜❣✉r❛ P♦rt❛♥t♦✱ ❛ ✈❛r✐❛çã♦ ❞❛ t❛①❛ ❞❛ ❞❡♥s✐❞❛❞❡ ❞❡ ❡♥❡r❣✐❛ ❡♠ ✉♠ ✈♦❧✉♠❡ V ♣❛r❛ ✉♠ ✢✉✐❞♦ ✐❞❡❛❧ é ❞❛❞❛ ♣♦r Z Z Z 2 2

  ρv ρv 3 2 2 t + ∂ + ρu r + ρu v r r = 0.

  ❞ pv · n❞

  • · n❞ ✭✷✳✶✾✮
  • V S S

      2

      2 ❋❛③❡♥❞♦ ♥♦✈❛♠❡♥t❡ ♦ ✉s♦ ❞♦ t❡♦r❡♠❛ ❞❛ ❞✐✈❡r❣ê♥❝✐❛ ❞❡ ●❛✉ss✱ ❡ ❝♦♠♦ ♦ ✈♦❧✉♠❡ V é ❛r❜✐trár✐♦✱ ❡♥❝♦♥tr❛♠♦s ❛ ❡q✉❛çã♦ ❞❛ ❝♦♥s❡r✈❛çã♦ ❞❡ ❡♥❡r❣✐❛✱ 2 2

      ρv ρv v ∂ t + ρu + ρu + p = 0,

      ✭✷✳✷✵✮

    • ∇ ·

      2

      2 ❖❜s❡r✈❡ q✉❡ ❛ q✉❛♥t✐❞❛❞❡ ❡♥tr❡ ❝♦❧❝❤❡t❡s é ✐❞❡♥t✐✜❝❛❞♦ ❝♦♠♦ ✈❡t♦r ✢✉①♦ ❞❡ ❡♥❡r❣✐❛✳

      ✷✳✷ Pr♦♣❛❣❛çã♦ ❞❛ ♦♥❞❛ ♥♦ ✢✉✐❞♦ ✐❞❡❛❧

      ❆ ❡q✉❛çã♦ ❞❛ ♦♥❞❛ q✉❡ r❡♣r❡s❡♥t❛ ❛ ❞✐♥â♠✐❝❛ ❞❡ ♣r♦♣❛❣❛çã♦ ❞❛ ♦♥❞❛ ❡♠ ✉♠ ✢✉✐❞♦ ✐❞❡❛❧ é ❞❡r✐✈❛❞❛ ❞❛ ❝♦♠❜✐♥❛çã♦ ❞❛ ❡q✉❛çã♦ ❞❡ ❡st❛❞♦ ❞❛ ♣r❡ssã♦ ❡♠ t❡r♠♦s ❞❛ ❞❡♥s✐❞❛❞❡ ❡ ❞❛ ❡q✉❛çã♦ ❞❡ ❝♦♥s❡r✈❛çã♦ ❞♦ ♠♦♠❡♥t♦ ❈♦♥s✐❞❡r❡ ✉♠ ✢✉✐❞♦ ❤♦♠♦❣ê♥❡♦✱ ❝✉❥❛ ❞❡♥s✐❞❛❞❡ ❛♠❜✐❡♥t❡ ρ ❡ ❛ ♣r❡ssã♦ ❛♠❜✐❡♥t❡ p sã♦ ❝♦♥st❛♥t❡s✳ ❆❧é♠ ❞✐ss♦✱ ❡①♣❛♥❞✐♠♦s ❛ ♣r❡ssã♦✱ q✉❡ ❞❡♣❡♥❞❡ ❞❛ ❞❡♥s✐❞❛❞❡✱ ❡♠ sér✐❡

      ❞❡ ❚❛②❧♦r ❡♠ t♦r♥♦ ❞❡ ρ ✳ ◆❡st❡ ❝❛s♦✱ ❡s❝r❡✈❡♠♦s ❛ sér✐❡ ❞❛ s❡❣✉✐♥t❡ ♠❛♥❡✐r❛✿ X (∂ p) n ρ n ρ = ) ,

      ✭✷✳✷✶✮ p − p (ρ − ρ ρ = ∂/∂ρ n=1 n! ♦♥❞❡ ∂ ❞❡s✐❣♥❛ ✈❛r✐❛çõ❡s ❞❡ ♣r❡ssã♦ ❝♦♠ ❛ ❞❡♥s✐❞❛❞❡ ❞♦ ✢✉✐❞♦✱ ♦✉ s❡❥❛✱ ♦♥❞❛s ❞❡ ρ ρ p) ♣r❡ssã♦✱ q✉❡ ♣♦r ❞❡✜♥✐çã♦ é ♦ s♦♠✳ ❆ss✐♠✱ ❛ ❞❡r✐✈❛❞❛ ♣❛r❝✐❛❧ ❞❡ ♣r✐♠❡✐r❛ ♦r❞❡♠ (∂ t❡♠ ❞✐♠❡♥sã♦ ❞❡ ✈❡❧♦❝✐❞❛❞❡ ❛♦ q✉❛❞r❛❞♦✳ P♦rt❛♥t♦✱ ❛ ✈❡❧♦❝✐❞❛❞❡ ❞♦ s♦♠ é ❞❡✜♥✐❞❛ ❞❛ s❡❣✉✐♥t❡ ♠❛♥❡✐r❛✱ 2 c = (∂ p) . ρ ρ ✭✷✳✷✷✮

      ❖ ❝♦♥❥✉♥t♦ ❞❡ ❡q✉❛çõ❡s ❞✐❢❡r❡♥❝✐❛✐s ♣❛r❝✐❛✐s ♥ã♦✲❧✐♥❡❛r❡s q✉❡ ❞❡s❝r❡✈❡♠ ❛ ❞✐♥â♠✐❝❛ ❞❛ ♦♥❞❛ ❡♠ ✉♠ ✢✉✐❞♦ ✐❞❡❛❧ sã♦ ♥♦t♦r✐❛♠❡♥t❡ ❞❡ ❞✐❢í❝✐❧ r❡s♦❧✉çã♦ ❛♥❛❧ít✐❝❛✳ ◆♦ ❡♥t❛♥t♦✱ s♦❧✉çõ❡s ❛♣r♦①✐♠❛❞❛s sã♦ út❡✐s ❡ ♣♦❞❡♠ s❡r ❡♥❝♦♥tr❛❞❛s ✉t✐❧✐③❛♥❞♦ ♦ ♠ét♦❞♦ ❞❡ ❛♣r♦①✐✲ ♠❛çõ❡s s✉❝❡ss✐✈❛s✳ ◆❡st❡ ♠ét♦❞♦✱ ❛ ♣r❡ssã♦✱ ❞❡♥s✐❞❛❞❡ ❡ ❛ ✈❡❧♦❝✐❞❛❞❡ sã♦ ❡①♣❛♥❞✐❞❛s ❡♠ ♣♦tê♥❝✐❛s ❞❡ ✉♠ ♣❡q✉❡♥♦ ♣❛râ♠❡tr♦ M✳ ❊ss❡ ♣❛râ♠❡tr♦ ❞❡ ❡①♣❛♥sã♦ M é ♦ ♥ú♠❡r♦ ❞❡ ▼❛❝❤✱ q✉❡ é ❛ ♠❡❞✐❞❛ ❛❞✐♠❡♥s✐♦♥❛❧ ❞❛ ✈❡❧♦❝✐❞❛❞❡ ❞❡✜♥✐❞❛ ❝♦♠♦ v

      M = , ✭✷✳✷✸✮ c

      ♦♥❞❡ v é ♠❛❣♥✐t✉❞❡ ♠á①✐♠❛ ❞❛ ✈❡❧♦❝✐❞❛❞❡ ❞♦ ❡❧❡♠❡♥t♦ ❞♦ ✢✉✐❞♦ ❡ c é ❛ ✈❡❧♦❝✐❞❛❞❡ ❞♦ s♦♠ ♥♦ ✢✉✐❞♦✳ P♦rt❛♥t♦✱ ♥♦ss❛ ❛♥á❧✐s❡ ❡stá ❝♦♥❞✐❝✐♦♥❛❞❛ ❛ ✉♠ r❡❣✐♠❡ ❞❡ ✈❡❧♦❝✐❞❛❞❡ ♠✉✐t♦ ❜❛✐①❛✱ M ≪ 1✳ ❉❡st❛ ❢♦r♠❛✱ ❛ ♣r❡ssã♦✱ ❛ ❞❡♥s✐❞❛❞❡ ❡ ❛ ✈❡❧♦❝✐❞❛❞❡ sã♦ ❡s❝r✐t❛s ❞❛ s❡❣✉✐♥t❡ ❢♦r♠❛✿ X n (n)

      = M p , ✭✷✳✷✹✮ p − p X n=1 n (n)

      = M ρ , ρ − ρ X n=1 n (n) ✭✷✳✷✺✮ v v

      = M , n=1 ✭✷✳✷✻✮ ♦♥❞❡ ♦ ✐♥❞✐❝❡ n ✐❞❡♥t✐✜❝❛ ❛ ♦r❞❡♠ ❞❛ ❡①♣❛♥sã♦✳ P❛r❛ ❛ ❞❡r✐✈❛çã♦ ❞♦ ♣r♦❜❧❡♠❛ ❞❡ ❡s✲ ♣❛❧❤❛♠❡♥t♦✱ ❛ ❛♣r♦①✐♠❛çã♦ ❧✐♥❡❛r ❞♦s ❝❛♠♣♦s ❛❝úst✐❝♦s é s✉✜❝✐❡♥t❡✳ ◆♦ ❡♥t❛♥t♦✱ ♣❛r❛ ❛ ❞❡r✐✈❛çã♦ ❞❛ ❢♦rç❛ ❡ t♦rq✉❡ ❞❡ r❛❞✐❛çã♦ s❡rá ♥❡❝❡ssár✐♦ ✉t✐❧✐③❛r ❝❛♠♣♦s ❛❝úst✐❝♦s ❞❡ s❡❣✉♥❞❛ ♦r❞❡♠✱ ❡ss❛ ❛♥á❧✐s❡ s❡rá ❛♣r❡s❡♥t❛ ♥♦ ❝❛♣ít✉❧♦ ✹✳

      ✷✳✷✳✶ ❊q✉❛çã♦ ❧✐♥❡❛r ❞❛ ♦♥❞❛

      ❆s ♣❡q✉❡♥❛s ♣❡rt✉r❜❛çõ❡s ❛❝úst✐❝❛s✱ ♥♦ q✉❛❧ ♦ ♣❛râ♠❡tr♦ ❞❡ ♣❡rt✉r❜❛çã♦ é M ≪ 1✱ ♣♦❞❡♠ s❡r ❞❡s❝r✐t❛s ♣❡❧❛ ❛♣r♦①✐♠❛çã♦ ❧✐♥❡❛r ❞♦s ❝❛♠♣♦s ❞❡ ♣r❡ssã♦✱ ❞❡♥s✐❞❛❞❡ ❡ ✈❡❧♦❝✐✲ ❞❛❞❡ ❞♦ ❡❧❡♠❡♥t♦ ❞❡ ✢✉✐❞♦ (1)

      = M p , ✭✷✳✷✼✮ p − p (1)

      = M ρ , ✭✷✳✷✽✮

      ρ − ρ (1) v = M v .

      ✭✷✳✷✾✮ ❙✉❜st✐t✉✐♥❞♦ ❛s ❊qs✳ ♣❛r❛ n = 1✱ ❡♥❝♦♥tr❛♠♦s q✉❡ ❛ ❛♣r♦①✐♠❛çã♦ ❧✐♥❡❛r ❞❛ ❡①♣❛♥sã♦ ❞❛ ♣r❡ssã♦ ❡♠ ❢✉♥çã♦ ❞❛ ❞❡♥s✐❞❛❞❡ é ❞❛❞❛ ♣♦r (1) 2 (1) p = c ρ .

      ✭✷✳✸✵✮

      ❆❣♦r❛ ❝♦♥s✐❞❡r❛♥❞♦ ❛s ❡q✉❛çõ❡s ❞❡ ❝♦♥s❡r✈❛çã♦ ❞❛ ♠❛ss❛ ❡ ❝♦♥s❡r✈❛çã♦ ❞❡ ♠♦♠❡♥t♦ ♦❜t❡♠♦s

      1 (1) (1) ∂ p + ρ = 0, t 2 ∇ · v ✭✷✳✸✶✮ c (1) (1) ρ ∂ v = 0. t

    • ∇p ✭✷✳✸✷✮ ❊ss❛s ❡q✉❛çõ❡s sã♦ ❛s ❡q✉❛çõ❡s ❧✐♥❡❛r❡s ❞❡ ❞✐♥â♠✐❝❛ ❞♦s ✢✉✐❞♦s✳

      ❚♦♠❛♥❞♦ ♦ r♦t❛❝✐♦♥❛❧ ❞❛ ❊q✳ ❡♥❝♦♥tr❛♠♦s q✉❡ ❛ ✈❡❧♦❝✐❞❛❞❡ ❞♦ ❡❧❡♠❡♥t♦ ❞♦ ✢✉✐❞♦ é ✐rr♦t❛❝✐♦♥❛❧✱ (1)

      = 0, ∇ × v

      ✭✷✳✸✸✮ ♦♥❞❡ ❡ss❡ r❡s✉❧t❛❞♦ é ♦❜t✐❞♦ ❞❡✈✐❞♦ ❛♦ ❢❛t♦ ❞❡ q✉❡ ♦ r♦t❛❝✐♦♥❛❧ ❞♦ ❣r❛❞✐❡♥t❡ é s❡♠♣r❡ ♥✉❧♦✳ ❆ss✐♠✱ ♣♦❞❡♠♦s ✐♥tr♦❞✉③✐r ✉♠❛ ❢✉♥çã♦ ♣♦t❡♥❝✐❛❧ ❡s❝❛❧❛r φ(r, t) ♣❛r❛ ❛ ✈❡❧♦❝✐❞❛❞❡ ❞♦ ❡❧❡♠❡♥t♦ ❞♦ ✢✉✐❞♦ é ❞❛❞♦ ❞❛ s❡❣✉✐♥t❡ ❢♦r♠❛✿ (1) v

      = ∇φ. ✭✷✳✸✹✮ ❆ ♣r❡ssã♦ é ❞❛❞❛ ❡♠ t❡r♠♦s ❞♦ ♣♦t❡♥❝✐❛❧ ❞❡ ✈❡❧♦❝✐❞❛❞❡ ❛ ♣❛rt✐r ❞❛ ❊q✳ (1) p ∂ φ. t

      = −ρ ✭✷✳✸✺✮ ❙✉❜st✐t✉✐♥❞♦ ❛ ✈❡❧♦❝✐❞❛❞❡ ❞♦ ❡❧❡♠❡♥t♦ ❞❡ ✢✉✐❞♦ ❡ ❛ ♣r❡ssã♦ ❡♠ t❡r♠♦s ❞♦ ♣♦t❡♥❝✐❛❧ ❞❡ ✈❡❧♦❝✐❞❛❞❡ ♥❛ ❊q✳ ❡♥❝♦♥tr❛♠♦s ❛ ❡q✉❛çã♦ ❧✐♥❡❛r ❞❛ ♦♥❞❛ ♣❛r❛ ♦ ♣♦t❡♥❝✐❛❧ ❞❡ ✈❡❧♦❝✐❞❛❞❡✱ 2

      1 2 ∂ φ(r, t) = 0. t ✭✷✳✸✻✮

      ∇ − 2 c ➱ ❢á❝✐❧ ♦❜s❡r✈❛r q✉❡ t❛♥t♦ ❛ ♣r❡ssã♦ ❝♦♠♦ ❛ ✈❡❧♦❝✐❞❛❞❡ ❞♦ ❡❧❡♠❡♥t♦ ❞♦ ✢✉✐❞♦ s❛t✐s❢❛③❡♠ ❛ ❡q✉❛çã♦ ❧✐♥❡❛r ❞❛ ♦♥❞❛✳

      ✷✳✷✳✷ ❊q✉❛çã♦ ❞❡ ❍❡❧♠❤♦❧t③

      ❖s ❢❡♥ô♠❡♥♦s ❞❡ ❢♦rç❛ ❡ t♦rq✉❡ ❞❡ r❛❞✐❛çã♦ sã♦ r❡❧❛❝✐♦♥❛❞♦s ❝♦♠ ♦s ❝❛♠♣♦s ❛❝úst✐❝♦s ✭♣r❡ssã♦✱ ❞❡♥s✐❞❛❞❡✱✈❡❧♦❝✐❞❛❞❡✮ q✉❡ t❡♥❤❛♠ ✈❛r✐❛çã♦ t❡♠♣♦r❛❧ ❤❛r♠ô♥✐❝❛ ❝♦♠ ❢r❡q✉ê♥❝✐❛ ❛♥❣✉❧❛r ω = 2πf✱ ♦♥❞❡ f é ❛ ❢r❡q✉ê♥❝✐❛✳ ❆ss✐♠✱ ❛ ✈❡❧♦❝✐❞❛❞❡ ❞♦ ❡❧❡♠❡♥t♦ ❞♦ ✢✉✐❞♦ é ❞❛❞❛

      ✐ωt

      ♣♦r φ(r)❡ ✳ ❊♥tã♦ ❛ ❡q✉❛çã♦ ❧✐♥❡❛r ❞❛ ♦♥❞❛ ♣♦❞❡ s❡r ❡①♣r❡ss❛ ❝♦♠♦ ❛ ❡q✉❛çã♦ ❞❡ ❍❡❧♠❤♦❧t③✿ 2 2

    • k )φ(r) = 0,

      ✭✷✳✸✼✮ (∇

      ♦♥❞❡ k = ω/c é ♦ ♥ú♠❡r♦ ❞❡ ♦♥❞❛ ❡ φ(r) é ❛ ❛♠♣❧✐t✉❞❡ ❞❡t❡r♠✐♥❛❞❛ ♣♦r ❝♦♥❞✐çõ❡s ❞❡ ❝♦♥t♦r♥♦ ❡s♣❡❝í✜❝❛s ❞❡ ✉♠ ❞❡t❡r♠✐♥❛❞♦ ♣r♦❜❧❡♠❛ ❛❝úst✐❝♦✳

      ❖ s✐st❡♠❛ ❞❡ ❝♦♦r❞❡♥❛❞❛s ❡s❝♦❧❤✐❞♦ ♣❛r❛ ♦❜t❡r ❛ s♦❧✉çã♦ ❞❛ ❡q✉❛çã♦ ❞❡ ❍❡❧♠❤♦❧t③ ❞❡♣❡♥❞❡ ❞♦ ♣r♦❜❧❡♠❛ q✉❡ s❡rá ❡st✉❞❛❞♦✳ P❛r❛ ♦ ♣r♦❜❧❡♠❛ ❞❡ ❡s♣❛❧❤❛♠❡♥t♦ ❛❝úst✐❝♦ ❡s❝♦❧❤❡r❡♠♦s ♦ s✐st❡♠❛ ❞❡ ❝♦♦r❞❡♥❛s ❡s❢ér✐❝❛s (r, θ, ϕ)✱ ♦♥❞❡ r é ❛ ❞✐stâ♥❝✐❛ r❛❞✐❛❧✱ θ é ♦

      ❈♦♦r❞❡♥❛❞❛s ❡s❢ér✐❝❛s (r, θ, ϕ)✿ r é ❛ ❞✐stâ♥❝✐❛ r❛❞✐❛❧✱ θ é ♦ â♥❣✉❧♦ ♣♦❧❛r ❡ ϕ é ♦

      ❋✐❣✉r❛ ✷✳✷✿

      â♥❣✉❧♦ ❛③✐♠✉t❛❧✳ z r (x,y,z)

      

    θ

    y

      φ x

      ❋♦♥t❡✿ ❆✉t♦r✱ ✷✵✶✺✳

      â♥❣✉❧♦ ♣♦❧❛r ❡ ϕ é ♦ â♥❣✉❧♦ ❛③✐♠✉t❛❧✱ ❝♦♥❢♦r♠❡ ✐❧✉str❛❞♦ ♥❛ ✜❣✉r❛ ❆ r❡❣r❛ ❞❡ tr❛♥s❢♦r♠❛çã♦ ❞❡ ❝♦♦r❞❡♥❛❞❛s ❡s❢ér✐❝❛s ❡♠ ❝❛rt❡s✐❛♥❛s ♣♦❞❡ s❡r ❞❡s❝r✐t❛

      ❞❡st❛ ❢♦r♠❛✿ x = r sin θ cos ϕ, ✭✷✳✸✽✮ y = r sin θ sin ϕ, ✭✷✳✸✾✮ z = r cos θ, ✭✷✳✹✵✮

      ♦♥❞❡ θ é ♠❡❞✐❞♦ ❛ ♣❛rt✐r ❞♦ ❡✐①♦ z ♣♦s✐t✐✈♦ ❝♦♠ 0 ≤ θ ≤ π ❡ ϕ ♥♦ ♣❧❛♥♦ xy ❝♦♠ 0 ≤ ϕ ≤ 2π✳

      ✷✳✷✳✸ ❙♦❧✉çã♦ ❞❛ ❡q✉❛çã♦ ❞❡ ❍❡❧♠❤♦❧t③

      ❯t✐❧✐③❛♥❞♦ ♦ ❧❛♣❧❛❝✐❛♥♦ ❡♠ ❝♦♦r❞❡♥❛❞❛s ❡s❢ér✐❝❛s✱ ❛ ❡q✉❛çã♦ ❞❡ ❍❡❧♠❤♦❧t③ ❛s✲ s✉♠❡ ❛ s❡❣✉✐♥t❡ ❢♦r♠❛✿

      1 2

      1

      1 2 2 2 r r θ θ + ∂ r ∂ φ ∂ (sin θ∂ φ) + ∂ φ + k φ = 0. 2 2 2 ϕ ✭✷✳✹✶✮ r r sin θ r sin θ

      ❖ ♠♦❞♦ ♠❛✐s s✐♠♣❧❡s ❞❡ r❡s♦❧✈❡r ❡ss❛ ❡q✉❛çã♦ ❞✐❢❡r❡♥❝✐❛❧ ♣❛r❝✐❛❧ é s✉❜❞✐✈✐❞✐✲❧❛ ❡♠ ✉♠ ❝♦♥❥✉♥t♦ ❞❡ ❡q✉❛çõ❡s ❞✐❢❡r❡♥❝✐❛✐s ♦r❞✐♥ár✐❛s✱ ❢❛r❡♠♦s ✐ss♦ ✉s❛♥❞♦ ♦ ♠ét♦❞♦ ❞❡ s❡♣❛r❛çã♦ ❞❡ ✈❛r✐á✈❡✐s ❡♠ ❝♦♦r❞❡♥❛❞❛s ❡s❢ér✐❝❛s ❞❛ s❡❣✉✐♥t❡ ❢♦r♠❛✿

      φ(r, θ, ϕ) = R(r)Θ(θ)Φ(ϕ), ✭✷✳✹✷✮

      ♦♥❞❡ R(r)✱ Θ(θ)✱ Φ(ϕ) sã♦ ❢✉♥çõ❡s ❛r❜✐trár✐❛s r❛❞✐❛❧✱ ♣♦❧❛r ❡ ❛③✐♠✉t❛❧✳ ❙✉❜st✐t✉✐♥❞♦ ❛

      ❊q✳ ❡ ❞✐✈✐❞✐♥❞♦ ♣♦r R(r)Θ(θ)Φ(ϕ) ✱ t❡♠♦s

      1 2

      1

      1 2 2 r r θ θ + r R (sin θ Θ) + Φ + k = 0, 2 ❞ ❞ ❞ ❞ ❞ ϕ ✭✷✳✹✸✮ 2 2 2 Rr Θr sin θ Φr sin θ 2 2 2 r θ = = = / ♦♥❞❡ ❞ ❞/❞r✱ ❞ ❞/❞θ ❡ ❞ ϕ ❞ ❞ϕ ✳ ❖❜s❡r✈❡ q✉❡✱ t♦❞❛s ❛s ❞❡r✐✈❛❞❛s ❛❣♦r❛ sã♦ 2 2 sin θ

      ❞❡r✐✈❛❞❛s ♦r❞✐♥ár✐❛s✱ ♥ã♦ ♠❛✐s ♣❛r❝✐❛✐s✳ ▼✉❧t✐♣❧✐❝❛♥❞♦ ♣♦r r ✱ ♣♦❞❡♠♦s ❡♥❝♦♥tr❛r✱

      1 2 2 2 2

      1 2

      1 Φ = r sin θ r R (sin θ Θ) . r r θ θ ❞ ϕ −k − ❞ ❞ − ❞ ❞ ✭✷✳✹✹✮ 2 2

      Φ Rr Θr sin θ ➱ ✐♠♣♦rt❛♥t❡ ✈❡r✐✜❝❛r q✉❡ ❛ ❊q✳ r❡❧❛❝✐♦♥❛ ✉♠❛ ❢✉♥çã♦ ❛♣❡♥❛s ❞❡ ϕ ❛ ✉♠❛ ❢✉♥çã♦ ❞❡ r ❡ ❞❡ θ ❛♣❡♥❛s✳ ❈♦♠♦ r✱θ ❡ ϕ sã♦ ✈❛r✐á✈❡✐s ✐♥❞❡♣❡♥❞❡♥t❡s✱ ♣♦❞❡♠♦s ✐❣✉❛❧❛r ❝❛❞❛ ❧❛❞♦ ❞❛ ❊q✳ ❛ ✉♠❛ ❝♦♥st❛♥t❡✳

      1 2 2 d , ϕ Φ(ϕ) = −m ✭✷✳✹✺✮ Φ 2 2 2

      1 2

      1 2 r sin θ r R (sin θ Θ) , r r θ θ −k − ❞ ❞ − ❞ ❞ = −m ✭✷✳✹✻✮ 2 2 Rr Θr sin θ

      ♦♥❞❡ m ❞❡✈❡ s❡r ✉♠ ✐♥t❡✐r♦✱ ❝✉❥♦ ♦ ✈❛❧♦r ❞❡♣❡♥❞❡ ❞♦s ❞❡t❛❧❤❡s ❞♦ ♣r♦❜❧❡♠❛✳ ▼✉❧t✐♣❧✐❝❛♥❞♦ 2 ❛ ❊q✳ ♣♦r r ❡ r❡❛rr✉♠❛♥❞♦ ♦s t❡r♠♦s✱ ♦❜t❡♠♦s 2 2 2

      1 2 1 m r k r R

    • (sin θ Θ) + . r r θ θ

      ❞ ❞ = − ❞ ❞ ✭✷✳✹✼✮ 2 R Θ sin θ sin θ ❉❡ ♠❛♥❡✐r❛ s✐♠✐❧❛r✱ ❝❛❞❛ ❧❛❞♦ ❞❛ ❡q✉❛çã♦ ♣♦❞❡ s❡r ✐❣✉❛❧❛❞♦ ❛ ✉♠❛ ❝♦♥st❛♥t❡✱ ❛ss✐♠ ♦❜t❡♠♦s✱ 2 1 m θ θ (sin θ Θ + n(n + 1)Θ = 0

      ❞ ❞ Θ) − 2 ✭✷✳✹✽✮ sin θ sin θ 2 1 n(n + 1)R 2 k R + r R = 0,

      ❞ r ❞ r ✭✷✳✹✾✮ 22 r r ♦♥❞❡ n ❞❛ ♠❡s♠❛ ❢♦r♠❛ q✉❡ m ❞❡✈❡ s❡r ✉♠ ✐♥t❡✐r♦ q✉❡ ♣♦❞❡ s❡r ❞❡t❡r♠✐♥❛❞♦✳

      ❆ s♦❧✉çã♦ ❞❛ ❊q✳ é ❞❛❞❛ ♣♦r imϕ Φ(ϕ) = Φ , 1 1 ❡ ✭✷✳✺✵✮

      ♦♥❞❡ Φ é ✉♠❛ ❝♦♥st❛♥t❡ ❡ m é ✉♠ ✐♥t❡✐r♦ ❞❡ ♠♦❞♦ q✉❡ ❛ s♦❧✉çã♦ é ❛ ♠❡s♠❛ ♣❛r❛ ϕ ❡ ϕ + 2π

      ✳ P❛r❛ ❡♥❝♦♥tr❛r ❛ s♦❧✉çã♦ ❞❛ ❊q✳ ♣r❡❝✐s❛♠♦s ❢❛③❡r ✉♠❛ s✉❜st✐t✉✐çã♦ ❞❡ ✈❛r✐❛✈❡❧

      η = cos θ ✳ ❉❡ ♠♦❞♦ q✉❡✱ ❛ t♦r♥❛✲s❡ 2 2 m η η + d )d Θ(η) Θ(η) = 0.

      ✭✷✳✺✶✮ (1 − η n(n + 1) − 2 1 − η

      ❆ ❊q✳ é ✐❞❡♥t✐✜❝❛❞❛ ❝♦♠♦ ❛ ❡q✉❛çã♦ ❛ss♦❝✐❛❞❛ ❞❡ ▲❡❣❡♥❞r❡✱ ❝✉❥❛ ❛ s♦❧✉çã♦ é ❞❛❞❛

      ♣♦r m m Θ(θ) = Θ P (cos θ), 1 n ✭✷✳✺✷✮ (cos θ) 1

      ♦♥❞❡ P n sã♦ ❛s ❢✉♥çõ❡s ❛ss♦❝✐❛❞❛s ❞❡ ▲❡❣❡♥❞r❡ ❡ Θ é ✉♠❛ ❝♦♥st❛♥t❡✳ ❆s ❢✉♥çõ❡s ❛♥❣✉❧❛r❡s Θ ❡ Φ sã♦ ❢r❡q✉❡♥t❡♠❡♥t❡ ❝♦♠❜✐♥❛❞❛s ❡♠ ✉♠❛ ❢✉♥çã♦ ❝♦♥❤❡❝✐❞❛ m

      ❝♦♠♦ ❤❛r♠ô♥✐❝♦s ❡s❢ér✐❝♦s Y n ✱ ♦♥❞❡ m (2n + 1)(n − m)! m imϕ s Y P (cos θ) , n (θ, ϕ) ≡ n n ≥ 0, −n ≤ m ≤ n. ❡ ✭✷✳✺✸✮

      4π(n + m)! ❆ ❡q✉❛çã♦ r❛❞✐❛❧ é ❝♦♥❤❡❝✐❞❛ ❝♦♠♦ ❡q✉❛çã♦ ❡s❢ér✐❝❛ ❞❡ ❇❡ss❡❧✱ ❝✉❥❛ ❛ s♦❧✉çã♦ é

      ❞❛❞❛ ♣♦r R(r) = R j (kr) + R y (kr), 1 n 2 n n n 1 2 ✭✷✳✺✹✮

      ♦♥❞❡ j ❡ y sã♦ ❢✉♥çõ❡s ❡s❢ér✐❝❛s ❞❡ ❇❡ss❡❧ ❡ ◆❡✉♠❛♥♥ ❞❡ ♦r❞❡♠ n✱ R ❡ R sã♦ ❝♦♥st❛♥t❡s✳ P♦❞❡♠♦s ❡s❝r❡✈❡r ❡ss❛ s♦❧✉çã♦ ❞❡ ❢♦r♠❛ ❛❧t❡r♥❛t✐✈❛ ❝♦♠♦ (1) (2) n n (1) (2) R(r) = R h (kr) + R h (kr), 3 n n ✭✷✳✺✺✮ 4

      ♦♥❞❡ h ❡ h sã♦ ❢✉♥çõ❡s ❡s❢ér✐❝❛s ❞❡ ❍❛♥❦❡❧ ❞❡ ♣r✐♠❡✐r♦ ❡ s❡❣✉♥❞♦ t✐♣♦ ❡ ❞❡ ♦r❞❡♠ n✱ 3 4 ❡ R ✱ R sã♦ ❝♦♥st❛♥t❡s✳

      ❯♠❛ ✈❡③ q✉❡ ♦❜t✐✈❡♠♦s ❛s s♦❧✉çõ❡s s❡♣❛r❛❞❛s ❞❛s ❢✉♥çõ❡s r❛❞✐❛❧✱ ♣♦❧❛r ❡ ❛③✐♠✉t❛❧✱ ❛ s♦❧✉çã♦ ❣❡r❛❧ ❞❛ ❡q✉❛çã♦ ❞❡ ❍❡❧♠❤♦❧t③ ❞❛❞❛ ♣❡❧❛ ❊q✳ t♦r♥❛✲s❡ X X n m φ(kr, θ, ϕ) = [a j (kr) + b y (kr)]Y (θ, ϕ). n=0 m=−n nm n nm n ✭✷✳✺✻✮ n ❉❡ ♠♦❞♦ ❛❧t❡r♥❛t✐✈♦✱ ♣♦❞❡♠♦s ❡s❝r❡✈❡r ❛ s♦❧✉çã♦ ❝♦♠♦ X ∞ n X (1) (2) m nm nm nm nm φ(kr, θ, ϕ) = [s h (kr) + c h (kr)]Y (θ, ϕ), n=0 m=−n nm nm n n n ✭✷✳✺✼✮

      ♦♥❞❡ a ✱ b ✱ s ❡ c sã♦ ❝♦❡✜❝✐❡♥t❡s q✉❡ s❡rã♦ ❞❡t❡r♠✐♥❛❞♦s ❛tr❛✈és ❞❛s ❝♦♥❞✐çõ❡s ❞❡ ❝♦♥t♦r♥♦ ❡s♣❡❝✐✜❝❛s ❞❡ ✉♠ ♣r♦❜❧❡♠❛ ❛❝úst✐❝♦✳

      ✷✳✸ ❊q✉❛çõ❡s ❞❡ ❝♦♥s❡r✈❛çã♦ ♣❛r❛ só❧✐❞♦s

      ◆❡st❛ s❡çã♦✱ ❛♣r❡s❡♥t❛r❡♠♦s ❛s ❡q✉❛çõ❡s ❜ás✐❝❛s ♣❛r❛ ✉♠ só❧✐❞♦ ❡❧ást✐❝♦ ✭❝♦♥s❡r✈❛çã♦ ❞❛ ♠❛ss❛✱ ❞♦ ♠♦♠❡♥t♦ ❡ ❞❡ ❡♥❡r❣✐❛✮✱ ✐♥❝❧✉✐♥❞♦ ♦s ❝♦♥❝❡✐t♦s ❞❡ ❞❡❢♦r♠❛çã♦ ❡ ♦ ❞❡s❡♥✈♦❧✲ ✈✐♠❡♥t♦ ❞❛s r❡❧❛çõ❡s ❝♦♥st✐t✉t✐✈❛s✳ ❆s ❡q✉❛çõ❡s ❛♣r❡s❡♥t❛❞❛s ♥❡st❛ s❡çã♦ s❡rã♦ r❡❢❡r✐❞❛s i (i = 1, 2, 3) ❡♠ s✐st❡♠❛ ❞❡ ❝♦♦r❞❡♥❛❞❛s ❝❛rt❡s✐❛♥❛s x ✱ ♣❡r♠✐t✐♥❞♦ q✉❡ ❛s ❝♦♠♣❧❡①✐❞❛❞❡s ❞♦ ❝á❧❝✉❧♦ t❡♥s♦r✐❛❧ s❡❥❛ ❡✈✐t❛❞♦✳

      ✷✳✸✳✶ ❉❡❢♦r♠❛çã♦

      ❈♦♥s✐❞❡r❡ ✉♠ ♠❡✐♦ ❝♦♥tí♥✉♦ ❞❡ ✈♦❧✉♠❡ V ❡ s✉♣❡r❢í❝✐❡ S s✉❥❡✐t♦ ❛ ✉♠❛ ❞❡❢♦r♠❛çã♦✳ 1 , x , x ) 2 3 1 ◆❛ s✐t✉❛çã♦ ✐♥✐❝✐❛❧✱ ♦ ♣♦♥t♦ P ❡stá ❧♦❝❛❧✐③❛❞♦ ♣❡❧♦ ✈❡t♦r R(x ✳ ❯♠ ♣♦♥t♦ P s✐✲ t✉❛❞♦ ♥❛ ✈✐③✐♥❤❛♥ç❛ ❞❡ P ✱ ❡♥❝♦♥tr❛✲s❡ ❧♦❝❛❧✐③❛❞♦ ♣❡❧♦ ✈❡t♦r dR ❡♠ r❡❧❛çã♦ ❛ P ✳ ❆♣ós ❛ ′ ′

      ❞❡❢♦r♠❛çã♦✱ ♦ ✈♦❧✉♠❡ ❡ ❛ s✉♣❡r❢í❝✐❡ s❡rã♦ ❞❡♥♦♠✐♥❛❞♦s ♣♦r V ❡ S ✱ ♦ ♣♦♥t♦ P s❡rá r❡♣r❡✲ 1 , x 2 , x 3 ) 1 s❡♥t❛❞♦ ♣♦r P ❡ ❡st❛rá ❧♦❝❛❧✐③❛❞♦ ♣❡❧♦ ✈❡t♦r r(x ✳ ❖ ♣♦♥t♦ P s❡rá r❡♣r❡s❡♥t❛❞♦ ♣♦r P 1 q✉❡ ❡st❛rá ❧♦❝❛❧✐③❛❞♦ ♣❡❧♦ ✈❡t♦r dx ❡♠ r❡❧❛çã♦ ❛ P ✱ ❝♦♠♦ ♠♦str❛ ❛ ✜❣✉r❛ 1 , w , w ) 2 3

      ❖ ❞❡s❧♦❝❛♠❡♥t♦ ❞❡ P ♣❛r❛ P é ♠❡❞✐❞♦ ♣❡❧♦ ✈❡t♦r w = (w ✳ ❉❡st❛ ❢♦r♠❛✱ ❛s r❡❧❛çõ❡s ❡♥tr❡ ♦s ✈❡t♦r❡s ♠♦str❛❞♦ ♥❛ ✜❣✉r❛ sã♦ ❞❛❞❛s ♣♦r r = R + w,

      ✭✷✳✺✽✮

    • w

      ❞r = ❞R + W . ✭✷✳✺✾✮ ◆♦ ❡♥t❛♥t♦✱ ❞❛ ❊q✳ ♦❜t❡♠♦s q✉❡ ❞r = ❞R + ❞w✳ ❯t✐❧✐③❛♥❞♦ ❡ss❛ ❡①♣r❡ssã♦✱ ♣♦❞❡♠♦s r❡❡s❝r❡✈❡r ❛ ❊q✳ ❝♦♠♦

      W = w + i ❞w. ✭✷✳✻✵✮ ❆❧é♠ ❞✐ss♦✱ ♣♦❞❡♠♦s ❡①♣r❡ss❛r ❞w ❞❛ s❡❣✉✐♥t❡ ❢♦r♠❛✿ i j i j = ∂ w

      ❞w ❞x ✭✷✳✻✶✮ j j = ∂/∂x ♦♥❞❡ ∂ ✳ ❊ss❛ ❡q✉❛çã♦ ♣♦❞❡ s❡r ❛♣r❡s❡♥t❛❞❛ ♥❛ ❢♦r♠❛

      1

      1 + i = (∂ j w i + ∂ i w j ) j (∂ j w i i w j ) j . ❞w ❞x ❞x ✭✷✳✻✷✮

      − ∂

      2

      2 ❆ss✐♠✱ ♣♦❞❡♠♦s ❞❡✜♥✐r ♦s t❡♥s♦r❡s ✐♥✜♥✐t❡s✐♠❛✐s ❞❡ ❞❡❢♦r♠❛çã♦ ❡ r♦t❛çã♦ ♣❛r❛ ♣❡q✉❡✲

      ♥❛s ❞❡❢♦r♠❛çõ❡s ❝♦♠♦

      1

      1 ǫ = (∂ w + ∂ w ), ω = (∂ w w ), ij j i i j ij j i i j

      − ∂ ✭✷✳✻✸✮

      2

      2 ♥❛ ❢♦r♠❛ ✈❡t♦r✐❛❧✱ ❛ ❞❡❢♦r♠❛çã♦ é ❞❛❞♦ ♣♦r

      1

      ❚

      ǫ = ,

      ∇w + ∇w ✭✷✳✻✹✮

      2

      ❚

      ♦♥❞❡ ∇w é ♦ t❡♥s♦r ❞❡ s❡❣✉♥❞❛ ♦r❞❡♠ ❡ ♦ s♦❜r❡s❝r✐t♦ ❞❡♥♦t❛ ❛ tr❛♥s♣♦st❛ ❞❡ ✉♠ t❡♥s♦r✳ ❖ r❡s✉❧t❛❞♦ ❞❛ ♠♦str❛ q✉❡ ❛ ❝✐♥❡♠át✐❝❛ ❞❡ ✉♠ ♣♦♥t♦ ❛r❜✐trár✐♦ ♥❛ ✈✐③✐♥❤❛♥ç❛ w

      ❞❡ P é ❣♦✈❡r♥❛❞❛ ♣❡❧♦ ❝❛♠♣♦ ❣r❛❞✐❡♥t❡ ❧♦❝❛❧ ∂ j i ❡ q✉❡ ♦ ♠♦✈✐♠❡♥t♦ é ✉♠❛ ❝♦♠❜✐♥❛çã♦ ❞❡ ❡❢❡✐t♦s ❧♦❝❛✐s ❞❡ ❞✐st♦rçã♦ ǫ ij ❡ r♦t❛çã♦ ❞❡ ❝♦r♣♦ rí❣✐❞♦ ω ij ✳

      ✷✳✸✳✷ ❚❡♥sã♦

      ❈♦♥s✐❞❡r❡ ✉♠ ♠❡✐♦ ❝♦♥tí♥✉♦ ❞❡ ✈♦❧✉♠❡ V ❡ s✉♣❡r❢í❝✐❡ S q✉❡ s♦❢r❡ ❛ ❛çã♦ ❞❡ ❢♦rç❛s i ❡①t❡r♥❛s f ✭i = 1, 2, 3✮ ❝♦♠♦ ♠♦str❛❞♦ ♥❛ ✜❣✉r❛ ◗✉❛♥❞♦ ✉♠ ♦❜❥❡t♦ é ❞❡❢♦r♠❛❞♦

      ′ ❉❡❢♦r♠❛çã♦ ❞❡ ✉♠ ♠❡✐♦ ❝♦♥tí♥✉♦ ❞❡ ✈♦❧✉♠❡ V ❡ s✉♣❡r❢í❝✐❡ S ♣❛r❛ ♦ ✈♦❧✉♠❡ V ❡

      ❋✐❣✉r❛ ✷✳✸✿

      s✉♣❡r❢í❝✐❡ S ✳ x 3 V S V ' S' R P P

    dR

    1 r w W P' dr P' 1 x 1 x 2 ❋♦♥t❡✿ ❆✉t♦r✱ ✷✵✶✺✳

      ❞❡✈✐❞♦ ❛ ❡ss❛s ❢♦rç❛s✱ ❛ ❞✐s♣♦s✐çã♦ ❞❛s ♠♦❧é❝✉❧❛s é ❛❧t❡r❛❞❛ ❡ ♦ ❝♦r♣♦ ❞❡✐①❛ ❞❡ ❡st❛r ♥♦ s❡✉ ❡st❛❞♦ ❞❡ ❡q✉✐❧í❜r✐♦ ♦r✐❣✐♥❛❧✳ ❈♦♠♦ r❡s✉❧t❛❞♦ ❞❡ss❛s ❢♦rç❛s ❡①t❡r♥❛s✱ ♦ ✈❡t♦r ❞❡ tr❛çã♦ ✐♥t❡r♥♦ q q✉❡ t❡♥❞❡ ❛ ❞❡✈♦❧✈❡r ♦ ❝♦r♣♦ ❛♦ ❡q✉✐❧í❜r✐♦ ✈❛✐ ❛t✉❛r s♦❜r❡ ✉♠ ❡❧❡♠❡♥t♦ ❞❡ s✉♣❡r❢í❝✐❡ ❛r❜✐trár✐❛ ❝♦♠ ♦ ✈❡t♦r ♥♦r♠❛❧ n ❛ ❡ss❛ s✉♣❡r❢í❝✐❡✱ ❝♦♠♦ ♠♦str❛❞♦ ♥❛ ✜❣✉r❛ ❖ ✈❡t♦r tr❛çã♦ é ❞❛❞♦ ♣♦r q

      ✭✷✳✻✺✮ = σ · n,

      ♦♥❞❡ σ é ♦ t❡♥s♦r ❞❡ t❡♥sõ❡s ❡ ❡stá ❞✐r❡t❛♠❡♥t❡ r❡❧❛❝✐♦♥❛❞♦ ❝♦♠ ❛s ❢♦rç❛s ✐♥t❡r♥❛s q✉❡ t❡♥❞❡♠ ❛ ❞❡✈♦❧✈❡r ♦ ❝♦r♣♦ ❛♦ ❡q✉✐❧í❜r✐♦✳ ❉❡st❛ ❢♦r♠❛✱ s❡ ♥ã♦ ❤á ❞❡❢♦r♠❛çã♦✱ ♥ã♦ ❡①✐st❡ t❡♥sã♦ ✐♥t❡r♥❛✳

      ❘❡♣r❡s❡♥t❛çã♦ ❞♦ ✈❡t♦r tr❛çã♦ q ❞❡ ✉♠ ♠❡✐♦ ❝♦♥tí♥✉♦ ❞❡ ✈♦❧✉♠❡ V ❡ s✉♣❡r❢í❝✐❡ S

      ❋✐❣✉r❛ ✷✳✹✿

      (i = 1, 2, 3) ❞❡✈✐❞♦ ❛ ❢♦rç❛s ❡①t❡r♥❛s f i ✳ S q n f 1 V f 3 f 2 ❋♦♥t❡✿ ❆✉t♦r✱ ✷✵✶✺✳

      ✷✳✸✳✸ ❈♦♥s❡r✈❛çã♦ ❞❛ ♠❛ss❛

      ❆ ♠❛ss❛ ❝♦♥t✐❞❛ ❡♠ ✉♠ ♠❡✐♦ ❝♦♥tí♥✉♦ ❞❡ ✈♦❧✉♠❡ V ❡ s✉♣❡r❢í❝✐❡ S ♣❛r❛ q✉❛❧q✉❡r ✐♥st❛♥t❡ ❞❡ t❡♠♣♦ t é ❞❛❞❛ ♣♦r Z 3 r m = ρ 1 , 1 = ρ 1 (r, t) V ❞ ✭✷✳✻✻✮

      ♦♥❞❡ ρ é ❛ ❞❡♥s✐❞❛❞❡ ❞❡ ♠❛ss❛ ❞❡ ✉♠ só❧✐❞♦✳ ❆ ❝♦♥s❡r✈❛çã♦ ❞❡ ♠❛ss❛ r❡q✉❡r q✉❡ t❛①❛ ❞❡ ✈❛r✐❛çã♦ ❞❛ ♠❛ss❛ s❡❥❛ ♥✉❧❛✱ ♦✉ s❡❥❛✱ dm/dt = 0✳ ❆ss✐♠✱ Z 3 3 r r V [∂ ρ + ρ ∂ ( )] = 0. t 1 ❞ 1 t ❞ ✭✷✳✻✼✮ ❖ ❞❡s❧♦❝❛♠❡♥t♦ ❞❡ ✉♠❛ ♣❛rtí❝✉❧❛ ❞❡✈✐❞♦ ❛ ✉♠❛ ❞❡❢♦r♠❛çã♦ ❞♦ ✈♦❧✉♠❡ V ❛♣ós ✉♠ ✐♥st❛♥t❡ ❞❡ t❡♠♣♦ dt é vdt✳ ❙❡ n é ♦ ✈❡t♦r ✉♥✐tár✐♦ ♥♦r♠❛❧ ❛ s✉♣❡r❢í❝✐❡ S ✱ ❡♥tã♦ ♦ ✈♦❧✉♠❡ ❞❡s❧♦❝❛❞♦ 2 3 2 r r r

      ♣❡❧❛s ♣❛rtí❝✉❧❛s ❞❡ ✉♠ ❡❧❡♠❡♥t♦ ❞❡ ár❡❛ d ♥❛ s✉♣❡r❢í❝✐❡ S é ❞ = v ·n❞t❞ ✳ P♦rt❛♥t♦✱ Z Z 3 2

      ∂ t ρ 1 ρ 1 = 0.

    • r v r

      ❞ ✭✷✳✻✽✮ V S · n❞

      ❯t✐❧✐③❛♠♦s ♦ t❡♦r❡♠❛ ❞❛ ❞✐✈❡r❣ê♥❝✐❛ ❞❡ ●❛✉ss✱ ❊q✳ ♥❛ ✐♥t❡❣r❛❧ ❞❡ s✉♣❡r❢í❝✐❡ ♣❛r❛ t♦r♥á✲❧❛ ✉♠❛ ✐♥t❡❣r❛❧ ❞❡ ✈♦❧✉♠❡✱ ❞❡st❛ ❢♦r♠❛ Z Z 3 3 ∂ ρ r = 0. t 1 + v r ❞ 1 V V ∇ · ρ ❞ ✭✷✳✻✾✮ ❈♦♠♦ ♦ ✈♦❧✉♠❡ V é ❛r❜✐trár✐♦✱ ❡♥tã♦ ❛ ❡q✉❛çã♦ ❞❡ ❝♦♥s❡r✈❛çã♦ ❞❛ ♠❛ss❛ é ❞❛❞❛ ♣♦r v

      ∂ t ρ 1 1 = 0.

      ✭✷✳✼✵✮

    • ∇ · ρ

      ✷✳✸✳✹ ❈♦♥s❡r✈❛çã♦ ❞♦ ♠♦♠❡♥t♦

      ❆ ✈❛r✐❛çã♦ ❞❛ t❛①❛ ❞♦ ♠♦♠❡♥t♦ ❧✐♥❡❛r ❡♠ ✉♠ ♠❡✐♦ ❝♦♥tí♥✉♦ ❞❡ ✈♦❧✉♠❡ V ❡ s✉♣❡r❢í❝✐❡ S

      é ✐❣✉❛❧ ❛ ❢♦rç❛ t♦t❛❧ ❛♣❧✐❝❛❞❛ s♦❜r❡ ❡ss❡ ✈♦❧✉♠❡✳ P❛r❛ q✉❛❧q✉❡r ✐♥st❛♥t❡ ❞❡ t❡♠♣♦ t✱ ♦ 1 v ♠♦♠❡♥t♦ ❧✐♥❡❛r ♥♦ ✈♦❧✉♠❡ ❝♦♥tí♥✉♦ V é ρ ✳ ❖ ❝♦r♣♦ ❡stá s✉❥❡✐t♦ ❛ ❢♦rç❛s ❞❡ s✉♣❡r❢í❝✐❡s ❡ ✈♦❧✉♠étr✐❝❛s✳ ❆ss✐♠✱ ❛ t❛①❛ ❞❡ ✈❛r✐❛çã♦ ❞♦ ♠♦♠❡♥t♦ ❧✐♥❡❛r ❞♦ ❝♦r♣♦ é ❞❛❞♦ ♣♦r Z Z Z 3 2 3

      ❱

      ∂ ρ v r = + σ r f r , t 1 V S ❞ · n ❞ ❞ ✭✷✳✼✶✮ V ♦♥❞❡ σ é ♦ t❡♥s♦r ❞❡ t❡♥sõ❡s ❡♠ ✉♠ só❧✐❞♦✳ ❘❡s♦❧✈❡♥❞♦ ❛ ❞❡r✐✈❛❞❛ t❡♠♣♦r❛❧ ♥❛ ✐♥t❡❣r❛❧ ❞♦ ❧❛❞♦ ❡sq✉❡r❞♦ ❞❛ ✐❣✉❛❧❞❛❞❡✱ ❡ ❧❡♠❜r❛♥❞♦ q✉❡ ❛ ❞❡r✐✈❛❞❛ ♥♦ t❡♠♣♦ ❞♦ ❡❧❡♠❡♥t♦ ❞❡ 3 2 t ( r r

      ✈♦❧✉♠❡ é ∂ ❞ ) = v · n ❞ ✱ t❡♠♦s Z Z Z Z 3 2 2 3

      ❱

      v r vv r σ r f r

      (v∂ t ρ 1 + ρ 1 ∂ t ) ρ 1 = ,

      ❞ ✭✷✳✼✷✮ V S S · n ❞ · n ❞ V t w s✉❜st✐t✉✐♥❞♦ ❛ ❡q✉❛çã♦ ❡ v = ∂ s✐♠♣❧✐✜❝❛♠♦s ♦ ❧❛❞♦ ❡sq✉❡r❞♦ ❞❛ ✐❣✉❛❧❞❛❞❡✳ ❯t✐❧✐③❛♥❞♦ ♥♦✈❛♠❡♥t❡ ♦ t❡♦r❡♠❛ ❞❡ ●❛✉ss✱ ❊q✳ ♥❛ ✐♥t❡❣r❛❧ ❞❡ s✉♣❡r❢í❝✐❡ ♣♦❞❡♠♦s r❡❡s❝r❡✈❡r ❡ss❛ ❡q✉❛çã♦ ❞❛ s❡❣✉✐♥t❡ ❢♦r♠❛ Z 2 3

      ❱ V (ρ ∂ w ) r = 0. 1 t − ∇ · σ − f ❞ ✭✷✳✼✸✮

      ❈♦♠♦ ♦ ✈♦❧✉♠❡ V é ❛r❜✐trár✐♦✱ ❛ ❡q✉❛çã♦ ❞❛ ❝♦♥s❡r✈❛çã♦ ❞♦ ♠♦♠❡♥t♦ é ❞❛❞❛ ♣♦r 2 ❱ w ρ 1 ∂ = 0. t

      ✭✷✳✼✹✮ − ∇ · σ − f

      ❱

      P❛r❛ ✉♠ ❝❛s♦ ♣❛rt✐❝✉❧❛r✱ ♦♥❞❡ ❛s ❢♦rç❛s ✈♦❧✉♠étr✐❝❛s f sã♦ ♠✉✐t♦ ♠❡♥♦r❡s q✉❡ ❛s ♦✉tr❛s ❢♦rç❛s ❞❛ ❊q✳ ❞❛ s❡❣✉✐♥t❡ ❢♦r♠❛✿ 2

      ρ ∂ w 1 t − ∇ · σ = 0. ✭✷✳✼✺✮ ❊ss❛ é ❛ ❡q✉❛çã♦ ❞❡ ❝♦♥s❡r✈❛çã♦ ❞♦ ♠♦♠❡♥t♦ ♣❛r❛ só❧✐❞♦s q✉❡ ✐r❡♠♦s ✉t✐❧✐③❛r ♥❛s ♣ró①✐♠❛s s❡çõ❡s✳

      ✷✳✸✳✺ ❈♦♥s❡r✈❛çã♦ ❞❛ ❡♥❡r❣✐❛

      ❆ ❈♦♥s❡r✈❛çã♦ ❞❡ ❡♥❡r❣✐❛ ❡st❛❜❡❧❡❝❡ q✉❡ ❛ ✈❛r✐❛çã♦ ❞❛ t❛①❛ ❞❛ ❡♥❡r❣✐❛ t♦t❛❧ é ✐❣✉❛❧ ❛♦ tr❛❜❛❧❤♦ r❡❛❧✐③❛❞♦ s♦❜r❡ ♦ ❝♦r♣♦ ❞❡✈✐❞♦ ❛ t♦❞❛s ❛s ❢♦rç❛s ❡①t❡r♥❛s ♣♦r ✉♥✐❞❛❞❡ ❞❡ t❡♠♣♦✳ c ❆ ❡♥❡r❣✐❛ ❝✐♥ét✐❝❛ E é ❞❡✜♥✐❞❛ ❝♦♠♦ Z

      1 2 3 r E = v ρ . c 1 ❞ ✭✷✳✼✻✮

      2 V ❆ ❡♥❡r❣✐❛ ✐♥t❡r♥❛ U é ❞❡✜♥✐❞❛ ❝♦♠♦ Z

      ✸ r

      U = uρ , V ✶ ❞ ✭✷✳✼✼✮ ♦♥❞❡ u é ❛ ❡♥❡r❣✐❛ ✐♥t❡r♥❛ ♣♦r ✉♥✐❞❛❞❡ ❞❡ ♠❛ss❛✳ ❉❡st❛ ❢♦r♠❛✱ ❛ ❝♦♥s❡r✈❛çã♦ ❞❡ ❡♥❡r❣✐❛ é ❞❛❞❛ ♣♦r Z Z Z

      1 2 ✸ ✈ ✸ 2 r r f r

      ∂ t ( v + u)ρ + 1 = .

      ❞ ✭✷✳✼✽✮

      (σ · n) · v ❞ · v❞ V S

      2 V ❉❡s❡♥✈♦❧✈❡♥❞♦ ♦ ❧❛❞♦ ❡sq✉❡r❞♦ ❞❡ss❛ ❡q✉❛çã♦✱ t❡♠♦s Z Z

      1 2 ✸

      1 2 2 t t 1 t + (v∂ v + ∂ u)ρ v + u + ∂ ρ r v + u ρ v r 1 1 ❞ · n ❞ V Z Z 2

      2 S

      2

      ✈ ✸ = + r f r . S (σ · n) · v ❞ · v❞ V

      ✭✷✳✼✾✮ ❆♣❧✐❝❛♥❞♦ ♦ t❡♦r❡♠❛ ❞❡ ●❛✉ss ♥♦✈❛♠❡♥t❡ ♥❛s ✐♥t❡❣r❛✐s ❞❡ s✉♣❡r❢í❝✐❡ ❡ ♦r❣❛♥✐③❛♥❞♦ ♦s t❡r♠♦s✱ ♦❜t❡♠♦s Z Z Z Z

      1 2 3 3

      ✸ ✈ ✸ (v∂ v +∂ u)ρ r v + u [∂ ρ v ] + r = r f r . t t 1 t 1 1

      ❞ + ∇ · ρ ❞

    • ∇·σv❞ ·v❞
    • V V

        2 V V ✭✷✳✽✵✮

        ◆♦t❡ q✉❡ ♦ t❡r♠♦ ❡♥tr❡ ❝♦❧❝❤❡t❡s é ③❡r♦✱ ♣♦✐s tr❛t❛✲s❡ ❞❛ ❡q✉❛çã♦ ❞❡ ❝♦♥s❡r✈❛çã♦ ❞❛ t w ♠❛ss❛ ❛❧é♠ ❞✐ss♦ s✉❜st✐t✉✐♥❞♦ v = ∂ ❡ r❡♦r❣❛♥✐③❛♥❞♦ ♦s t❡r♠♦s✱ r❡❡s❝r❡✈❡♠♦s ❡ss❛ ❡q✉❛çã♦ ❞❛ s❡❣✉✐♥t❡ ❢♦r♠❛ Z 2 ✈ 3 w w r

        [(ρ 1 ∂ )∂ t + (ρ t 1 ∂ t = 0.

        ✭✷✳✽✶✮ V − ∇ · σ − f u − σ∇ · v)]❞ ❖ ♣r✐♠❡✐r♦ ♣❛r❡♥t❡s❡s é ③❡r♦✱ ♣♦✐s tr❛t❛✲s❡ ❞❛ ❡q✉❛çã♦ ❞❡ ❝♦♥s❡r✈❛çã♦ ❞♦ ♠♦♠❡♥t♦ ♣♦rt❛♥t♦ ❛ ❡q✉❛çã♦ ❛❝✐♠❛ r❡❞✉③ ♣❛r❛

        ρ ∂ 1 t u − σ∇ · v = 0. ✭✷✳✽✷✮ ❊ss❛ é ❛ ❡q✉❛çã♦ ❞❡ ❝♦♥s❡r✈❛çã♦ ❞❛ ❡♥❡r❣✐❛ ♣❛r❛ só❧✐❞♦s✳

        ✷✳✹ ▼♦❞❡❧♦ ❞❡ ❑❡❧✈✐♥✲❱♦✐❣t

        ❊♠ ✉♠ ♠❛t❡r✐❛❧ ❡❧ást✐❝♦✱ ❛ t❡♥sã♦ ❡ ❛ ❞❡❢♦r♠❛çã♦ ❡stã♦ r❡❧❛❝✐♦♥❛❞♦s ♣❡❧❛ r✐❣✐❞❡③✱ q✉❡ é ✉♠❛ ♠❡❞✐❞❛ ❞❛ r❡s✐stê♥❝✐❛ ❞♦ ♠❛t❡r✐❛❧ ❛ ❞❡❢♦r♠❛çã♦ ❡♠ r❡s♣♦st❛ ❛ ✉♠❛ ❢♦rç❛ ❛♣❧✐❝❛❞❛✳ P❛r❛ ✉♠ ♠❡✐♦ ✐s♦tró♣✐❝♦✱ ❈❛✉❝❤② ❣❡♥❡r❛❧✐③♦✉ ❛ ❧❡✐ ❞❡ ❍♦♦❦❡ ❛ss✉♠✐♥❞♦ q✉❡ ❛s ❝♦♠♣♦♥❡♥t❡s ❞❛ t❡♥sã♦ sã♦ ❧✐♥❡❛r♠❡♥t❡ r❡❧❛❝✐♦♥❛❞❛s ❝♦♠ ❛s ❝♦♠♣♦♥❡♥t❡s ❞❛ ❞❡❢♦r♠❛çã♦✳ ❉❡st❛ ❢♦r♠❛✱ ♣♦❞❡♠♦s ❡s❝r❡✈❡r ❛ ❡q✉❛çã♦ t❡♥s♦r✐❛❧ ♥❛ ♥♦t❛çã♦ ❞❡ ❊✐♥st❡✐♥✱ ♦♥❞❡ ✉♠ í♥❞✐❝❡ r❡♣❡t✐❞♦ ✐♠♣❧✐❝❛ s❡♠♣r❡ ✉♠ s♦♠❛tór✐♦ s♦❜r❡ ❡ss❡ í♥❞✐❝❡✱ ❞❛ s❡❣✉✐♥t❡ ❢♦r♠❛✿ ij kl ijkl σ = C ǫ , ij ijkl kl ✭✷✳✽✸✮

        ♦♥❞❡ ♦ σ é ♦ t❡♥s♦r ❞❡ t❡♥sõ❡s✱ ǫ é ♦ t❡♥s♦r ❞❡❢♦r♠❛çã♦ ❡ C é ♦ t❡♥s♦r ❞❛s ❝♦♥st❛♥t❡s ❡❧ást✐❝❛s✱ ♦✉ ♠ó❞✉❧♦s✱ ❞♦ ♠❛t❡r✐❛❧ ❡ é ❝❤❛♠❛❞♦ ♦ t❡♥s♦r ❞♦ ♠ó❞✉❧♦ ❞❡ ❡❧❛st✐❝✐❞❛❞❡✳

        P❛r❛ ❝♦♥s✐❞❡r❛r ♦ ❝♦♠♣♦rt❛♠❡♥t♦ ✈✐s❝♦❡❧ást✐❝♦ ♥♦ q✉❛❧ ❛ t❛①❛ ❞❡ ❞❡❢♦r♠❛çã♦ ❞❡♣❡♥✲ ❞❡♥t❡ ❞♦ t❡♠♣♦ é ♦❜s❡r✈❛❞❛ ❡♠ r❡s♣♦st❛ ❛ ✉♠❛ ❢♦rç❛ ❛♣❧✐❝❛❞❛✱ ❛ ♣♦❞❡ s❡r ❣❡♥❡r❛❧✐③❛❞❛ ❛❞✐❝✐♦♥❛♥❞♦ t❡r♠♦s ♣r♦♣♦r❝✐♦♥❛✐s ❛ ❞❡r✐✈❛❞❛s t❡♠♣♦r❛❧ ❞❛ t❡♥sã♦ ❡ ❞❛ ❞❡✲ ❢♦r♠❛çã♦✱ " # " # X M M 1 m m m m X 2 m m m=1 m=1 A + 1 + ∂ σ = C B ∂ ǫ , ijkl t ijkl t ✭✷✳✽✹✮ ij ijkl kl

        = ∂ /∂t ♦♥❞❡ ∂ t ✳ ❊st❛ ❡①♣r❡ssã♦ é r❡s♣♦♥sá✈❡❧ ♣❛r❛ ♦s q✉❛tr♦ t✐♣♦s ❞❡ ❝♦♠♣♦rt❛✲ ♠❡♥t♦ ✈✐s❝♦❡❧ást✐❝♦ ❝❧áss✐❝♦ ✭❑❡❧✈✐♥✲❱♦✐❣t✱ ▼❛①✇❡❧❧✱ ❩❡♥❡r ♦✉ ❛♥t✐✲❩❡♥❡r✮ ❞❡♣❡♥❞❡♥❞♦ m m

        , B , M = 0 = 1 1 2 1 2 ❞♦s ✈❛❧♦r❡s ❞❡ A ijkl ijkl ❡ M ✳ ◗✉❛♥❞♦ M ❡ M ✱ ♦❜t❡♠♦s ♦ ♠♦❞❡❧♦ ❞❡ ❑❡❧✈✐♥✲❱♦✐❣t σ = C ǫ + B ∂ ǫ . ij ijkl kl ijkl t kl

        ✭✷✳✽✺✮ P❛r❛ ❞❡s❡♥✈♦❧✈❡r ❛ r❡❧❛çã♦ t❡♥sã♦✲❞❡❢♦r♠❛çã♦✱ ❊q✳ D = λδ δ + µ(δ δ + δ δ ) + κ(δ δ δ ), ijkl ij kl ik jl il jk ik jl il jk ij − δ ✭✷✳✽✻✮

        ♦♥❞❡ λ✱ µ ❡ κ sã♦ ❝♦♥st❛♥t❡s ❡ δ é ❛ ❢✉♥çã♦ ❞❡❧t❛ ❞❡ ❑r♦♥❡❝❦❡r✳ ❯t✐❧✐③❛♥❞♦ ❡ss❡ t❡♦r❡♠❛ ♣❛r❛ ♦s t❡♥s♦r❡s ❞❡ q✉❛rt❛ ♦r❞❡♠ ❞❛ ❛ r❡❧❛çã♦ t❡♥sã♦✲❞❡❢♦r♠❛çã♦ ❞♦ ♠♦❞❡❧♦ ❑❡❧✈✐♥✲❱♦✐❣t ♣♦❞❡ s❡r ❡s❝r✐t❛ ❝♦♠♦

        σ = λ δ ǫ + 2µ ǫ + λ δ ∂ ǫ + 2µ ∂ ǫ , ij ❡ ij kk ❡ ij ✈ ij t kk ✈ t ij ✭✷✳✽✼✮

        ♦✉ s❡❥❛✱ σ = [(λ + λ ∂ ) + µ ∂ )ǫ,

        ❡ ✈ t ❡ ✈ t

        tr ǫ] I + 2(µ ✭✷✳✽✽✮

        ❡ ❡

        ♦♥❞❡✱ I é ♦ t❡♥s♦r ✉♥✐tár✐♦ ❞❡ s❡❣✉♥❞❛ ♦r❞❡♠✱ ❵tr✬ s✐❣♥✐✜❝❛ ♦ tr❛ç♦ ❞❡ ✉♠ t❡♥s♦r✱ λ ❡ µ sã♦ ❛s ❝♦♥st❛♥t❡s ❡❧ást✐❝❛s ❧♦♥❣✐t✉❞✐♥❛❧ ❡ ❝✐s❛❧❤❛♠❡♥t♦ ❡♠ ✉♥✐❞❛❞❡s ❞❡ P❛ · s✱ ❡♥q✉❛♥t♦

        ✈ ✈

        q✉❡ λ ❡ µ sã♦ ♦s ❝♦❡✜❝✐❡♥t❡s ❞❡ ✈✐s❝♦s✐❞❛❞❡ ❧♦♥❣✐t✉❞✐♥❛❧ ❡ ❝✐s❛❧❤❛♠❡♥t♦ ❡♠ ✉♥✐❞❛❞❡s ❞❡ P❛ · s✳

        ❯t✐❧✐③❛♥❞♦ ♦ t❡♥s♦r ❞❡❢♦r♠❛çã♦ r❡❡s❝r❡✈❡♠♦s ❡ss❛ r❡❧❛çã♦ ❝♦♠♦ ❢✉♥çã♦ ❞♦ ❞❡s❧♦❝❛♠❡♥t♦ w✱ 2 2 2 2 2

        ρ 1 ∂ w i = (λ + µ )∂ w j + µ ∂ w i + (λ + µ )∂ ∂ t w j + µ ∂ ∂ t w j

      t ij j ij j

      ❡ ❡ ❡ ✈ ✈ ✈ ✭✷✳✽✾✮ ◆❛ ❢♦r♠❛ ✈❡t♦r✐❛❧✱ ❡ss❛ ❡q✉❛çã♦ é ❡q✉✐✈❛❧❡♥t❡ ❛ 2 2 2 w w w w

        ρ 1 ∂ = (λ + µ + (λ + µ t ) + µ ∂ t . t )∇(∇ · w) + µ ∇ )∇ (∇ · ∂ ∇ ❡ ❡ ❡ ✈ ✈ ✈ ✭✷✳✾✵✮ 2 w

        ❆❣♦r❛✱ ✉t✐❧✐③❛♠♦s ❛ ✐❞❡♥t✐❞❛❞❡ ✈❡t♦r✐❛❧ ∇ = ∇(∇ · w) − ∇ × ∇ × w ♣❛r❛ r❡❡s❝r❡✈❡r

        ❡ss❛ ❡q✉❛çã♦ ❝♦♠♦ 2 2 2 2 2 w w w

        ∂ = c ℓ c t c t , t ℓ ∇(∇ · w) − c s ∇ × ∇ × w + τ ℓ ∇ (∇ · ∂ ) − τ s ∇ × ∇ × ∂ s ✭✷✳✾✶✮ ♦♥❞❡ ❛ ✈❡❧♦❝✐❞❛❞❡ ❞♦ s♦♠ ❧♦♥❣✐t✉❞✐♥❛❧ ❡ ❝✐s❛❧❤❛♠❡♥t♦ sã♦ ❞❛❞❛s ♣♦r s

        (λ + 2µ )

        ❡ ❡

        c ℓ = , ✭✷✳✾✷✮ r ρ 1

        µ

        

      c = . s

        ✭✷✳✾✸✮ ρ 1

        ❆❧é♠ ❞✐ss♦✱ ♦s t❡♠♣♦s ❞❡ r❡❧❛①❛çã♦ ❧♦♥❣✐t✉❞✐♥❛❧ ❡ ❝✐s❛❧❤❛♠❡♥t♦ sã♦ ❝♦♥✈❡♥✐❡♥t❡♠❡♥t❡

        ❞❡✜♥✐❞♦s ❝♦♠♦ λ + 2µ

        ✈ ✈

        τ = , 2 ✭✷✳✾✹✮ ρ c 1

        ❞

        µ

        

      τ = . s 2 ✭✷✳✾✺✮

        ρ c 1

        

      s

        ◆♦t❛♠♦s ❛ ♥❛t✉r❡③❛ ❝♦♠♣❧❡①❛ ❞❛ ❡q✉❛çã♦ ❞❡ ♠♦✈✐♠❡♥t♦ ❞♦ ❞❡s❧♦❝❛♠❡♥t♦ ❉❡st❛ ❢♦r♠❛✱ é ♥❡❝❡ssár✐♦ ❡s❝r❡✈❡r ❡ss❛ ❡q✉❛çã♦ ❞❡ ❢♦r♠❛ ♠❛✐s s✐♠♣❧❡s✱ ♣❛r❛ ✐ss♦ ❢❛r❡♠♦s ✉s♦ ❞♦ t❡♦r❡♠❛ ❞❛ ❞❡❝♦♠♣♦s✐çã♦ ❞❡ ❍❡❧♠❤♦❧t③ ❡①♣r❡ss❛♥❞♦ ♦ ✈❡t♦r ❞❡s❧♦❝❛♠❡♥t♦ w ❡♠ t❡r♠♦s ❞♦ ♣♦t❡♥❝✐❛❧ ❡s❝❛❧❛r φ ❡ ❞♦ ♣♦t❡♥❝✐❛❧ ✈❡t♦r A✱

        ✐ωt

        w , = (∇φ + ∇ × A)e ∇ · A = 0, ✭✷✳✾✻✮ ♦♥❞❡ φ ❡ A sã♦ ❛s ❛♠♣❧✐t✉❞❡s ❞❛s ❢✉♥çõ❡s ♣♦t❡♥❝✐❛✐s ❡s❝❛❧❛r ❡ ✈❡t♦r ❝♦rr❡s♣♦♥❞❡♥t❡s ❛s ♦♥❞❛s ❧♦♥❣✐t✉❞✐♥❛✐s ❡ ❞❡ ❝✐s❛❧❤❛♠❡♥t♦✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✳ ❙✉❜st✐t✉✐♥❞♦ ❊q✳ ♥❛ ❊q✳ 2 2 2 2 2 − 2 2 2 2 2 −

        ✐ωt ✐ωt ∂ φ φ c ∂ φ e ∂ A A c ∂ A e = 0. ℓ ℓ ℓ t ℓ s t

        ∇ t − c ℓ ∇ − τ ℓ ∇ + ∇ × t − c ∇ − τ ∇ ✭✷✳✾✼✮

        s s

        ❊st❛ ❡q✉❛çã♦ s❡rá s❛t✐s❢❡✐t❛ s❡ ❝❛❞❛ t❡r♠♦ ❡♥tr❡ ♣❛r❡♥t❡s❡s ❞❡s❛♣❛r❡❝❡r✱ ❛ss✐♠ ♣♦❞❡♠♦s ❡s❝r❡✈❡r ❛ ❡q✉❛çã♦ ❞❛ ♦♥❞❛ ❡♠ ✉♠ só❧✐❞♦ ♣❛r❛ ❝❛❞❛ ♣♦t❡♥❝✐❛❧ s❡♣❛r❛❞❛♠❡♥t❡✱ 2 − − 2 2 ✐ωt 1 + τ ℓ ∂ t ∂ φ ℓ e = 0,

        ✭✷✳✾✽✮ ∇ − c ℓ t 2 − 2 2 −

        ✐ωt 1 + τ ∂ ∂ A e = 0. t s ∇ − c t ✭✷✳✾✾✮ s

        ❯♠❛ ✈❡③ q✉❡ ❛ ❢✉♥çã♦ ♣♦t❡♥❝✐❛❧ A é ✉♠ ❝❛♠♣♦ ✈❡t♦r✐❛❧ s♦❧❡♥♦✐❞❛❧✱ ♣♦❞❡♠♦s ❡①♣r❡ssá✲

        

      s,1 s,2

        ❧♦ ❡♠ t❡r♠♦ ❞❡ ❞♦✐s ♣♦t❡♥❝✐❛✐s ❡s❝❛❧❛r❡s ψ ❡ ψ ✱ ✐st♦ é✱ ♦s ❝❤❛♠❛❞♦s ♣♦t❡♥❝✐❛✐s ❡s❝❛❧❛r❡s ❞❡ ❉❡❜②❡

        A ).

        s,1 s,2 ✭✷✳✶✵✵✮

        = ∇ × (rψ ) + ∇ × ∇ × (rψ ❆ss✐♠✱ s✉❜st✐t✉✐♥❞♦ ♦ ♣♦t❡♥❝✐❛❧ ✈❡t♦r ❊qs✳ ❡ ♦❜t❡♠♦s ❛s ❡q✉❛çõ❡s ❞❡ ❍❡❧♠❤♦❧t③ ❡s❝❛❧❛r❡s 2 2

        φ

      • k ℓ = 0,

        ✭✷✳✶✵✶✮ ∇ ℓ 2 2 s,1 ψ

      • k = 0, ∇

        ✭✷✳✶✵✷✮

        s

        ψ

        s,2

        ♦♥❞❡ k ❡ k s sã♦ ♦s ♥ú♠❡r♦s ❞❡ ♦♥❞❛ ❧♦♥❣✐t✉❞✐♥❛❧ ❡ ❝✐s❛❧❤❛♠❡♥t♦✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✳ ❆s r❡❧❛çõ❡s ❞❡ ❞✐s♣❡rsã♦ ❞❛s ♦♥❞❛s ❧♦♥❣✐t✉❞✐♥❛❧ ❡ ❝✐s❛❧❤❛♠❡♥t♦ sã♦ 2 2 ω 2 k = n (ωτ ), j j ∈ {ℓ, s}, ✭✷✳✶✵✸✮ j

        ❝

        c j

        (ωτ ) j ♦♥❞❡ n ❝ é r❡❝♦♥❤❡❝✐❞♦ ❝♦♠♦ ♦ í♥❞✐❝❡ ❞❡ r❡❢r❛çã♦ ❝♦♠♣❧❡①♦ ❞♦ ♠❛t❡r✐❛❧ ✈✐s❝♦❡❧ást✐❝♦✱

        1 n (ωτ ) = ,

        ❝ j

        ✭✷✳✶✵✹✮ j ∈ {ℓ, s}. j ) p1 + (−✐ωτ j = (ωτ j ) ➱ út✐❧ ❞❡✜♥✐r ♦ t❡♠♣♦ ❞❡ r❡❧❛①❛çã♦ ❛❞✐♠❡♥s✐♦♥❛❧ ε ✳ ❆❧é♠ ❞✐ss♦✱ ♣♦❞❡♠♦s R

        ❡s❝r❡✈❡r ♦ í♥❞✐❝❡ ❞❡ r❡❢r❛çã♦ ❝♦♠♣❧❡①♦ ❡♠ t❡r♠♦s ❞❡ s✉❛ ♣❛rt❡ r❡❛❧ n ❡ ♣❛rt❡ ✐♠❛❣✐♥ár✐❛ α ˜

        ✱ n (ε ) = n (ε ) + ).

        

      ❝ j ❘ j j

        ✐˜α(ε ✭✷✳✶✵✺✮ ◆♦t❡ q✉❡ ♦ ♠❡✐♦ é ❞✐s♣❡rs✐✈♦ ❡ ❛❜s♦r✈❡❞♦r✱ ❛ ♣❛rt❡ ✐♠❛❣✐♥ár✐❛ ❡stá r❡❧❛❝✐♦♥❛❞❛ ❝♦♠ q✉❛♥t✐❞❛❞❡ ˜α q✉❡ ❝♦rr❡s♣♦♥❞❡ ❛ ❢✉♥çã♦ ❞❡ ❛❜s♦rçã♦ ❛❞✐♠❡♥s✐♦♥❛❧✱ ❡♥q✉❛♥t♦ q✉❡ ❛ ♣❛rt❡ R r❡❛❧ ❡stá r❡❧❛❝✐♦♥❛❞❛ ❝♦♠ ❞✐s♣❡rsã♦✱ n s❡rá r❡❢❡r✐❞♦ ❝♦♠♦ ♦ í♥❞✐❝❡ ❞❡ r❡❢r❛çã♦ ❞♦ ♠❛t❡r✐❛❧✳

        ✷✳✺ ▼♦❞❡❧♦ ❞❡ ❑❡❧✈✐♥✲❱♦✐❣t ❋r❛❝✐♦♥ár✐♦

        ❆ ♣❛rtí❝✉❧❛ ❡s♣❛❧❤❛❞♦r❛ é ❢❡✐t❛ ❞❡ ✉♠ ♠❛t❡r✐❛❧ só❧✐❞♦ ✈✐s❝♦❡❧ást✐❝♦✳ ❏á ❡stá ❜❡♠ ❡st❛❜❡✲ ❧❡❝✐❞♦ q✉❡ ❛ ❛❜s♦rçã♦ ❛❝úst✐❝❛ ❞❡ ♦♥❞❛s ❧♦♥❣✐t✉❞✐♥❛✐s ♦✉ ❞❡ ❝✐s❛❧❤❛♠❡♥t♦ ❡♠ ✉♠❛ ❣r❛♥❞❡ ✈❛r✐❡❞❛❞❡ ❞❡ ♠❛t❡r✐❛✐s ✈✐s❝♦❡❧ást✐❝♦s ♦❜❡❞❡❝❡ ❛ ✉♠❛ ❧❡✐ ❞❡ ♣♦tê♥❝✐❛ ❞❛ ❢r❡q✉ê♥❝✐❛ y

        α(ω) = α ω , ✭✷✳✶✵✻✮

        ♦♥❞❡ α é ♦ ❝♦❡✜❝✐❡♥t❡ ❞❡ ❛❜s♦rçã♦ ❡ y é ♦ ❡①♣♦❡♥t❡ ❞❛ ❧❡✐ ❞❡ ♣♦tê♥❝✐❛ q✉❡ ✈❛r✐❛ ❡♥tr❡ 0 ❡ 2✳ ❊st❡s ♣❛râ♠❡tr♦s ❡stã♦ r❡❧❛❝✐♦♥❛❞♦s ❝♦♠ ❛s ♣r♦♣r✐❡❞❛❞❡s ❞♦ ♠❛t❡r✐❛❧ ✈✐s❝♦❡❧ást✐❝♦✳

        ❊st❡ ♠♦❞❡❧♦ ❢♦✐ ❡s❝♦❧❤✐❞♦ ♣♦rq✉❡ ❞❡s❝r❡✈❡ ❛ ❞❡♣❡♥❞ê♥❝✐❛ ❞❛ ❧❡✐ ❞❡ ♣♦tê♥❝✐❛ ❞❛ ❢r❡q✉ê♥✲ ❝✐❛ ♦❜s❡r✈❛❞❛ ❡①♣❡r✐♠❡♥t❛❧♠❡♥t❡ ❡♠ ✈ár✐♦s ♠❛t❡r✐❛✐s ✈✐s❝♦❡❧ást✐❝♦s ❉❡st❛ ❢♦r♠❛✱ ❛♣r❡s❡♥t❛♠♦s ✉♠❛ ❣❡♥❡r❛❧✐③❛çã♦ ❞♦ ♠♦❞❡❧♦ ❞❡ ❑❡❧✈✐♥✲❱♦✐❣t ❛♣r❡s❡♥t❛❞♦ ❛♥t❡r✐♦r♠❡♥t❡✳ ❊ss❛ ❣❡♥❡r❛❧✐③❛çã♦ é ❢❡✐t❛ ✐♥tr♦❞✉③✐♥❞♦ ✉♠ ♦♣❡r❛❞♦r ❞❡ ❝á❧❝✉❧♦ ❢r❛❝✐♦♥ár✐♦✱ ❝❤❛♠❛❞❛ ❞❡ ❞❡r✐✈❛❞❛ ❢r❛❝✐♦♥ár✐❛✱ ♥❛ r❡❧❛çã♦ ❡♥tr❡ ❛ t❡♥sã♦ ❡ ❞❡❢♦r♠❛çã♦✳ ❉❡st❛ ❢♦r♠❛✱ ❛ ❊q✳ ♣♦❞❡ s❡r ❣❡♥❡r❛❧✐③❛❞❛ s✉❜st✐t✉✐♥❞♦ ❛s ❞❡r✐✈❛❞❛s t❡♠♣♦r❛✐s ❝♦♠ ❡①♣♦❡♥t❡s ✐♥t❡✐r♦s ♣♦r ❞❡r✐✈❛❞❛s t❡♠♣♦r❛✐s ❢r❛❝✐♦♥ár✐❛s " # " # X M M 1 m m+ν−1 m m+ν−1 X 2 1 + A + ∂ σ ij = C ijkl B ∂ ǫ kl , ν m=1 m=1 ijkl t ijkl t ✭✷✳✶✵✼✮

        ♦♥❞❡ ∂ ✭ν > 0✮ é ♦ ♦♣❡r❛❞♦r ❞❡r✐✈❛❞❛ ❢r❛❝✐♦♥ár✐❛ t❡♠♣♦r❛❧✳ ❊st❡ ♦♣❡r❛❞♦r ❛t✉❛♥❞♦ t ❡♠ ✉♠❛ ❢✉♥çã♦ ❞♦ t❡♠♣♦✱ f(t)✱ é ❞❡✜♥✐❞♦ ❛q✉✐ ❛tr❛✈és ❞❛ tr❛♥s❢♦r♠❛❞❛ ❞❡ ❋♦✉r✐❡r ❞❛ s❡❣✉✐♥t❡ ❢♦r♠❛ Z ∞ ν − ν

        ✐ωt −∞ [∂ f (t)] F (ω), t ❡ ❞t = (−iω) ✭✷✳✶✵✽✮

        ♦♥❞❡ F (ω) é ❛ tr❛♥s❢♦r♠❛❞❛ ❞❡ ❋♦✉r✐❡r✳ ❆ ❞❡r✐✈❛❞❛ ❢r❛❝✐♦♥ár✐❛ ❞❡ ♦r❞❡♠ ❛r❜✐trár✐❛ ν ♣♦❞❡ s❡r ❡♥t❡♥❞✐❞❛ ❝♦♠♦ ✉♠❛ ❣❡♥❡r❛❧✐③❛çã♦ ❞❛ ❞❡r✐✈❛❞❛ ❞❡ m✲és✐♠❛ ♦r❞❡♠✱ s✉❜st✐t✉✐♥❞♦ ♦ ♥ú♠❡r♦ ✐♥t❡✐r♦ m ♣♦r ✉♠ ♥ú♠❡r♦ r❡❛❧ ν✳ 1 2

        ◆♦✈❛♠❡♥t❡✱ ❞❡♣❡♥❞❡♥❞♦ ❞♦s ✈❛❧♦r❡s ❞❡ M ❡ M ❡st❛ ❡①♣r❡ssã♦ é r❡s♣♦♥sá✈❡❧ ♣♦r

        1 = 0, M = 1 2 q✉❛tr♦ t✐♣♦s ❞❡ ❝♦♠♣♦rt❛♠❡♥t♦ ✈✐s❝♦❡❧ást✐❝♦ ❝❧áss✐❝♦ ✳ P❛r❛ ♦ ❝❛s♦✱ M ✱ ♦❜t❡♠♦s ♦ ♠♦❞❡❧♦ ❞❡ ❑❡❧✈✐♥✲❱♦✐❣t ❢r❛❝✐♦♥ár✐♦ ν σ = C ǫ + B ∂ ǫ . ij ijkl kl ijkl kl t ✭✷✳✶✵✾✮

        ❉❛ ♠❡s♠❛ ❢♦r♠❛ q✉❡ ♦ ♠♦❞❡❧♦ ❑❡❧✈✐♥✲❱♦✐❣t✱ ♣❛r❛ ♦ ❝❛s♦ ✐s♦tró♣✐❝♦✱ ♦ ♠♦❞❡❧♦ ❢r❛❝✐♦✲ ♥ár✐♦ ♣♦❞❡ s❡r ❡s❝r✐t♦ ❞❛ s❡❣✉✐♥t❡ ❢♦r♠❛✿ ν ν

        σ = λ δ ǫ + 2µ ǫ + λ δ ∂ ǫ + 2µ ∂ ǫ , ij ij kk ij ij kk ij

        ❡ ❡ ✈ t ✈ t ✭✷✳✶✶✵✮

        ♦✉ s❡❥❛✱ ν ν σ = [(λ + λ ∂ ) + µ ∂ )ǫ,

        ❡ ✈ ❡ ✈ t tr ǫ] I + 2(µ t ✭✷✳✶✶✶✮

        ♥♦t❡ q✉❡ ❛❣♦r❛ ♦s ❝♦❡✜❝✐❡♥t❡s ❞❡ ✈✐s❝♦s✐❞❛❞❡ ❧♦♥❣✐t✉❞✐♥❛❧ λ ✈ ❡ ❝✐s❛❧❤❛♠❡♥t♦ µ ✈ t❡♠ ✉♥✐✲ ν )

        ❞❛❞❡s ❞❡ (P❛ · s ❞❡✈✐❞♦ ❛ ❞❡r✐✈❛❞❛ ❢r❛❝✐♦♥ár✐❛✳ ❙❡❣✉✐♥❞♦ ♦s ♠❡s♠♦s ♣❛ss♦s ✉s❛❞♦s ♥❛ s❡çã♦ ❡ss❛ ❡①♣r❡ssã♦ t❛♠❜é♠ ♣♦❞❡ s❡r ❡s❝r✐t❛ ❝♦♠♦ ✉♠❛ ❡q✉❛çã♦ ❞❡ ♦♥❞❛ ♣❛r❛ ❝❛❞❛ ♣♦t❡♥❝✐❛❧ s❡♣❛r❛❞❛♠❡♥t❡✱ ν ν 2 − − 2 2 ✐ωt 1 + τ ∂ ∂ φ = 0, ℓ t ∇ − c ℓ t ν ν 2 − − 2 2 ✐ωt ℓ ❡ ✭✷✳✶✶✷✮ 1 + τ ∂ ∂ A = 0. t ∇ − c t ❡ ✭✷✳✶✶✸✮

        s s

        ❆❣♦r❛✱ ♦s t❡♠♣♦s ❞❡ r❡❧❛①❛çã♦ ❧♦♥❣✐t✉❞✐♥❛❧ ❡ ❝✐s❛❧❤❛♠❡♥t♦ sã♦ ❝♦♥✈❡♥✐❡♥t❡♠❡♥t❡ ❞❡✜♥✐❞♦s ❝♦♠♦ 1/ν

        λ + 2µ

        ✈ ✈

        τ = , 2 ✭✷✳✶✶✹✮ ρ 1 c

        ❞ 1/ν

        µ

        ✈ τ = . s 2 ✭✷✳✶✶✺✮

        ρ c 1

        

      s

        ◆♦t❡ q✉❡ ν = 1✱ r❡t♦r♥❛r❡♠♦s ❛♦ ♠♦❞❡❧♦ ❝❧áss✐❝♦ ❞❡ ❑❡❧✈✐♥✲❱♦✐❣t✳ ℓ , ψ P❛r❛ ♦❜t❡r ❛s ❡q✉❛çõ❡s ♣❛r❛ ❛s ❛♠♣❧✐t✉❞❡s ❞♦s ♣♦t❡♥❝✐❛✐s φ s,1 ❡ ψ s,2 ✱ ♣♦❞❡✲s❡ ♠♦s✲ ν − ν −

        ✐ωt ✐ωt

        tr❛r ❛ ♣❛rt✐r ❞❛ ❊q✳ = (−✐ω)

        ♦❜t❡♠♦s ♥♦✈❛♠❡♥t❡ ❛s ❡q✉❛çõ❡s ❞❡ ❍❡❧♠❤♦❧t③ 2 2

      • k φ = 0,

        ∇ ℓ ✭✷✳✶✶✻✮ 2 2 s,1 ψ + k = 0. ∇ s

        ✭✷✳✶✶✼✮ ψ

        s,2

        ❆❣♦r❛✱ ❝♦♠ ♦s í♥❞✐❝❡ ❞❡ r❡❢r❛çã♦ ❝♦♠♣❧❡①♦ ❞♦ ♠❛t❡r✐❛❧ ✈✐s❝♦❡❧ást✐❝♦ ❞❛s ♦♥❞❛s ❧♦♥❣✐✲ t✉❞✐♥❛❧ ❡ ❞❡ ❝✐s❛❧❤❛♠❡♥t♦ s❡♥❞♦ ❡s❝r✐t❛ ❝♦♠♦ 1 n (ωτ ) = , j

        ❝ j ∈ {ℓ, s}. ✭✷✳✶✶✽✮ ν

        ) p1 + (−✐ωτ j

        ❆ s♦❧✉çã♦ ❞❛ ❡q✉❛çã♦ ❞❡ ❍❡❧♠❤♦❧t③ é ❞❛❞❛ ♥❛ s❡çã♦ ❆s ❛♠♣❧✐t✉❞❡s ❞♦s ♣♦t❡♥❝✐❛✐s ❞❡♥tr♦ ❞❛ ♣❛rtí❝✉❧❛ ❞❡✈❡ s❡r ❢✉♥çõ❡s r❡❣✉❧❛r❡s ✭✜♥✐t❛s✮✳ P♦r ✐ss♦✱ ♦s ♣♦t❡♥❝✐❛✐s ❧♦♥❣✐t✉❞✐♥❛❧ ❡ ❝✐s❛❧❤❛♠❡♥t♦ ❞❡ ❉❡❜②❡ sã♦ ❡①♣r❡ss♦s ♥❛ ❜❛s❡ ❞❡ ♦♥❞❛ ♣❛r❝✐❛❧ ❝♦♠♦ X m

        φ (k r, θ, ϕ) = b a j (k r)Y (θ, ϕ), ℓ ℓ n nm n ℓ X n,m n ✭✷✳✶✶✾✮ m ψ (k r, θ, ϕ) = c a j (k r)Y (θ, ϕ), n nm n

        s,1 s s n ✭✷✳✶✷✵✮ X n,m m

        ψ (k r, θ, ϕ) = d a j (k r)Y (θ, ϕ), n nm n

        s,2 s s n ✭✷✳✶✷✶✮ n n n n n,m

        ♦♥❞❡✱ ♦s ❝♦❡✜❝✐❡♥t❡s ❞❡ ❡①♣❛♥sã♦ b ✱ c ✱ d ❡ s s❡rã♦ ❞❡t❡r♠✐♥❛❞♦s ♣❡❧❛ r❡s♦❧✉çã♦ ❞♦ s✐st❡♠❛ ❧✐♥❡❛r ♦❜t✐❞♦ ❛ ♣❛rt✐r ❞❛s ❝♦♥❞✐çõ❡s ❞❡ ❝♦♥t♦r♥♦ ♥❛ s✉♣❡r❢í❝✐❡ ❞❛ ♣❛rtí❝✉❧❛✱ q✉❡ s❡rã♦ ❞✐s❝✉t✐❞❛s ♥♦ ♣ró①✐♠♦ ❝❛♣ít✉❧♦✳

        ◆❡st❡ ❝❛♣ít✉❧♦✱ ❞❡s❝r❡✈❡♠♦s ❛s ❡q✉❛çõ❡s ❞❡ ❝♦♥s❡r✈❛çã♦ ❡♠ ✉♠ ❞❡t❡r♠✐♥❛❞♦ ♠❡✐♦✱ ❜❡♠ ❝♦♠♦ ❛ ♣r♦♣❛❣❛çã♦ ❞❛ ♦♥❞❛ ♥❡ss❡ ♠❡✐♦ ❡ ❛♣r❡s❡♥t❛♠♦s ❛ s♦❧✉çã♦ ❞❛ ❡q✉❛çã♦ ❞❡ ❍❡❧♠❤♦❧t③✳ ❊ss❡s ❡q✉❛çõ❡s ❞♦ ♠♦❞❡❧♦ s❡rã♦ ✉s❛❞❛s ♥♦ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❞❡st❛ t❡s❡✳

        ◆♦ ♣ró①✐♠♦ ❝❛♣ít✉❧♦ ✈❛♠♦s ❛❜♦r❞❛r ♦s ❝♦♥❝❡✐t♦s q✉❡ ❢✉♥❞❛♠❡♥t❛♠ ♦ ❡s♣❛❧❤❛♠❡♥t♦ ❛❝úst✐❝♦ ♣♦r ✉♠❛ ♣❛rtí❝✉❧❛ ❡s❢ér✐❝❛✱ ❡ ♥♦ ❝❛♣ít✉❧♦ ✹✱ ✐r❡♠♦s ❛♣r❡s❡♥t❛r ✉♠ ❡st✉❞♦ s♦❜r❡ ❢♦rç❛ ❡ t♦rq✉❡ ❞❡ r❛❞✐❛çã♦ q✉❡ ✉t✐❧✐③❛♠ ♦s ❢✉♥❞❛♠❡♥t♦s ❞❡ ❡s♣❛❧❤❛♠❡♥t♦✳

        3

      ❊s♣❛❧❤❛♠❡♥t♦ ❛❝úst✐❝♦ ♥♦ ❧✐♠✐t❡ ❞❡ ❘❛②❧❡✐❣❤

        ◗✉❛♥❞♦ ✉♠❛ ♦♥❞❛ s♦♥♦r❛ ❡♥❝♦♥tr❛ ✉♠ ♦❜stá❝✉❧♦✱ ♣❛rt❡ ❞❛ ♦♥❞❛ é ❞❡s✈✐❛❞❛ ❞♦ s❡✉ ❝❛✲ ♠✐♥❤♦ ♦r✐❣✐♥❛❧✳ ➱ ✉s✉❛❧ ❞❡✜♥✐r ❛ ❞✐❢❡r❡♥ç❛ ❡♥tr❡ ❛ ♦♥❞❛ ❞❡s✈✐❛❞❛ ❡ ❛ ♦♥❞❛ ♥ã♦ ♣❡rt✉r❜❛❞❛✱ q✉❡ ❡st❛r✐❛ ♣r❡s❡♥t❡ s❡ ♥ã♦ ❤♦✉✈❡ss❡ ♦ ♦❜stá❝✉❧♦✱ ❝♦♠♦ ♦♥❞❛ ❡s♣❛❧❤❛❞❛✳

        ❚r❛❜❛❧❤♦s ♣✐♦♥❡✐r♦s ❡st❛❜❡❧❡❝❡r❛♠ ❛s ❜❛s❡s t❡ór✐❝❛s ♣❛r❛ ❛♥❛❧✐s❛r ♦ ❡s♣❛❧❤❛♠❡♥t♦ ❡♠ ✉♠❛ ♣❛rtí❝✉❧❛ ❡s❢ér✐❝❛ ✐♠❡rs❛ ❡♠ ✉♠ ✢✉✐❞♦ ✐❞❡❛❧✱ ❞❡✈✐❞♦ ❛ ♦♥❞❛ ♣❧❛♥❛ ✐♥❝✐❞❡♥t❡✳ ❖ ❡s✲ t✉❞♦ ♣✐♦♥❡✐r♦ s♦❜r❡ ❡s♣❛❧❤❛♠❡♥t♦ ❛❝úst✐❝♦ ❢♦✐ ❞❡s❡♥✈♦❧✈✐❞♦ ♣♦r ▲♦r❞ ❘❛②❧❡✐❣❤ ◆❡ss❡ ❡st✉❞♦✱ ❘❛②❧❡✐❣❤ ❝♦♥s✐❞❡r♦✉ ♦ ❝❛s♦ ❧✐♠✐t❡ ♥♦ q✉❛❧ ♦ ❡s♣❛❧❤❛❞♦r ✭❡s❢❡r❛ rí❣✐❞❛✮ é ♠✉✐t♦ ♠❡♥♦r q✉❡ ♦ ❝♦♠♣r✐♠❡♥t♦ ❞❡ ♦♥❞❛✳ ❯♠ ❝❛s♦ ❣❡r❛❧ ♣❛r❛ ❡s♣❛❧❤❛♠❡♥t♦ ♣♦r ✉♠❛ ❡s❢❡r❛ ♦✉ ❝✐❧✐♥❞r♦ rí❣✐❞♦ ❢♦✐ ❞❡s❡♥✈♦❧✈✐❞♦ ♣♦r ▼♦rs❡ ♥♦ q✉❛❧ ❛ s♦❧✉çã♦ ❞♦ ❡s♣❛❧❤❛♠❡♥t♦ ❞❡s❡♥✲ ✈♦❧✈✐❞❛ ♣♦r ❡❧❡ ♥ã♦ ♥❡❝❡ss❛r✐❛♠❡♥t❡ ❝♦♥s✐❞❡r❛ q✉❡ ♦ ❡s♣❛❧❤❛❞♦r é ♠✉✐t♦ ♠❡♥♦r ❞♦ q✉❡ ♦ ❝♦♠♣r✐♠❡♥t♦ ❞❡ ♦♥❞❛✳ ◆❡ss❡ ❡st✉❞♦✱ ♦ ❡s♣❛❧❤❛❞♦r ♣♦❞❡ s❡r ❝♦♥s✐❞❡r❛❞♦ ❛♣r♦①✐♠❛❞❛♠❡♥t❡ ❞♦ ♠❡s♠♦ t❛♠❛♥❤♦ ❞♦ ❝♦♠♣r✐♠❡♥t♦ ❞❡ ♦♥❞❛✳ ❊♠ ❣❡r❛❧✱ ❞❡♣❡♥❞❡♥❞♦ ❞♦ ♠❛t❡r✐❛❧ q✉❡ é ❢❡✐t♦ ♦ ❡s♣❛❧❤❛❞♦r✱ ❛s ♦♥❞❛s s♦♥♦r❛s ♣♦❞❡♠ s❡r ❡s♣❛❧❤❛❞❛s ❡ ❛❜s♦r✈✐❞❛s✳ ❉❡st❛ ❢♦r♠❛✱ ▼♦rs❡ ❡ ❝♦❧❛❜♦r❛❞♦r❡s ❣❡♥❡r❛❧✐③❛r❛♠ ♦ ❡st✉❞♦ ❛♥t❡r✐♦r✱ ❝♦♥s✐❞❡r❛♥❞♦ ♦s ❡❢❡✐t♦s ❞❛ ♦♥❞❛ ❧♦♥❣✐t✉❞✐♥❛❧ ❞❡♥tr♦ ❞❡ ❡s♣❛❧❤❛❞♦r❡s ✢✉✐❞♦s ✭❝✐❧✐♥❞r♦ ❡ ❡s❢❡r❛✮✳ ❆ t❡♦r✐❛ ❞❡ ❡s♣❛❧❤❛♠❡♥t♦ ♣❛r❛ ❝✐❧✐♥❞r♦ ❡ ❡s❢❡r❛ ❢❡✐t♦ ❞❡ ♠❛t❡r✐❛❧ só❧✐❞♦ ❡❧ást✐❝♦ s✉❜♠❡rs♦ ❡♠ ✉♠ ✢✉✐❞♦✱ q✉❡ s✉✲ ♣♦rt❛♠ ♦♥❞❛s ❞❡ ❝✐s❛❧❤❛♠❡♥t♦ ❡ ♦♥❞❛s ❧♦♥❣✐t✉❞✐♥❛✐s ♦❜t❡✈❡ ❛✈❛♥ç♦s s✐❣♥✐✜❝❛t✐✈♦s ❛tr❛✈és ❞♦ tr❛❜❛❧❤♦ ♣✐♦♥❡✐r♦ ❞❡ ❏✳ ❋❛r❛♥ ❖ ❡s♣❛❧❤❛♠❡♥t♦ ❛❝úst✐❝♦ ❛ ♣❛rt✐r ❞❡ ✉♠❛ ❡s❢❡r❛ ✈✐s❝♦❡❧ást✐❝❛ ❢♦✐ ❞❡s❡♥✈♦❧✈✐❞♦ ♣♦r ❆②r❡s ❡ ●❛✉♥❛✉r❞ q✉❡ ❝♦♥s✐❞❡r♦✉ ✉♠❛ ♦♥❞❛ ♣❧❛♥❛ ✐♥❝✐❞❡♥t❡ s♦❜r❡ ✉♠ ♦❜❥❡t♦ ✈✐s❝♦❡❧ást✐❝♦ ❡ ✉t✐❧✐③♦✉ ♦ ♠♦❞❡❧♦ ❞❡ ❑❡❧✈✐♥✲❱♦✐❣t ♣❛r❛ ♠♦❞❡❧❛r ❛ ♣r♦♣❛❣❛çã♦ ❞❡ ♦♥❞❛s ❞❡♥tr♦ ❞❛ ❡s❢❡r❛✳

        ◆❡st❡ ❝❛♣ít✉❧♦✱ ❛♣r❡s❡♥t❛r❡♠♦s ❛ t❡♦r✐❛ ❞❡ ❡s♣❛❧❤❛♠❡♥t♦ ❛❝úst✐❝♦ ♣❛r❛ ✉♠❛ ♣❛rtí❝✉❧❛ ❡s❢ér✐❝❛ ❢❡✐t❛ ❞❡ ✉♠❛ ♠❛t❡r✐❛❧ só❧✐❞♦ ✈✐s❝♦❡❧ást✐❝♦ s✉s♣❡♥s❛ ❡♠ ✉♠ ✢✉✐❞♦ ✐❞❡❛❧ ❝♦♠ ✉♠❛ ♦♥❞❛ ❛❝úst✐❝❛ ❛r❜✐trár✐❛✳

        ✸✳✶ ❊s♣❛❧❤❛♠❡♥t♦ ♣♦r ✉♠❛ ❡s❢❡r❛ ✈✐s❝♦❡❧ást✐❝❛

        ❈♦♥s✐❞❡r❡ ✉♠❛ ♦♥❞❛ ❛❝úst✐❝❛ ❤❛r♠ô♥✐❝❛ ❛r❜✐trár✐❛ ❝♦♠ ❢r❡q✉ê♥❝✐❛ ❛♥❣✉❧❛r ω ♣r♦♣❛✲ ❣❛♥❞♦ ❡♠ ✉♠ ✢✉✐❞♦ ❤♦♠♦❣ê♥❡♦ ❞❡ ❞❡♥s✐❞❛❞❡ ρ ❡ ✈❡❧♦❝✐❞❛❞❡ ❞♦ s♦♠ c ✳ ❯♠❛ ♣❛rtí❝✉❧❛ 1

        ❡s❢ér✐❝❛ ❞❡ r❛✐♦ a ❡ ❞❡♥s✐❞❛❞❡ ρ ❢❡✐t❛ ❞❡ ♠❛t❡r✐❛❧ ✈✐s❝♦❡❧ást✐❝♦ ❡stá s✉s♣❡♥s❛ ♥❡ss❡ ✢✉✐❞♦✳ ❈♦♥s✐❞❡r❛♠♦s ❛ ♣❛rtí❝✉❧❛ ❝❡♥tr❛❞❛ ♥❛ ♦r✐❣❡♠ ❞❡ ✉♠ s✐st❡♠❛ ❞❡ ❝♦♦r❞❡♥❛❞❛s ❝♦♠♦ ♣♦❞❡ s❡r ✈✐st♦ ♥❛ ✜❣✉r❛ ❆♦ ✐♥❝✐❞✐r ✉♠❛ ♦♥❞❛ s♦❜r❡ ❛ ♣❛rtí❝✉❧❛ ✈✐s❝♦❡❧ást✐❝❛✱ ♣❛rt❡ ❞❛ ♦♥❞❛ ❛❝úst✐❝❛ é ❛❜s♦r✈✐❞❛ ❡ ♣❛rt❡ é ❡s♣❛❧❤❛❞❛✳ ❆s ♦♥❞❛s ✐♥❝✐❞❡♥t❡ ❡ ❡s♣❛❧❤❛❞❛ sã♦ ❞❡s✲ − −

        ✐ωt ✐ωt

        (r) (r)

        ✐♥ s❝

        ❝r✐t❛s ❡♠ t❡r♠♦s ❞♦s ♣♦t❡♥❝✐❛✐s ❞❡ ✈❡❧♦❝✐❞❛❞❡ φ ❡ ❡ φ ❡ ✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✳ ❆s ❛♠♣❧✐t✉❞❡s ❞♦s ♣♦t❡♥❝✐❛s ❞❡ ✈❡❧♦❝✐❞❛❞❡ s❛t✐s❢❛③❡♠ ❛ ❡q✉❛çã♦ ❞❡ ♦♥❞❛ ❞❡ ❍❡❧♠❤♦❧t③✱

        φ 2 2 ✐♥

      • k ) = 0, (∇

        ✭✸✳✶✮ φ

        s❝

        ♦♥❞❡ k = ω/c é ♦ ♥ú♠❡r♦ ❞❡ ♦♥❞❛✳ ◆❛ ❛♣r♦①✐♠❛çã♦ ❧✐♥❡❛r✱ ❞❡ ❛❝♦r❞♦ ❝♦♠ ❛s ❊qs✳ ❡ ❛ ♣r❡ssã♦ ✐♥❝✐❞❡♥t❡ ✭❡s♣❛❧❤❛❞❛✮ ❡ ❛ ✈❡❧♦❝✐❞❛❞❡ ❞♦ ✢✉✐❞♦ sã♦ r❡❧❛❝✐♦♥❛❞♦s ♣♦r p = ωφ ,

        ✐♥✭s❝✮ ✐ρ ✐♥✭s❝✮ ✭✸✳✷✮ v . ✐♥✭s❝✮ = ∇φ ✐♥✭s❝✮ ✭✸✳✸✮

        ❈♦♥s✐❞❡r❛♠♦s q✉❡ ❛ ♦r✐❣❡♠ ❞♦ s✐st❡♠❛ ❞❡ ❝♦♦r❞❡♥❛❞❛s ❡stá s✐t✉❛❞♦ ♥♦ ❝❡♥tr♦ ❞❛ ♣❛r✲ tí❝✉❧❛ ❡s❢ér✐❝❛✳ ❉❡✈✐❞♦ à s✐♠❡tr✐❛ ❞♦ ♣r♦❜❧❡♠❛✱ ❞❡s❝r❡✈❡♠♦s ♦s ♣♦t❡♥❝✐❛✐s ✐♥❝✐❞❡♥t❡ ❡ ❡s♣❛❧❤❛❞♦ ❝♦♠♦ ❢✉♥çõ❡s ❞❡ ❝♦♦r❞❡♥❛❞❛s ❡s❢ér✐❝❛s✿ ❞✐stâ♥❝✐❛ r❛❞✐❛❧ r ♣❛r❛ ♦ ♣♦♥t♦ ♦❜✲ s❡r✈❛çã♦ r = (x, y, z)✱ â♥❣✉❧♦ ♣♦❧❛r θ✱ ❡ â♥❣✉❧♦ ❛③✐♠✉t❛❧ ϕ✳ P♦rt❛♥t♦✱ ❞❡ ❛❝♦r❞♦ ❝♦♠ ❛ s♦❧✉çã♦ ❞❛ ❡q✉❛çã♦ ❞❡ ❍❡❧♠❤♦❧t③ ❞❛❞❛ ♥❛ s❡çã♦ ❛s ❛♠♣❧✐t✉❞❡s ❞♦s ♣♦t❡♥❝✐❛✐s ✐♥❝✐❞❡♥t❡ ❡ ❡s♣❛❧❤❛❞♦✱ ♣♦❞❡♠ s❡r ❡①♣❛♥❞✐❞♦ ❡♠ ✉♠❛ sér✐❡ ❞❡ ♦♥❞❛s ♣❛r❝✐❛✐s ❞❛ s❡❣✉✐♥t❡ ❢♦r♠❛✿ X m

        φ (kr, θ, ϕ) = a j (kr)Y (θ, ϕ),

        ✐♥ nm n ✭✸✳✹✮ X n,m (1) m n

        φ (kr, θ, ϕ) = a s h (kr)Y (θ, ϕ), P ∞ P n s❝ nm n ✭✸✳✺✮ n,m n n = nm n

        ♦♥❞❡ P ✱ a é ♦ ❝♦❡✜❝✐❡♥t❡ ❞❡ ❢♦r♠❛ ❞♦ ❢❡✐①❡ ❞❛ ♦♥❞❛ ✐♥❝✐❞❡♥t❡ ❡ s n,m n=0 m=−n é ♦ ❝♦❡✜❝✐❡♥t❡ ❞❡ ❡s♣❛❧❤❛♠❡♥t♦ ❡s❝❛❧❛r✳

        P❛r❛ ❛ ♦♥❞❛ ✐♥❝✐❞❡♥t❡✱ ❡st❛♠♦s ♣r❡♦❝✉♣❛❞♦s ❡♠ ❡♥❝♦♥tr❛r ❛ s♦❧✉çã♦ ❞❡ ✉♠❛ ♦♥❞❛ q✉❡ s❡ ♣r♦♣❛❣❛ ♥✉♠❛ r❡❣✐ã♦ q✉❡ ✐♥❝❧✉✐ ❛ ♦r✐❣❡♠ ❞♦ s✐st❡♠❛ ❞❡ ❝♦♦r❞❡♥❛❞❛s✱ ♣♦r ✐ss♦ ❡st❛ s♦❧✉çã♦ ❞❡✈❡ s❡r r❡❣✉❧❛r ♥❛ ♦r✐❣❡♠ ✭✜♥✐t❛ ❞❡♥tr♦ ❞❛ r❡❣✐ã♦ ❞❡ ♣r♦♣❛❣❛çã♦✮✳ P♦r ♦✉tr♦ ❧❛❞♦✱ ♣❛r❛ ❛ ♦♥❞❛ ❡s♣❛❧❤❛❞❛ ♥ã♦ ❝♦♥s✐❞❡r❛♠♦s ❛ r❡❣✐ã♦ q✉❡ ✐♥❝❧✉✐ ❛ ♦r✐❣❡♠ ❞♦ s✐st❡♠❛ ❞❡

        s❝

        ❝♦♦r❞❡♥❛❞❛s✳ ❆❧é♠ ❞✐ss♦✱ ♥♦t❡ q✉❡ ❛ ❛♠♣❧✐t✉❞❡ ❞❛ ♦♥❞❛ ❡s♣❛❧❤❛❞❛ φ ❞❡✈❡ s❛t✐s❢❛③❡r ❛ ❝♦♥❞✐çã♦ ❞❡ r❛❞✐❛çã♦ ❙♦♠♠❡r❢❡❧❞ ♥♦ ✐♥✜♥✐t♦✱ lim r (∂ = 0. r s❝ r→∞ − ik) φ ✭✸✳✻✮ ❊st❛ ❝♦♥❞✐çã♦ ✐♠♣õ❡ q✉❡ ❛ ♦♥❞❛ ❡s♣❛❧❤❛❞❛ ♥ã♦ ❞❡✈❡ s❡r r❡✢❡t✐❞❛ ♥♦ ✐♥✜♥✐t♦✳

        ❖ ❝♦❡✜❝✐❡♥t❡ ❞❡ ❢♦r♠❛ ❞♦ ❢❡✐①❡ ♣♦❞❡ s❡r ❞❡t❡r♠✐♥❛❞♦ ❡♠ t❡r♠♦s ❞♦ ♣♦t❡♥❝✐❛❧ ❞❡ ✈❡❧♦❝✐❞❛❞❡ ✐♥❝✐❞❡♥t❡ ❛✈❛❧✐❛❞♦s s♦❜r❡ ✉♠❛ s✉♣❡r❢í❝✐❡ ❡s❢ér✐❝❛ ❞❡ ❝♦♥tr♦❧❡ ❞❡ r❛✐♦ R ≫ a✳ m

        (θ, ϕ) ❆ss✐♠✱ ♠✉❧t✐♣❧✐❝❛♥❞♦ ❛ ❊q✳ ♣♦r Y n ✱ ✐♥t❡❣r❛♥❞♦ s♦❜r❡ ✉♠ â♥❣✉❧♦ só❧✐❞♦ ✭Ω =

        ❯♠ ❢❡✐①❡ ✐♥❝✐❞❡♥t❡ ✭❜❛rr❛s ✈❡rt✐❝❛✐s ✈❡r❞❡✲❡s❝✉r♦✮✱ ❝♦♠ ❝♦♠♣r✐♠❡♥t♦ ❞❡ ♦♥❞❛ λ

        ❋✐❣✉r❛ ✸✳✶✿

        

      ❡ r❡♣r❡s❡♥t❛❞♦ ♣❡❧❛ ❛♠♣❧✐t✉❞❡ ❞♦ ♣♦t❡♥❝✐❛❧ ❞❡ ✈❡❧♦❝✐❞❛❞❡ φ ✐♥ ✱ ❛t✐♥❣❡ ✉♠❛ ♣❡q✉❡♥❛ ♣❛rtí❝✉❧❛

      ✈✐s❝♦❡❧ást✐❝❛ ✭❝ír❝✉❧♦ ✈❡r♠❡❧❤♦✮ ❞❡ r❛✐♦ a s✉s♣❡♥s♦ ♥✉♠ ✢✉✐❞♦✱ ❝♦♠ ❞❡♥s✐❞❛❞❡ ρ ❡ ❛ ✈❡❧♦❝✐❞❛❞❡

      ❞❡ s♦♠ c ✳ ❯♠ s✐st❡♠❛ ❞❡ ❝♦♦r❞❡♥❛❞❛s ❝❛rt❡s✐❛♥❛ ❡stá s✐t✉❛❞♦ ♥♦ ❝❡♥tr♦ ❞❛ ♣❛rtí❝✉❧❛ ❝♦♠

      ✉♠ ♣♦♥t♦ ❞❡ ♦❜s❡r✈❛çã♦ ❞❡s✐❣♥❛❞♦ ♣♦r r = (x, y, z)✳ ❈♦♦r❞❡♥❛❞❛s ❡s❢ér✐❝❛s (r, θ, ϕ) t❛♠❜é♠

      sã♦ ✐❧✉str❛❞♦s✳ ❆s ♦♥❞❛s ❡s♣❛❧❤❛❞❛s ✭❛r❝♦s ✈❡r❞❡ ❝❧❛r♦✮ sã♦ r❡♣r❡s❡♥t❛❞♦s ♣❡❧❛ ❛♠♣❧✐t✉❞❡ ❞♦

      ♣♦t❡♥❝✐❛❧ ❞❡ ✈❡❧♦❝✐❞❛❞❡ φ s❝ ✳ ❙❡t❛s ♣r❡t❛s ✐♥❞✐❝❛♠ ❛s ❞✐r❡çõ❡s ❞❡ ♣r♦♣❛❣❛çã♦ ❞❛ ♦♥❞❛✳ y x a z

        ❋♦♥t❡✿ ❆✉t♦r✱ ✷✵✶✺✳

        sin θ ❞θ❞ϕ✮ ❡ ✉s❛♥❞♦ ❛ r❡❧❛çã♦ ❞❡ ♦rt♦❣♦♥❛❧✐❞❛❞❡ ❞♦s ❤❛r♠ô♥✐❝♦s ❡s❢ér✐❝♦s✱ ♦❜t❡♠♦s Z π Z 2π

        1 m∗ a nm = φ (kR, θ, ϕ)Y (θ, ϕ) sin θ

        ✐♥ n ❞θ❞ϕ.

        j n (kR) n P♦r ♦✉tr♦ ❧❛❞♦✱ ♦ ❝♦❡✜❝✐❡♥t❡ ❞❡ ❡s♣❛❧❤❛♠❡♥t♦ ❡s❝❛❧❛r s s❡rá ❞❡t❡r♠✐♥❛❞♦ ❛♣❧✐❝❛♥❞♦ ❛s ❝♦♥❞✐çõ❡s ❞❡ ❝♦♥t♦r♥♦ ❛♣r♦♣r✐❛❞❛s s♦❜r❡ ❛ ✐♥t❡r❢❛❝❡ ✢✉✐❞♦✲✈✐s❝♦❡❧ást✐❝❛ ♥❛ s✉♣❡r❢í❝✐❡ ❞❛ ♣❛rtí❝✉❧❛✳

        ❆ ❛♣r♦①✐♠❛çã♦ ❞❡ ❝❛♠♣♦ ❞✐st❛♥t❡ (kr ≫ 1) ♣❛r❛ ❛ ♦♥❞❛ ✐♥❝✐❞❡♥t❡ é ♦❜t✐❞❛ s✉❜st✐t✉✐♥❞♦ ❛ ❢♦r♠❛ ❛ss✐♥tót✐❝❛ ❞❛ ❢✉♥çã♦ ❞❡ ❇❡ss❡❧ ❡s❢ér✐❝❛

        1 j n (kr) ∼

        ✭✸✳✼✮ sin(kr − nπ/2), kr

        ♥❛ ❊q✳ ❆ss✐♠✱ X

        1 m φ a (θ, ϕ),

        ✐♥ (kr, θ, ϕ) ∼ nm

        ✭✸✳✽✮ sin(kr − nπ/2)Y n kr n,m

        ❆ ♦♥❞❛ ❡s♣❛❧❤❛❞❛ ♥♦ ❝❛♠♣♦ ❞✐st❛♥t❡ ✭kr ≫ 1✮✱ ♣♦❞❡ s❡r ♦❜t✐❞❛ s✉❜st✐t✉✐♥❞♦ ❛ r❡❧❛çã♦ ❛ss✐♥tót✐❝❛

      (1) − n−1 ❡

      ✐kr h , n (kr) ∼ ✐

        ✭✸✳✾✮ kr

        ♥❛ ❊q✳ ❉❡st❛ ❢♦r♠❛✱ ❡♥❝♦♥tr❛♠♦s

        ✐kr X

        ✐❡ − φ s n a nm Y (θ, ϕ).

        s❝ ✐ ✭✸✳✶✵✮

        (kr, θ, ϕ) ∼ − n kr n,m ❆ ♣❛rtí❝✉❧❛✱ ♣♦r s❡r ❝♦♠♣♦st❛ ❞❡ ✉♠ ♠❛t❡r✐❛❧ ✈✐s❝♦❡❧ást✐❝♦✱ ♣❡r♠✐t❡ ❛❜s♦rçã♦ ❞❡ ♦♥❞❛s

        ❧♦♥❣✐t✉❞✐♥❛❧ ❡ ❝✐s❛❧❤❛♠❡♥t♦ ❡♠ s❡✉ ✐♥t❡r✐♦r✳ ❆ ♣r♦♣❛❣❛çã♦ ❞❡ ♦♥❞❛s ❡♠ ✉♠ ♠❛t❡r✐❛❧ ✈✐s❝♦❡❧ást✐❝♦ ❢♦✐ ❛♣r❡s❡♥t❛❞♦ ♥❛ s❡çã♦ ❆s ❛♠♣❧✐t✉❞❡s ❞♦s ♣♦t❡♥❝✐❛✐s ❞❡♥tr♦ ❞❛ ♣❛rtí❝✉❧❛ ❞❡✈❡ s❡r ❢✉♥çõ❡s r❡❣✉❧❛r❡s ✭✜♥✐t❛s✮✳ P♦r ✐ss♦✱ ♦s ♣♦t❡♥❝✐❛✐s ❧♦♥❣✐t✉❞✐♥❛❧ ❡ ❞❡ ❝✐s❛❧❤❛♠❡♥t♦ ❞❡ ❉❡❜②❡ sã♦ ❡①♣r❡ss♦s ♥❛ ❜❛s❡ ❞❡ ♦♥❞❛ ♣❛r❝✐❛❧ ❝♦♠♦ X m

        φ = b a j (k r)Y (θ, ϕ), ℓ n nm n ℓ X n,m n ✭✸✳✶✶✮ m ψ = c a j (k r)Y (θ, ϕ),

        s,1 n nm n s X n,m n ✭✸✳✶✷✮ m

        ψ = d a j (k r)Y (θ, ϕ), ℓ s s,2 n nm n s n,m n ✭✸✳✶✸✮ n n n n ♦♥❞❡ k ✱ k sã♦ ♦s ♥ú♠❡r♦s ❞❡ ♦♥❞❛ ❧♦♥❣✐t✉❞✐♥❛❧ ❡ ❞❡ ❝✐s❛❧❤❛♠❡♥t♦ ❡ b ✱ c ✱ d ✱ s sã♦ ♦s ❝♦❡✜❝✐❡♥t❡s ❞❡ ❡①♣❛♥sã♦ q✉❡ s❡rã♦ ❞❡t❡r♠✐♥❛❞♦s ♣❡❧❛ r❡s♦❧✉çã♦ ❞♦ s✐st❡♠❛ ❧✐♥❡❛r ♦❜t✐❞♦ ❛ ♣❛rt✐r ❞❛s ❝♦♥❞✐çõ❡s ❞❡ ❝♦♥t♦r♥♦ ♥❛ s✉♣❡r❢í❝✐❡ ❞❛ ♣❛rtí❝✉❧❛✳

        ❆s ❝♦♥❞✐çõ❡s ❞❡ ❝♦♥t♦r♥♦ ❡①✐❣❡♠ q✉❡ ♦ t❡♥s♦r ❞❡ t❡♥sã♦ ❡ ♦ ✈❡t♦r ❞❡ ❞❡s❧♦❝❛♠❡♥t♦ ❞❡✈❡♠ s❡r ❝♦♥tí♥✉♦ ❡♠ t♦❞❛ ❛ s✉♣❡r❢í❝✐❡ ❞❛ ♣❛rtí❝✉❧❛✳ ❆ss✐♠✱ ♥❛ s✉♣❡r❢í❝✐❡ ❞❛ ♣❛rtí❝✉❧❛ r = a

        ✱ t❡♠♦s✿ r

      • ❆ ❝♦♠♣♦♥❡♥t❡ r❛❞✐❛❧ ❞♦ ✈❡t♦r ❞❡s❧♦❝❛♠❡♥t♦ ❞❛ ♣❛rtí❝✉❧❛ w ♥❛ ✐♥t❡r❢❛❝❡ é ✐❣✉❛❧
        • v )

        ✐♥✱r s❝✱r

        à ❝♦♠♣♦♥❡♥t❡ r❛❞✐❛❧ ❞♦ ❞❡s❧♦❝❛♠❡♥t♦ ❞♦ ✢✉✐❞♦ ✐❞❡❛❧ (✐/ω)(v ✳ ❆ss✐♠✱ ✉t✐❧✐③❛♥❞♦ ❛ ❊q✳ ♦❜t❡♠♦s

        ✐ w (a, θ, ϕ) = ∂ [φ (r, θ, ϕ) + φ . r r ✐♥ s❝ r=a (r, θ, ϕ)]| ✭✸✳✶✹✮

        ω rr ♥❛ ✐♥t❡r❢❛❝❡ ❞❛ ♣❛rtí❝✉❧❛ ✈✐s❝♦❡❧ást✐❝❛ é ✐❣✉❛❧ à

      • ❆ ❝♦♠♣♦♥❡♥t❡ r❛❞✐❛❧ ❞❛ t❡♥sã♦ σ
        • p ) ♣r❡ssã♦ ❡①t❡r♥❛ ♥♦ ✢✉✐❞♦ ✐❞❡❛❧ (p ✐♥ s❝ ✳ ❉❡st❛ ❢♦r♠❛✱ ❞❛ ❊q✳ ❡♥❝♦♥tr❛♠♦s σ ω[φ (a, θ, ϕ) + φ (a, θ, ϕ)]. rr ✐♥ s❝ ✭✸✳✶✺✮

        (a, θ, ϕ) = −✐ρ rθ ❡ σ rϕ ❞❡✈❡♠ s❡r ♥✉❧❛s

      • ❆s ❝♦♠♣♦♥❡♥t❡s t❛♥❣❡♥❝✐❛✐s ❞❛ t❡♥sã♦ ❞❡ ❝✐s❛❧❤❛♠❡♥t♦ σ

        ♥❛ s✉♣❡r❢í❝✐❡ ❞❛ ♣❛rtí❝✉❧❛✱ σ (a, θ, ϕ) = 0,

        ✭✸✳✶✻✮ σ rϕ (a, θ, ϕ) = 0. ✭✸✳✶✼✮

        ❆ ❝♦♠♣♦♥❡♥t❡ r❛❞✐❛❧ ❞♦ ✈❡t♦r ❞❡s❧♦❝❛♠❡♥t♦ ❞❡♥tr♦ ❞❛ ♣❛rtí❝✉❧❛ ❡♠ t❡r♠♦s ❞♦s ♣♦t❡♥❝✐❛✐s

        ❡s❝❛❧❛r❡s é ♦❜t✐❞❛ ❝♦♠❜✐♥❛♥❞♦ ❛ ❊q✳ ♦ r❡s✉❧t❛❞♦ é 2

        s,1 s,1 ✭✸✳✶✽✮ s

        P♦r ♦✉tr♦ ❧❛❞♦✱ ♣❛r❛ ♦❜t❡r ❛s ❝♦♠♣♦♥❡♥t❡s ❞♦ t❡♥s♦r ❞❡ t❡♥sã♦ ❡♠ ❝♦♦r❞❡♥❛❞❛s ❡s❢ér✐❝❛s✱ ❞❡r✐✈❛♠♦s ♦ ✈❡t♦r ❞❡s❧♦❝❛♠❡♥t♦ ❡♠ t❡r♠♦s ❞♦s ♣♦t❡♥❝✐❛✐s ❡s❝❛❧❛r❡s ❞❡ ❉❡❜②❡ φ ✱ ψ s,1 ❡ ψ s,2 ✳ ❙✉❜st✐t✉í♠♦s ♦ r❡s✉❧t❛❞♦ ❞❡♥tr♦ ❞❛ ❊q✳ ♦❜t❡♠♦s ♦ t❡♥s♦r ❞❡❢♦r♠❛çã♦ ǫ✳ ❋✐♥❛❧♠❡♥t❡✱ s✉❜st✐t✉✐♥❞♦ ❛ ❞❡❢♦r♠❛çã♦ ♥❛ ❊q✳ ❡♥❝♦♥tr❛♠♦s ν 2 ν

        σ ) ]k φ + 2[µ ) ] rr ❡ ✈ ℓ ❡ ✈ = −[λ + (−✐ωλ ℓ + (−✐ωµ 2 2

        ∂ (φ + ∂ (rψ )) + k ∂ (rψ ) , ℓ r s,1 r s,1 × r s ✭✸✳✶✾✮ ν

        1 σ = [µ ) ] 2∂ ∂ (φ + ∂ (rψ )) rθ ❡ ✈ r θ ℓ r s,1

      • (−✐ωµ r 1 ψ
      • 2k ∂ + ψ ∂ ∂ ψ , θ s,1 ϕ r s,2 ✭✸✳✷✵✮

        s −

        sin θ r ν

        2

        1 σ = [µ ) ] ∂ ∂ (φ + ∂ (rψ )) rϕ r ϕ ℓ r

        ❡ + (−✐ωµ ✈ s,1

        sin θ r 1 ψ 2 s,2 + k ∂ ψ ∂ ψ . ϕ s,1 θ r s,2

        s − ∂ − ✭✸✳✷✶✮

        sin θ r ❆s ❝♦♥❞✐çõ❡s ❞❡ ❝♦♥t♦r♥♦✱ sã♦ ❛♣❧✐❝❛❞❛s ✉t✐❧✐③❛♥❞♦ ❛ ❝♦♠♣♦♥❡♥t❡

        ❞♦ ✈❡t♦r ❞❡s❧♦❝❛♠❡♥t♦ ❊q✳ ❆ss✐♠✱ ❛♣❧✐❝❛♥❞♦ ❛ ♣r✐♠❡✐r❛ ❝♦♥❞✐çã♦ ❞❡ ❝♦♥t♦r♥♦ ❊q✳ ♦❜t❡♠♦s

        ✐ b [nj (k r)j (k r)] + c n(n + 1)j (k r) = (kr) n n ℓ ℓ n+1 ℓ n n s n n+1 r) − (k {nj (kr) − (kr)j ω (1) (1) nh

        }| ✭✸✳✷✷✮

      • s n (kr) r=a , n (kr) − (kr)h n

        ✉t✐❧✐③❛♥❞♦ ❛ r❡❧❛çã♦ ❞❡ r❡❝♦rrê♥❝✐❛ n j (x) = j (x), n+1 n ✭✸✳✷✸✮ (x) − j n x

        = ∂ j x ♦♥❞❡ j ✱ ❞❡♥tr♦ ❞❛ ❡q✉❛çã♦ ❡♥❝♦♥tr❛♠♦s ❛ ♣r✐♠❡✐r❛ ❡q✉❛çã♦ ❞♦ s✐st❡♠❛ ❧✐♥❡❛r

        ✐(ka) (1) ′ ✐(ka) ′ s h (ka) + c n(n + 1)j (k n n n s n ℓ ℓ

        a) + b (k a)j (k a) = j (ka). − n n n ✭✸✳✷✹✮

        ω ω

        ❯t✐❧✐③❛♥❞♦ ♦s ♣♦t❡♥❝✐❛✐s ❞❡ ✈❡❧♦❝✐❞❛❞❡ φ ✐♥✭s❝✮ ❡ ❛ ❝♦♠♣♦♥❡♥t❡ r❛❞✐❛❧ ❞❛ t❡♥sã♦ ❊q✳

      • (−✐ωλ
      • (−✐ωµ
      • c n
      • (−✐ωλ
      • (−✐ωµ
      • 2n(n + 1)[µ
      • (−✐ωµ

      • (−✐ωλ
      • (−✐ωµ
      • {[2n(n + 1) − (k
      • 2n(n + 1) [(k
      • c n
      • c n

        s

        a) − (k

        s

        [2j n (k

        a) − (k ℓ a)j n (k ℓ a)]∂ θ Y m n (θ, ϕ)

        a)]Y m n (θ, ϕ) = 2b n [j n (k ℓ

        s

        a)j n (k

        s

        a) − (k 2

        s

        a) 2 j n (k

        s

        ◆❛ t❡r❝❡✐r❛ ❝♦♥❞✐çã♦ ❞❡ ❝♦♥t♦r♥♦ ❡♠ ❢✉♥çã♦ ❞♦s ♣♦t❡♥❝✐❛✐s ❞❡ ❉❡❜②❡✳ ❉❡st❛ ❢♦r♠❛✱ ❡♥❝♦♥tr❛♠♦s ✐m sin θ d n [(k

        ✭✸✳✸✶✮ ❊st❛ é ❛ s❡❣✉♥❞❛ ❡q✉❛çã♦ ❞♦ s✐st❡♠❛ ❧✐♥❡❛r✳

        a) 2 ρ 1 ω j n (ka).

        s

        ✐ρ (k

        a)] c n = −

        s

        a) − j n (k

        s

        a)j n (k

        s

        a) − 4(k ℓ a)j n (k ℓ a)}b n

        a) 2 ]j n (k ℓ

        s

        a) 2 ρ 1 ω h (1) n (ka)s n

        s

        a) 2 j n (k

        s

        a) − 2(k

        s

        s a)]}Y m n (θ, ϕ).

        a) 2 j ′′ n (k

        a) − 2(k s

        s

        a) − 2(k s a)j n (k

        s

        a) 2 j n (k

        a) − (k s

        s

        [2j n (k

        a) − (k a)j n (k a)]

        {2b n [j n (k

        a)]∂ θ Y m n (θ, ϕ) = − ✐m sin θ

        a)j n (k

        s

        s

        a) − (k 2

        s

        a) 2 j n (k

        s

        ✭✸✳✸✷✮ ❯t✐❧✐③❛♥❞♦ ❛ ❛ q✉❛rt❛ ❝♦♥❞✐çã♦ ❞❡ ❝♦♥t♦r♥♦ é ♦❜t✐❞❛ ❞❛ s❡❣✉✐♥t❡ ❢♦r♠❛✿ d n [(k

        a)]∂ θ Y m n (θ, ϕ).

        s

        a) 2 j ′′ n (k

        s

        a) − 2(k

        s

        a)j n (k

        (k

        ❆❣♦r❛✱ ❛ ❊q✳ ♣♦❞❡ s❡r r❡❡s❝r✐t❛ ❝♦♠♦✱ ✐ρ

        , ✭✸✳✸✵✮

        s

        ✐ρ ωa 2 s n h (1) n (ka) − {[λ

        ✭✸✳✷✼✮ ♥❛ ❡♥❝♦♥tr❛♠♦s

        − n(n + 1)]j n (x) + 2xj n (x) = 0,

        (x) = 0, ✭✸✳✷✻✮ x 2 j ′′′ n (x) + 4xj ′′ n (x) + [2 + x 2

        ✉s❛♥❞♦ ❛s r❡❧❛çõ❡s ❞✐❢❡r❡♥❝✐❛✐s x 2 j ′′ n (x) + 2xj n (x) + [x 2 − n(n + 1)]j n

        , ✭✸✳✷✺✮

        r) }}| r=a

        s

        j n (k

        s

        r) + rk 3

        s

        j n (k

        r) + k 2

        ✈

        s

        j ′′′ n (k

        s

        r) + rk 3

        s

        j ′′ n (k

        s

        3k 2

        {b 2 ′′ n

        ✈ ν

        ✈

      ν

      2

        = −[λ ❡

        ❡♠ t❡r♠♦s ❞♦s ♣♦t❡♥❝✐❛✐s ❡s❝❛❧❛r❡s ♥❛ s❡❣✉♥❞❛ ❝♦♥❞✐çã♦ ❞❡ ❝♦♥t♦r♥♦ ♦❜t❡♠♦s −✐ρ j (1) n

        ❡

        ) ν ](k

        s

        ωa 2 j n (ka).

        ) ν ] = ρ 1 ω 2 k 2

        ✈

        ❡

        ✭✸✳✷✾✮ [µ

        k 2 ,

        s − 2k 2

        k 2

        s

        ) ν ] = ρ 1 ω 2 k 2

        ✈

        ❡

        [λ

        ✭✸✳✷✽✮ ❈♦♠❜✐♥❛♥❞♦ ❛s ✈❡❧♦❝✐❞❛❞❡s ❞♦ s♦♠ ♦❜t❡♠♦s

        a)] c n = −✐ρ

        a) 2 j n (k

        s

        a) − j n (k

        s

        a)j n (k

        s

        ) ν ] [(k

        ✈

        ❡

        a) 2 j ′′ n (k a)}b n

        ) ν ](k

        ✈

        ❡

        a) − 2[µ

        ✭✸✳✸✸✮

        ❈♦♠❜✐♥❛♥❞♦ ❛ ❊q✳ ♦❜t❡♠♦s 2

        s n s s s ✭✸✳✸✹✮

        a) − (k n ❆ss✐♠✱ ♦ ❝♦❡✜❝✐❡♥t❡ ❞❡ ❡s♣❛❧❤❛♠❡♥t♦ d q✉❡ ❛♣❛r❡❝❡ ♥❛ s❡rá ♥✉❧♦✱ ❞❡ ♠♦❞♦ q✉❡ ♦ ♣♦t❡♥❝✐❛❧ ❡s❝❛❧❛r ❞❡ ❉❡❜②❡ ♣❛r❛ ♦♥❞❛s ❞❡ ❝✐s❛❧❤❛♠❡♥t♦ ψ s,2 é ③❡r♦✳ ❈♦♥s❡q✉❡♥t❡✲

        ♠❡♥t❡✱ ❛♣❡♥❛s ✉♠ ♣♦t❡♥❝✐❛❧ ❞❡ ❉❡❜②❡ ψ s,1 é ♥❡❝❡ssár✐♦ ♣❛r❛ ❞❡s❝r❡✈❡r ❛ ♣r♦♣❛❣❛çã♦ ❞❡ ♦♥❞❛s ❞❡ ❝✐s❛❧❤❛♠❡♥t♦ ♥♦ ✐♥t❡r✐♦r ❞❡ ✉♠❛ ♣❛rtí❝✉❧❛ ❡s❢ér✐❝❛✳ n = 0

        ❉❡st❛ ❢♦r♠❛✱ s✉❜st✐t✉✐♥❞♦ d ❡ ✉t✐❧✐③❛♥❞♦ ❛ r❡❧❛çã♦ ❞✐❢❡r❡♥❝✐❛❧ ❝♦♠♦ ′ ′ 2 2b [j (k a)j (k a)]+c a)j (k a)+[(k n n ℓ ℓ ℓ n s s s n s

        a) (k a)−(k n {2(k n −2n(n+1)+2]j a)} = 0. ✭✸✳✸✺✮ ❆ss✐♠✱ ❛ ❛♣❧✐❝❛çã♦ ❞❛s ❝♦♥❞✐çõ❡s ❞❡ ❝♦♥t♦r♥♦ ♣r♦❞✉③ ✉♠ ❝♦♥❥✉♥t♦ ❞❡ três ❡q✉❛çõ❡s

        ❛❧❣é❜r✐❝❛s ❧✐♥❡❛r❡s q✉❡ ♣♦❞❡♠ s❡r ❡①♣r❡ss❛s ❞❛ s❡❣✉✐♥t❡ ♠❛♥❡✐r❛✿             d d d s e 11 12 13 n 1 d d d b = e . 21 22 23 n 2 ✭✸✳✸✻✮ d d c 32 33 n n

        P♦rt❛♥t♦✱ ♦❜t❡♠♦s ♦ ❝♦❡✜❝✐❡♥t❡ ❞❡ ❡s♣❛❧❤❛♠❡♥t♦ ❡s❝❛❧❛r s ✉t✐❧✐③❛♥❞♦ ❛ r❡❣r❛ ❞❡ ❈r❛✲ ♠❡r ♣❛r❛ r❡s♦❧✈❡r ♦ s✐st❡♠❛ ❧✐♥❡❛r ❞❛ s❡❣✉✐♥t❡ ❢♦r♠❛✿     1     e d d d d d 1 12 13 11 12 13 s = e d d d d d , n 2 22 23 21 22 23

        ❞❡t ❞❡t ✭✸✳✸✼✮ d d d d 32 33 32 33 ♦♥❞❡ ♦s ❡❧❡♠❡♥t♦s ❞❛ ♠❛tr✐③ sã♦ ❞❛❞♦s ♥❛ t❛❜❡❧❛ ➱ ✐♠♣♦rt❛♥t❡ r❡ss❛❧t❛r q✉❡ ❡ss❛ é ❛ s♦❧✉çã♦ ❣❡r❛❧ s❡♠ ❛♣r♦①✐♠❛çõ❡s ❞♦ ♣r♦❜❧❡♠❛ ❞❡ ❡s♣❛❧❤❛♠❡♥t♦ ♣♦r ✉♠❛ ❡s❢❡r❛ ✈✐s❝♦❡✲ ❧ást✐❝❛✳

        ✸✳✷ ▲✐♠✐t❡ ❞❡ ❘❛②❧❡✐❣❤

        ❊s♣❛❧❤❛♠❡♥t♦ ❞❡ ✉♠❛ ♦♥❞❛ ♣♦r ✉♠❛ ♣❡q✉❡♥❛ ♣❛rtí❝✉❧❛ ❞❡ r❛✐♦ ♠✉✐t♦ ♠❡♥♦r ❞♦ q✉❡ ♦ ❝♦♠♣r✐♠❡♥t♦ ❞❡ ♦♥❞❛ é ❝♦♥❤❡❝✐❞♦ ❝♦♠♦ ❡s♣❛❧❤❛♠❡♥t♦ ❘❛②❧❡✐❣❤✳ ◆❛ ❛♥á❧✐s❡ ❞❡ ❡s♣❛✲ ❧❤❛♠❡♥t♦ ❘❛②❧❡✐❣❤✱ é ♥❛t✉r❛❧ ❞❡✜♥✐r ✉♠ ♣❛râ♠❡tr♦ ❞❡ ❡s❝❛❧❛ ❡♠ t❡r♠♦s ❞♦ t❛♠❛♥❤♦ ❞❛ ♣❛rtí❝✉❧❛✱ ε = ka ≪ 1✳ ❯s✉❛❧♠❡♥t❡✱ ❛ ❡①♣❛♥sã♦ ❞❡ ♦♥❞❛s ♣❛r❝✐❛✐s ❞❛ ♦♥❞❛ ❡s♣❛❧❤❛❞❛ ❞❛❞❛ ♥❛ ❊q✳ é r❡❞✉③✐❞❛ ♣❛r❛ t❡r♠♦s q✉❡ ❡♥✈♦❧✈❡♠ s♦♠❡♥t❡ ♦s ❝♦❡✜❝✐❡♥t❡s ❞❡ ❡s♣❛❧❤❛♠❡♥t♦ 1

        ❞❡ ♠♦♥♦♣♦❧♦ s ❡ ❞✐♣♦❧♦ s P❛r❛ ❛♥❛❧✐s❛r ❛ ❢♦rç❛ ❡ t♦rq✉❡ ❞❡ r❛❞✐❛çã♦ ❛❝úst✐❝♦ s♦❜r❡ ♣❛rtí❝✉❧❛s ♥♦ ❧✐♠✐t❡ ❞❡ ❡s♣❛✲ 1

        ❧❤❛♠❡♥t♦ ❘❛②❧❡✐❣❤✱ ❡①♣❛♥❞✐♠♦s ♦s ❝♦❡✜❝✐❡♥t❡s ❞❡ ❡s♣❛❧❤❛♠❡♥t♦ s ❡ s ♥❛ ❊q✳ ❉❡st❛ ❢♦r♠❛✱

        ❚❛❜❡❧❛ ✸✳✶✿ ❈♦❡✜❝✐❡♥t❡s ❞❡ ❡s♣❛❧❤❛♠❡♥t♦ ❞❛❞♦s ♥❛ ❊q✳ sã♦ ♦❜t✐❞♦s r❡s♦❧✈❡♥❞♦ ♦ s✐st❡♠❛ ❞❡ ❡q✉❛çõ❡s ❧✐♥❡❛r❡s ♣❛r❛ ❝♦♥❞✐çõ❡s ❞❡ ❝♦♥t♦r♥♦ ♥❛ s✉♣❡r❢í❝✐❡ ❞❛ ♣❛rtí❝✉❧❛✳ ❖s ❊❧❡♠❡♥t♦s ❞❛ ♠❛tr✐③ ❞❡ss❛ ❡q✉❛çã♦ sã♦ ❞❛❞♦s ❛q✉✐✳ e 1

        (ka) ✐(ka/ω)j n 2 2 e /ρ ω)k a j (ka) 2 1 n

        −✐(ρ s (1) d 11 n (ka) −✐(ka/ω)h d k aj (k 12 ℓ ℓ n

        a) d n(n + 1)j (k 13 n

        a) 2 2 (1) s d /ρ ω)k a h n (ka) 21 ✐(ρ 1 2 s 2 d aj (k a )]j (k 22 ℓ ℓ n ℓ

        a) −4k n

        a) + [(2n(n + 1) − k s d 2n(n + 1)[k aj (k (k a)] 23 s s n s n

        a) − j d 31 d 32 2[j n (k ℓ ℓ aj (k ℓ a)] 2

        a) − k n d 2k aj (k 33 s s s n s n − 2n(n + 1) + 2]j

        a) + [(k

        a) (k

        a)

        ❋♦♥t❡✿ ❆✉t♦r✱ ✷✵✶✺✳

        t❡♠♦s 2 f f 3 5 6 8 7 s ε ε ε

      • ), = −✐ ✐g − ❖(ε ✐ε ✭✸✳✸✽✮

        3 2

        9 f f 1 3 5 1 6 8 7 s = + + + ε ε ε ), 1 1 ✐ ✐g − ❖(ε ✐ε ✭✸✳✸✾✮

        6

        36 ♦♥❞❡ f ✱ f 1 ✱ g ✱ ❡ g 1 sã♦ ❝♦❡✜❝✐❡♥t❡s ❝♦♠♣❧❡①♦s q✉❡ ❞❡♣❡♥❞❡♠ ❞❛s ♣r♦♣r✐❡❞❛❞❡s ✈✐s❝♦❡✲ ❧ást✐❝❛s ❞❛ ♣❛rtí❝✉❧❛✱ ❡ s❡rã♦ ❞❡t❡r♠✐♥❛❞❛s ♣♦st❡r✐♦r♠❡♥t❡✳ ❆♥t❡❝✐♣❛♠♦s q✉❡ ♣❛r❛ ✉♠ 1 1

        ✢✉✐❞♦ ✐❞❡❛❧ ♦ ❝♦❡✜❝✐❡♥t❡ f é r❡❛❧✱ ❡♥q✉❛♥t♦ q✉❡ ❛ ♣❛rt❡ ✐♠❛❣✐♥ár✐❛ ❞❡ f ✱ g ✱ ❡ g ❡stã♦ r❡❧❛❝✐♦♥❛❞❛s ❝♦♠ ❛s ♣r♦♣r✐❡❞❛❞❡s ❞❡ ❛❜s♦rçã♦ ❞❛ ♣❛rtí❝✉❧❛ ❡s♣❛❧❤❛❞♦r❛ ➱ ✐♠♣♦r✲ 8 7 2 = ) t❛♥t❡ ♥♦t❛r q✉❡ ❛ ♦r❞❡♠ ❞♦ ❝♦❡✜❝✐❡♥t❡ ❞❡ ❡s♣❛❧❤❛♠❡♥t♦ q✉❛❞r✉♣♦❧♦ é s ❖(ε ✐ε ✳

      • ❆ss✐♠✱ ❝♦❡✜❝✐❡♥t❡s ♠✉❧t✐♣♦❧❛r❡s ♠❛✐s ❡❧❡✈❛❞♦s ❝♦♠ n = 2, 3, ... ♣♦❞❡♠ s❡r ❞❡s❝❛rt❛❞♦s ❝♦♠ s❡❣✉r❛♥ç❛ ♥❛ ❛♥á❧✐s❡ ❞❡ ❡s♣❛❧❤❛♠❡♥t♦ ❘❛②❧❡✐❣❤✱ ✐st♦ é✱ ε ≪ 1✳

        , f , g , 1 ➱ út✐❧ ❡s❝r❡✈❡r ❡①♣❧✐❝✐t❛♠❡♥t❡ ❛s ♣❛rt❡s r❡❛✐s ❡ ✐♠❛❣✐♥ár✐❛s ❞♦s ❝♦❡✜❝✐❡♥t❡s f 1

        ✭❘✮

        ❡ g ♥❛s ❊qs✳ ■st♦ é ❢❡✐t♦ ❛tr❛✈és ❞❛ ✐♥❝❧✉sã♦ ❞♦s í♥❞✐❝❡s s✉♣❡r✐♦r❡s

        ✭■✮

        ❡ ♣❛r❛✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✱ ❛ ♣❛rt❡ r❡❛❧ ❡ ✐♠❛❣✐♥ár✐❛ ❞❡ ❝❛❞❛ ❝♦❡✜❝✐❡♥t❡✱ ♣♦r ❡①❡♠♣❧♦✱

        ✭❘✮ ✭■✮

      • f = f

        ✐f ✳ ❆❣♦r❛✱ ♣♦❞❡♠♦s ❡s❝r❡✈❡r ♦s ❝♦❡✜❝✐❡♥t❡s ❞❡ ❡s♣❛❧❤❛♠❡♥t♦ ❡s❝❛❧❛r❡s ❝♦♠♦ 2

        ✭■✮ ✭❘✮ ✭❘✮

        f f f 3 6 3 s = ε ε ε , − − ✐ ✭✸✳✹✵✮

        3

        9

        3 2

        ✭■✮ ✭❘✮ ✭❘✮

        f f f 1 3 ✭■✮ 5 1 6 1 3 s + ε ε ε ε . 1 = − − g 1 − ✐ ✭✸✳✹✶✮

        6

        36

        6 ◆♦t❡ q✉❡ s♦♠❡♥t❡ ♦s t❡r♠♦s ❞❡ ♠❡♥♦r❡s ♦r❞❡♥s ❞❡ ε ❢♦r❛♠ ♠❛♥t✐❞♦s ♥❡ss❛s ❡q✉❛çõ❡s✱

        ❛ss✐♠ ❛ ❝♦♥tr✐❜✉✐çã♦ ❞❡ g ❢♦✐ ♥❡❣❧✐❣❡♥❝✐❛❞❛✳ P❛r❛ ✉♠❛ ♣❛rtí❝✉❧❛ ♥ã♦ ❛❜s♦r✈❡❞♦r❛✱ ♦s 1 1 5 ❝♦❡✜❝✐❡♥t❡s f ✱ f ✱ ❡ g sã♦ q✉❛♥t✐❞❛❞❡s r❡❛✐s✳ ◆❡ss❡ ❝❛s♦✱ ❛ ❝♦♥tr✐❜✉✐çã♦ ❞♦ t❡r♠♦ ε 3 1

        ♣♦❞❡ s❡r ❞❡s♣r❡③❛❞❛ q✉❛♥❞♦ ❝♦♠♣❛r❛❞❛ ❝♦♠ ♦ t❡r♠♦ ❞❡ ε ✳ ❈♦♠♦ f é r❡❛❧ ❡♠ ✉♠ ✢✉✐❞♦

        ( ■)

        ♥ã♦ ✈✐s❝♦s♦ ♦ ❝♦❡✜❝✐❡♥t❡ f é ♥✉❧♦✳ ❯♠❛ ✈❡③ q✉❡ ❡st❛♠♦s ❧✐❞❛♥❞♦ ❝♦♠ ✢✉✐❞♦s ✐❞❡❛✐s ❛q✉✐✱ 1 ❡ss❡ ❝♦❡✜❝✐❡♥t❡ s❡rá ♦♠✐t✐❞♦ ♣♦st❡r✐♦r♠❡♥t❡✳

        ◆❡st❡ ❝❛♣ít✉❧♦✱ ❢♦✐ ❞✐s❝✉t✐❞♦ ❛ t❡♦r✐❛ ❞❡ ❡s♣❛❧❤❛♠❡♥t♦ ❛❝úst✐❝♦ ♣♦r ✉♠❛ ♣❛rtí❝✉❧❛ ❡s❢ér✐❝❛ ✈✐s❝♦❡❧ást✐❝❛✳ ❆❧é♠ ❞✐ss♦✱ ❞❡r✐✈❛♠♦s ♦ ❝♦❡✜❝✐❡♥t❡ ❞❡ ❡s♣❛❧❤❛♠❡♥t♦ ❡s❝❛❧❛r ♣❛r❛ ❛♣r♦①✐♠❛çã♦ ♠♦♥♦♣♦❧♦ ❡ ❞✐♣♦❧♦ q✉❡ s❡rá ✉t✐❧✐③❛❞❛ ♥❛s ❢ór♠✉❧❛s ❞❡ ❢♦rç❛ ❡ t♦rq✉❡ ❞❡ r❛❞✐❛çã♦ ❛❝úst✐❝♦✳ ➱ ✐♠♣♦rt❛♥t❡ r❡ss❛❧t❛r q✉❡ ♦ ❡s♣❛❧❤❛♠❡♥t♦ ❛❝úst✐❝♦ ❡stá ❞✐r❡t❛♠❡♥t❡ r❡❧❛❝✐♦♥❛❞♦ ❝♦♠ ♦ ❢❡♥ô♠❡♥♦ ❞❡ ❢♦rç❛ ❡ t♦rq✉❡ ❞❡ r❛❞✐❛çã♦ ❛❝úst✐❝♦✳

        ◆♦ ♣ró①✐♠♦ ❝❛♣ít✉❧♦✱ ✐r❡♠♦s ❞❡s❝r❡✈❡r ❝♦♠♦ ♦❜t❡r ❡①♣r❡ssõ❡s ❛♥❛❧ít✐❝❛s ♣❛r❛ ❢♦rç❛ ❡ t♦rq✉❡ ❞❡ r❛❞✐❛çã♦ ❛❝úst✐❝♦ q✉❡ ✉t✐❧✐③❛♠ ♦s ❢✉♥❞❛♠❡♥t♦s ❞❡ ❡s♣❛❧❤❛♠❡♥t♦✳

        4

      ❋♦rç❛ ❡ t♦rq✉❡ ❞❡ r❛❞✐❛çã♦ ❛❝úst✐❝♦ ♥♦ ❧✐♠✐t❡

      ❞❡ ❘❛②❧❡✐❣❤

        ❆ ❢♦rç❛ ❞❡ r❛❞✐❛çã♦ ❛❝úst✐❝❛ s✉r❣❡ q✉❛♥❞♦ ✉♠ ❝❛♠♣♦ ❛❝úst✐❝♦ ✐♥t❡r❛❣❡ ❝♦♠ ✉♠ ♦❜❥❡t♦ s✉s♣❡♥s♦ ❡♠ ✉♠ ✢✉✐❞♦✳ ❉❡✈✐❞♦ ❛ ❡ss❛ ✐♥t❡r❛çã♦✱ ♦ ♦❜❥❡t♦ ❛❜s♦r✈❡ ❡✴♦✉ ❡s♣❛❧❤❛ ❛ ♦♥❞❛ ❛❝úst✐❝❛ ♦❝❛s✐♦♥❛♥❞♦ ✉♠❛ tr❛♥s❢❡rê♥❝✐❛ ❞❡ ♠♦♠❡♥t♦ ❞❛ ♦♥❞❛ ♣❛r❛ ♦ ♦❜❥❡t♦✳ ❆ss✐♠✱ ♦ ✢✉①♦ ❞♦ ♠♦♠❡♥t♦ ❧✐♥❡❛r ♠é❞✐♦ ❡ ❛ t❡♥sã♦ q✉❡ ❛t✉❛♠ s♦❜r❡ ♦ ♦❜❥❡t♦ é ❝♦♥❤❡❝✐❞♦ ❝♦♠♦ ❢♦rç❛ ❞❡ r❛❞✐❛çã♦ ❛❝úst✐❝❛✳ ❉❛ ♠❡s♠❛ ❢♦r♠❛✱ ♦ t♦rq✉❡ ❞❡ r❛❞✐❛çã♦ ❛❝úst✐❝♦ ♣♦❞❡ s✉r❣✐r ❞❡✈✐❞♦ ❛ tr❛♥s❢❡rê♥❝✐❛ ❞♦ ♠♦♠❡♥t♦ ❛♥❣✉❧❛r ❞❛ ♦♥❞❛ ♣❛r❛ ♦ ♦❜❥❡t♦✳ ❉❡st❛ ❢♦r♠❛✱ ♦ ✢✉①♦ ❞♦ ♠♦♠❡♥t♦ ❛♥❣✉❧❛r ♠é❞✐♦ ❡①❡r❝✐❞♦ s♦❜r❡ ♦ ♦❜❥❡t♦ é ❞❡♥♦♠✐♥❛❞♦ ❝♦♠♦ t♦rq✉❡ ❞❡ r❛❞✐❛çã♦ ❛❝úst✐❝♦✳ ◆❡st❡ ❝❛♣ít✉❧♦✱ ✈❛♠♦s ❛♣r❡s❡♥t❛r ❡①♣r❡ssõ❡s ❛♥❛❧ít✐❝❛s ♣❛r❛ ❢♦rç❛ ❡ t♦rq✉❡ ❞❡ r❛❞✐❛çã♦ ❛❝úst✐❝♦ ❡①❡r❝✐❞❛ s♦❜r❡ ✉♠❛ ♣❛rtí❝✉❧❛ ❡s❢ér✐❝❛ ❞❡✈✐❞♦ ❛ ✉♠❛ ❢r❡♥t❡ ❞❡ ♦♥❞❛ ❛r❜✐trár✐❛✳

        ✹✳✶ ❊❢❡✐t♦s ♥❡❣❧✐❣❡♥❝✐❛❞♦s

        ◆❡st❡ tr❛❜❛❧❤♦ ♥ã♦ ✐r❡♠♦s ❝♦♥s✐❞❡r❛r ♦s ❡❢❡✐t♦s t❡r♠♦✲✈✐s❝♦s♦s ♣♦r tr❛t❛r ❛♣❡♥❛s ❞❡ ✉♠ ✢✉✐❞♦ ✐❞❡❛❧✳ ❈♦♥s✐❞❡r❡ ✉♠❛ ♦♥❞❛ ❤❛r♠ô♥✐❝❛ ❛❝úst✐❝❛ ❝♦♠ ❢r❡q✉ê♥❝✐❛ ❛♥❣✉❧❛r ω ✐♥t❡r❛✲ ❣✐♥❞♦ ❝♦♠ ✉♠❛ ♣❛rtí❝✉❧❛ ✈✐s❝♦❡❧ást✐❝❛ ❡s❢ér✐❝❛ ❞❡ r❛✐♦ a✱ s✉s♣❡♥s❛ ❡♠ ✉♠ ✢✉✐❞♦ ❤♦♠♦❣ê✲ ♥❡♦ ❞❡ ❞❡♥s✐❞❛❞❡ ρ ❡ ✈❡❧♦❝✐❞❛❞❡ ❞♦ s♦♠ c ✳ ■r❡♠♦s ♥❡❣❧✐❣❡♥❝✐❛r ♦s ❡❢❡✐t♦s t❡r♠♦✲✈✐s❝♦s♦s ♥♦ ♣r♦❝❡ss♦ ❞❡ ❢♦rç❛ ❡ t♦rq✉❡ ❞❡ r❛❞✐❛çã♦ ❛❝úst✐❝♦✳ ❉❡st❛ ❢♦r♠❛✱ ♦ ✢✉✐❞♦ é ❝♦♥s✐❞❡r❛❞♦ ♥ã♦✲✈✐s❝♦s♦✳ ❆ss✐♠✱ é ♥❡❝❡ssár✐♦ ❛ss✉♠✐r q✉❡ ♦ r❛✐♦ ❞❛ ♣❛rtí❝✉❧❛ ❡ ♦ ❝♦♠♣r✐♠❡♥t♦ ❞❡

        = p2D /ω ♦♥❞❛ ✐♥❝✐❞❡♥t❡ λ ❞❡✈❡♠ s❡r ♠✉✐t♦ ♠❛✐♦r q✉❡ ♦s ❧✐♠✐t❡s tér♠✐❝♦s δ t t ❡ ✈✐s❝♦s♦s δ = p2ν /ω

        ✈ ✵ ✐st♦ é✱

        , δ ,

        a, λ ≫ δ t ✈ ✭✹✳✶✮ ♦♥❞❡ D t é ❛ ❞✐❢✉s✐✈✐❞❛❞❡ tér♠✐❝❛ ❡ ν ✵ é ❛ ✈✐s❝♦s✐❞❛❞❡ ❝✐♥ét✐❝❛✳ ❱❛❧♦r❡s tí♣✐❝♦s ❞❡ss❡s ❧✐♠✐t❡s ♥❛ á❣✉❛ ♣❛r❛ ✉♠❛ ♦♥❞❛ ❝♦♠ ❢r❡q✉ê♥❝✐❛ ❞❡ 2 ▼❍③ ❡ ❝♦♠♣r✐♠❡♥t♦ ❞❡ ♦♥❞❛ λ = 0.75 ♠♠ sã♦ δ = 0.15 µ = 0.38 µ

        t ✈

        ♠ ❡ δ ♠✳ P♦rt❛♥t♦✱ ♣♦❞❡♠♦s ♥❡❣❧✐❣❡♥❝✐❛r ♦s ❡❢❡✐t♦s t❡r♠♦✲✈✐s❝♦s♦s ♣❛r❛ ♣❛rtí❝✉❧❛s ❝♦♠ r❛✐♦ a ≫ 3.8 µ♠✳

        ■❧✉str❛çã♦ ❞❛ ✐♥t❡r❛çã♦ ❡♥tr❡ ❛ ♦♥❞❛ ✐♥❝✐❞❡♥t❡ φ ✭❜❛rr❛s ✈❡rt✐❝❛✐s ✈❡r❞❡✲❡s❝✉r♦✮ ❞❡

        ❋✐❣✉r❛ ✹✳✶✿

        ✐♥

      ❝♦♠♣r✐♠❡♥t♦ ❞❡ ♦♥❞❛ λ ❡ ❛ ♣❛rtí❝✉❧❛ ✈✐s❝♦❡❧ást✐❝❛ ✭❝ír❝✉❧♦ ✈❡r♠❡❧❤♦✮ ❞❡ r❛✐♦ a s✉s♣❡♥s❛ ❡♠ ✉♠

      ✢✉✐❞♦ ✐❞❡❛❧✳ ❆ ♦♥❞❛ ❡s♣❛❧❤❛❞❛ ❞❡✈✐❞♦ ❡ss❛ ✐♥t❡r❛çã♦ é ❞❡♥♦t❛❞❛ ♣♦r φ ✭❛r❝♦s ✈❡r❞❡ ❝❧❛r♦✮✳ s❝ n

      dS

      a

        ❋♦♥t❡✿ ❆✉t♦r✱ ✷✵✶✺✳ ✹✳✷ ❋♦rç❛ ❞❡ r❛❞✐❛çã♦ ❛❝úst✐❝❛ ♥♦ ❧✐♠✐t❡ ❞❡ ❘❛②❧❡✐❣❤ ✹✳✷✳✶ ❚❡♥s♦r t❡♥sã♦ ❞❡ r❛❞✐❛çã♦

        ❙✉♣♦♥❤❛ q✉❡ ✉♠❛ ♦♥❞❛ ✐♥❝✐❞❡♥t❡ ❛t✐♥❣❡ ✉♠❛ ♣❡q✉❡♥❛ ♣❛rtí❝✉❧❛ ❡s❢ér✐❝❛ ❞❡ s✉♣❡r❢í❝✐❡

        r❛❞

        S ❝♦♠♦ ♣♦❞❡ s❡r ✈✐st♦ ♥❛ ✜❣✉r❛ ❆ ❢♦rç❛ ❞❡ r❛❞✐❛çã♦ ❛❝úst✐❝❛ ♠é❞✐❛ F ❡①❡r❝✐❞❛

        ♣❡❧❛ ♦♥❞❛ s♦❜r❡ ❛ ♣❛rtí❝✉❧❛ é ❡♥❝♦♥tr❛❞❛ ✐♥t❡❣r❛♥❞♦ ♦ t❡♥s♦r ❞❡ t❡♥sã♦ S q✉❡ ❛❣❡ s♦❜r❡ ❛ ♣❛rtí❝✉❧❛✱ ♦✉ s❡❥❛✱ Z

        r❛❞ S 2 F r

        = , S · n❞ ✭✹✳✷✮ ♦♥❞❡ ❛ ❜❛rr❛ ❞❡♥♦t❛ ♠é❞✐❛ t❡♠♣♦r❛❧✳ ❆ ♠é❞✐❛ t❡♠♣♦r❛❧ ❞❡ ✉♠❛ ❢✉♥çã♦ f ❛♦ ❧♦♥❣♦ ❞❡ ✉♠ ♣❡rí♦❞♦ ❞❡ ♦s❝✐❧❛çã♦ ❝♦♠♣❧❡t♦ 2π/ω é ❞❡✜♥✐❞♦ ❞❛ s❡❣✉✐♥t❡ ❢♦r♠❛✱ Z 2π/ω

        ω f = f (t) ❞t. ✭✹✳✸✮

        2π ❉❡st❛ ❢♦r♠❛✱ ♣r❡❝✐s❛♠♦s ❡♥❝♦♥tr❛r ♦ t❡♥s♦r ❞❡ t❡♥sã♦ ♠é❞✐♦ ❞❡✈✐❞♦ ❛ ✐♥t❡r❛çã♦ ❞❛

        ♦♥❞❛ ❛❝úst✐❝❛ ❝♦♠ ❛ ♣❛rtí❝✉❧❛✳ ➱ ❝♦♥✈❡♥✐❡♥t❡ r❡❧❡♠❜r❛r q✉❡ ❞❡✜♥✐♠♦s ♦ t❡♥s♦r ❞❡ t❡♥sã♦ ❞❡ r❛❞✐❛çã♦ ❛♦ ❞❡♠♦♥str❛r ❛ ❡q✉❛çã♦ ❞❡ ❝♦♥s❡r✈❛çã♦ ❞♦ ♠♦♠❡♥t♦ ❊q✳ ❞❛ s❡❣✉✐♥t❡ ❢♦r♠❛✿ S = pI + ρvv.

        ✭✹✳✹✮ ◆♦r♠❛❧♠❡♥t❡✱ ♦s ❝❛♠♣♦s ❞❡ s❡❣✉♥❞❛ ♦r❞❡♠ s❡r✐❛♠ ♥❡❣❧✐❣❡♥❝✐❛❞♦s ❝♦♠♣❛r❛❞♦s ❝♦♠

        ♦s ❝❛♠♣♦s ❞❡ ♣r✐♠❡✐r❛ ♦r❞❡♠✳ ◆♦ ❡♥t❛♥t♦✱ s❡ ♦s ❝❛♠♣♦s ❧✐♥❡❛r❡s tê♠ ✉♠❛ ❞❡♣❡♥❞ê♥❝✐❛ ❤❛r♠ô♥✐❝❛ ♥♦ t❡♠♣♦✱ ❡❧❡s ♥ã♦ ❝♦♥tr✐❜✉❡♠ ♣❛r❛ ❛ ♠é❞✐❛ t❡♠♣♦r❛❧✳ P♦r ❡①❡♠♣❧♦✱ ❛ ♣❛rt✐r ❞❛ ❊q✳ ♦❜t❡♠♦s cos ωt = 0 ♦✉ sin ωt = 0✳ ◆♦ ❡♥t❛♥t♦✱ ❛ ♠é❞✐❛ t❡♠♣♦r❛❧ ❞❡ t❡r♠♦s ❝♦♥t❡♥❞♦ ♣r♦❞✉t♦s ❞❡ ❢✉♥çõ❡s ❤❛r♠ô♥✐❝❛s ♥♦ t❡♠♣♦ ❞❡ ♣r✐♠❡✐r❛ ♦r❞❡♠ ♥ã♦ é ♥✉❧❛✱ ♣♦r 2

        ωt = 1/2 ❡①❡♠♣❧♦ cos ✳ P♦rt❛♥t♦✱ ❛ ❢♦rç❛ ❞❡ r❛❞✐❛çã♦ ❛❝úst✐❝❛ é ♥♦ ♠í♥✐♠♦ ✉♠ ❡❢❡✐t♦ ❞❡ s❡❣✉♥❞❛ ♦r❞❡♠✳ ❉❡st❛ ❢♦r♠❛✱ é ♥❡❝❡ssár✐♦ ❛♣r❡s❡♥t❛r ❛❧❣✉♠❛s ♣r♦♣r✐❡❞❛❞❡s ❞❛s ♠é❞✐❛s t❡♠♣♦r❛✐s ❞❡ ♣r♦❞✉t♦s ❞❡ ❢✉♥çõ❡s ❤❛r♠ô♥✐❝❛s ❡ ❝♦♠♣❧❡①❛s✳ ❊♠ ❣❡r❛❧✱ ❡st❛♠♦s ✐♥t❡r❡ss❛❞♦ ❡♠ ♠é❞✐❛ t❡♠♣♦r❛❧ ❞❡ q✉❛♥t✐❞❛❞❡s ❞❡ s❡❣✉♥❞❛ ♦r✲

        ❞❡♠ q✉❡ s❡rã♦ ✉s❛❞♦s ❡①t❡♥s✐✈❛♠❡♥t❡ ♥❛s ♣ró①✐♠❛s s❡çõ❡s✳ ❈♦♥s✐❞❡r❡ q✉❡ ❞✉❛s ❢✉♥çõ❡s ❤❛r♠ô♥✐❝❛s ❞❡ ♣r✐♠❡✐r❛ ♦r❞❡♠ sã♦ ❞❡✜♥✐❞❛s ❞❛ s❡❣✉✐♥t❡ ❢♦r♠❛✱

        ✐ωt

        f (t) = ], 1 ❘❡[f ❡ ✭✹✳✺✮

        ✐ωt

        g(t) = ], 1 1 1 ❘❡[g ❡ ✭✹✳✻✮ ♦♥❞❡ f ❡ g sã♦ ❛♠♣❧✐t✉❞❡s ❞❛s ❢✉♥çõ❡s ❡ “❘❡” é ❛ ♣❛rt❡ r❡❛❧ ❞❡ ✉♠❛ q✉❛♥t✐❞❛❞❡ ❝♦♠♣❧❡①❛✳ ❆ ♠é❞✐❛ t❡♠♣♦r❛❧ ❞♦ ♣r♦❞✉t♦ ❞❛s ❢✉♥çõ❡s s♦❜r❡ ✉♠ ♣❡rí♦❞♦ ❝♦♠♣❧❡t♦ T é ❞❡✜♥✐❞♦ ❝♦♠♦ Z T

        1

        1 f (t)g(t) = f (t)g(t) g ], 1 ❞t = ❘❡[f 1 ✭✹✳✼✮ (∗) T

        2 ♦♥❞❡ ♦ ❛st❡r✐s❝♦ ✐♥❞✐❝❛ ❝♦♠♣❧❡①♦ ❝♦♥❥✉❣❛❞♦✳

        ❆ t❡♦r✐❛ ❞❡ ❢♦rç❛ ❞❡ r❛❞✐❛çã♦ ❞❡♣❡♥❞❡ ❞❛ ❡①♣❛♥sã♦ ❞❡ s❡❣✉♥❞❛ ♦r❞❡♠ ❞♦s ❝❛♠♣♦s ❛❝úst✐❝♦s ♥♦ ✢✉✐❞♦✳ ❊st❛s ❡①♣❛♥sõ❡s sã♦ ✈✐st❛s ❞❡ ❢♦r♠❛ ❣❡r❛❧ ♥❛s ❊qs✳ P❛r❛ ❛ ❛♣r♦①✐♠❛çã♦ ❞❡ s❡❣✉♥❞❛ ♦r❞❡♠✱ ❡♥❝♦♥tr❛♠♦s q✉❡ ♦ t❡♥s♦r t❡♥sã♦ ❞❡ r❛❞✐❛çã♦ é ❞❛❞♦ ♣♦r S (2) (2) (1) (1) I v v

        = p + ρ , (2) (1) ✭✹✳✽✮ ♦♥❞❡ p é ❛ ♣r❡ssã♦ ❞❡ s❡❣✉♥❞❛ ♦r❞❡♠✱ I é ♦ t❡♥s♦r ✉♥✐tár✐♦ ❡ v é ❛ ✈❡❧♦❝✐❞❛❞❡ ❞❡ ♣r✐♠❡✐r❛ ♦r❞❡♠✳ ❖s ❝❛♠♣♦s ❛❝úst✐❝♦s ♥❛ ❛♣r♦①✐♠❛çã♦ ❞❡ s❡❣✉♥❞❛ ♦r❞❡♠ ♥❛s ❊qs✳ t♦r♥❛♠✲s❡ (1) 2 (2) p = p + M p + M p , (1) 2 (2) ✭✹✳✾✮

        ρ = ρ + M ρ + M ρ , (1) 2 (2) ✭✹✳✶✵✮ v = M v + M v .

        ✭✹✳✶✶✮ ❈♦♠❜✐♥❛♥❞♦ ❛s ❊qs✳ ♦❜t❡♠♦s ❛ ❡q✉❛çã♦ ❞❡ s❡❣✉♥❞❛ ♦r❞❡♠ ❞♦ ♠♦✈✐♠❡♥t♦ ♣❛r❛ ♦ ❝❛♠♣♦ ❛❝úst✐❝♦ (2) (2) (1) (1) (1) (1) ρ ∂ v ∂ v (v . t t

        = −∇p − ρ − ρ · ∇)v ✭✹✳✶✷✮ ◆♦t❡ q✉❡ ♦s ❞♦✐s ú❧t✐♠♦s t❡r♠♦s ❞❛ ❊q✳ sã♦ ♣r♦❞✉t♦s ❞❡ ❞♦✐s ❝❛♠♣♦s ❞❡ ♣r✐♠❡✐r❛ ♦r❞❡♠✳ ❈♦♥s✐❞❡r❛♥❞♦ ❛ ♠é❞✐❛ t❡♠♣♦r❛❧ ❞❛ ❊q✳ ❜t❡♠♦s (2) (1) (1) (1) (1) ∂ v (v . t

        ∇p = −ρ − ρ · ∇)v ✭✹✳✶✸✮ ❆ss✐♠✱ ♣♦❞❡♠♦s ❡s❝r❡✈❡r ❛ ❊q✳ ❡♠ t❡r♠♦s ❞♦ ♣♦t❡♥❝✐❛❧ ❞❡ ✈❡❧♦❝✐❞❛❞❡✱ ❢❛③❡♥❞♦ (1) v

        = ∇φ✱ (2) (1)t

        ∇p = −ρ ∇φ − ρ ∇φ · ∇∇φ, ✭✹✳✶✹✮

        ♦♥❞❡✱ (1) 2 ! 1 v ,

        ✭✹✳✶✺✮ ∇φ · ∇∇φ = ∇(∇φ · ∇φ) = ∇

        2

        2 ✉t✐❧✐③❛♥❞♦ ❛s ❡①♣r❡ssõ❡s ❞❛ ❛♣r♦①✐♠❛çã♦ ❞❡ ♣r✐♠❡✐r❛ ♦r❞❡♠ ❊qs✳ ❡♥❝♦♥✲ tr❛♠♦s ❛ ♣r❡ssã♦ ❞❡ s❡❣✉♥❞❛ ♦r❞❡♠ ❡♠ t❡r♠♦s ♣r❡ssã♦ ❡ ✈❡❧♦❝✐❞❛❞❡ ❞♦ ✢✉✐❞♦ ❞❡ ♣r✐♠❡✐r❛ ♦r❞❡♠✱ (1) (1) 2 2 (2) p ρ v p = ,

        ✭✹✳✶✻✮ 2(1) (1) (1) 2 2ρ c

        2 = v

        ♦♥❞❡ v · v ✳ ◆♦t❡ q✉❡ ♦ ♣r✐♠❡✐r♦ ❡ ♦ s❡❣✉♥❞♦ t❡r♠♦ ❞♦ ❧❛❞♦ ❞✐r❡✐t♦ ❞❡ss❛ ❡q✉❛çã♦ é ❛ ❡♥❡r❣✐❛ ♣♦t❡♥❝✐❛❧ ❡ ❝✐♥ét✐❝❛✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✳ ❋✐♥❛❧♠❡♥t❡✱ s✉❜st✐t✉✐♥❞♦ ❛ ❊q✳ ♦❜t❡♠♦s ❛ t❡♥sã♦ ❞❡ r❛❞✐❛çã♦ ❞❡ s❡❣✉♥❞❛ ♦r❞❡♠✱ (1) (1) 2 2 ! S (2) (1) (1) p ρ v I v v

        = + ρ , 2 − ✭✹✳✶✼✮

        2ρ c

        2 ♦✉✱ S (2) (1) (1) v v ,

        = −LI + ρ ✭✹✳✶✽✮ ♦♥❞❡ ♦s t❡r♠♦s ❞❡♥tr♦ ❞♦ ♣❛rê♥t❡s❡s ❞❛ ❊q✳ é ✐❞❡♥t✐✜❝❛❞♦ ❝♦♠♦ ❞❡♥s✐❞❛❞❡ ▲❛❣r❛♥✲ ❣✐❛♥❛ ❛❝úst✐❝❛ L✱ ✈✐st♦ q✉❡ ♦ ♣r✐♠❡✐r♦ t❡r♠♦ é ❛ ❡♥❡r❣✐❛ ♣♦t❡♥❝✐❛❧ ❡ ♦ s❡❣✉♥❞♦ t❡r♠♦ é ❛ ❡♥❡r❣✐❛ ❝✐♥ét✐❝❛✳

        ✹✳✷✳✷ ▼ét♦❞♦ ❞❡ ❝❛♠♣♦ ❞✐st❛♥t❡

        ❆ ❢♦rç❛ ❞❡ r❛❞✐❛çã♦ ❛❝úst✐❝❛ ♠é❞✐❛ ♣♦❞❡ s❡r ❡s❝r✐t❛ ✉t✐❧✐③❛♥❞♦ ♦ t❡♥s♦r ❞❡ t❡♥sã♦ ♠é❞✐♦ ❊q✳ ❞❛ s❡❣✉✐♥t❡ ♠❛♥❡✐r❛✿ Z " ! # (1) (1) 2 2 p ρ v

        r❛❞ I (1) (1) 2 F v v r = + ρ .

        ✭✹✳✶✾✮ 2 − · n❞ S 2ρ c

        2 ❖s ❝❛♠♣♦s ❡s♣❛❧❤❛❞♦s ❞❡ ♣r✐♠❡✐r❛ ♦r❞❡♠ ❢♦r❛♠ ❞❡t❡r♠✐♥❛❞♦s ♣❛r❛ ✉♠ ❞❛❞♦ ❝❛♠♣♦

        ✐♥❝✐❞❡♥t❡ ❞❡ ♣r✐♠❡✐r❛ ♦r❞❡♠ ❛tr❛✈és ❞♦ ♣r♦❜❧❡♠❛ ❞❡ ❡s♣❛❧❤❛♠❡♥t♦ ❢♦✐ ❛♣r❡s❡♥t❛❞♦ ♥♦ ❝❛♣ít✉❧♦ ❆ ❢♦rç❛ ❞❡ r❛❞✐❛çã♦ ❛❝úst✐❝❛ s♦❜r❡ ✉♠❛ ♣❛rtí❝✉❧❛ é ❡♥tã♦ ❝❛❧❝✉❧❛❞❛ ❝♦♥❤❡❝❡♥❞♦ ❛ ♣r❡ssã♦ t♦t❛❧ ✭♦♥❞❛s ✐♥❝✐❞❡♥t❡ ❡ ❡s♣❛❧❤❛❞❛✮ ❡ ❛ ✈❡❧♦❝✐❞❛❞❡ ❞♦ ✢✉✐❞♦ ❛❣✐♥❞♦ s♦❜r❡ ❛ s✉♣❡r❢í❝✐❡ ❞❛ ♣❛rtí❝✉❧❛✳

        ❈♦♠♦ ♥ã♦ ❡①✐st❡ ❢♦rç❛s ✈♦❧✉♠étr✐❝❛s ♥❡st❡ ♣r♦❜❧❡♠❛✱ ❞❡ ❛❝♦r❞♦ ❝♦♠ ❛ ❝♦♥s❡r✈❛çã♦ ❞♦ ♠♦♠❡♥t♦ ♣❛r❛ ✢✉✐❞♦ ✐❞❡❛❧✱ ❊q✳ ♦ ❞✐✈❡r❣❡♥t❡ ❞♦ t❡♥s♦r t❡♥sã♦ ❞❡ r❛❞✐❛çã♦ s❡rá ③❡r♦ (2)

        = 0 ✭∇ · S ✮✱ ❞❡✈✐❞♦ ❛ ❡st❡ ❢❛t♦✱ é ♣♦ssí✈❡❧ ❡①♣r❡ss❛r ❛ ❢♦rç❛ s♦❜r❡ ✉♠ ♦❜❥❡t♦ ❝♦♠♦ ❛ ✐♥t❡❣r❛❧ s♦❜r❡ ✉♠❛ s✉♣❡r❢í❝✐❡ ❢❡❝❤❛❞❛ q✉❡ ❡♥❣❧♦❜❡ ♦ ♠❡s♠♦✱ ❞❡♥♦♠✐♥❛❞❛ ❞❡ s✉♣❡r❢í❝✐❡ ❞❡

        ❝

        ❝♦♥tr♦❧❡ S ✳ ❈♦♥s✐❞❡r❛♥❞♦ ♦ r❡s✉❧t❛❞♦ ❞❛ ❛♥á❧✐s❡ ❞❛ t❡♦r✐❛ ❞❡ ❡s♣❛❧❤❛♠❡♥t♦✱ é ✈❛♥t❛❥♦s♦ ❛ ❡s❝♦❧❤❛ ❞❛ ❛♣r♦①✐♠❛çã♦ ❞❡ ❝❛♠♣♦ ❞✐st❛♥t❡✱ kr ≫ 1✳ ◆❡ss❛ r❡❣✐ã♦ ❞❡ ❝❛♠♣♦ ❞✐st❛♥t❡ r❡♣r❡s❡♥t❛❞❛ ♣❡❧♦ ❝ír❝✉❧♦ tr❛❝❡❥❛❞♦ ♥❛ ✜❣✉r❛ ♦s ❝á❧❝✉❧♦s sã♦ s✐❣♥✐✜❝❛♥t❡♠❡♥t❡ s✐♠✲ ♣❧✐✜❝❛❞♦s ❞❡✈✐❞♦ ♦s ♣♦t❡♥❝✐❛✐s ❞❡ ✈❡❧♦❝✐❞❛❞❡ φ ✐♥ ❡ φ s❝ s❡r❡♠ ❡①♣r❡ss♦s ♣❡❧❛s ❊qs✳ ❡ r❡s♣❡❝t✐✈❛♠❡♥t❡✳

        ❙✐st❡♠❛ ❞❡ ❝♦♦r❞❡♥❛❞❛s ♣❛r❛ ❝♦♠♣✉t❛r ❛ ❢♦rç❛ ❞❡ r❛❞✐❛çã♦ ❛❝úst✐❝❛ ❡♠ ✉♠❛ ♣❛rtí✲

        ❋✐❣✉r❛ ✹✳✷✿

        ❝✉❧❛ ❡s❢ér✐❝❛ ❞❡ r❛✐♦ a ✉t✐❧✐③❛♥❞♦ ✉♠❛ s✉♣❡r❢í❝✐❡ ❞❡ ❝♦♥tr♦❧❡ S ❝ ✭❧✐♥❤❛s tr❛❝❡❥❛❞❛s✮ ❞❡ r❛✐♦ R ❝ S c n R c dS

      x

      S z y a

        ❋♦♥t❡✿ ❆✉t♦r✱ ✷✵✶✺✳

        ❈♦♥s✐❞❡r❡ ❛ ❝♦♥✜❣✉r❛çã♦ ❞❛❞❛ ♥❛ ✜❣✳ ✱ ♦♥❞❡ ❛ s✉♣❡r❢í❝✐❡ ❞❡ ❝♦♥tr♦❧❡ S ❝ r❡♣r❡s❡♥✲ t❛❞❛ ♣❡❧♦ ❝ír❝✉❧♦ tr❛❝❡❥❛❞♦ ❡ ❝♦♠ r❛✐♦ R ❝ ❡stá ❝❡♥tr❛❞❛ ♥❛ ♦r✐❣❡♠ ❞♦ s✐st❡♠❛ ❞❡ ❝♦♦r❞❡✲ ♥❛❞❛s✳ P❛r❛ ❡♥❝♦♥tr❛r ❛ ❢♦rç❛ s♦❜r❡ ❛ s✉♣❡r❢í❝✐❡ ❞❡ ❝♦♥tr♦❧❡✱ ✐r❡♠♦s ✐♥t❡❣r❛r ♦ ❞✐✈❡r❣❡♥t❡ ❞❛ t❡♥sã♦ ❞❡ r❛❞✐❛çã♦ ❞❡ s❡❣✉♥❞❛ ♦r❞❡♠ s♦❜r❡ ♦ ✈♦❧✉♠❡ ❞❡ ❝♦♥tr♦❧❡ V ❝ ✱ Z Z Z 3 2 2 (2) (2) (2) r = r r = 0. S S V S S

        ∇ · S ❞ · n❞ · n❞ ✭✹✳✷✵✮ ◆♦t❡ q✉❡ ✉t✐❧✐③❛♠♦s ♦ t❡♦r❡♠❛ ❞❛ ❞✐✈❡r❣ê♥❝✐❛ ♣❛r❛ ❡♥❝♦♥tr❛r ❡ss❛ ❡q✉❛çã♦✳ P♦rt❛♥t♦✱ ❡♥❝♦♥tr❛♠♦s q✉❡ ❛ ❢♦rç❛ ❞❡ r❛❞✐❛çã♦ ❛❝úst✐❝❛ ♣♦❞❡ s❡r r❡❡s❝r✐t❛ ✉t✐❧✐③❛♥❞♦ ❛ ❊q✳ ❞❛ s❡❣✉✐♥t❡ ❢♦r♠❛✿ Z Z

      • ❝ ❝

        r❛❞ (2) S 2 (1) (1) 2 F r v v r

        = , ✭✹✳✷✶✮

        = − · n❞ LI − ρ · n❞ S S ❝ ❝ 2 r ✉t✐❧✐③❛♥❞♦ ❛ ❡q✉❛çã♦ q✉❡ r❡❧❛❝✐♦♥❛ ♦ â♥❣✉❧♦ só❧✐❞♦ ❞Ω ❝♦♠ ♦ ❡❧❡♠❡♥t♦ ❞❡ ár❡❛ ❞ ✱ 2 2 r r /R

        ❞Ω = e · ❞ ❝ ✱ ♦❜t❡♠♦s 2 Z

        r❛❞ (1) (1)

        F = R v v r LI − ρ · e ❞Ω, ✭✹✳✷✷✮

        ❝ r

        ♦♥❞❡ ❞Ω = sin θ❞θ❞ϕ é ♦ â♥❣✉❧♦ só❧✐❞♦ ❞✐❢❡r❡♥❝✐❛❧ ❡♠ ❝♦♦r❞❡♥❛❞❛s ❡s❢ér✐❝❛s ❡ e é ♦ ✈❡t♦r ✉♥✐tár✐♦ ♥❛ ❞✐r❡çã♦ r❛❞✐❛❧ ❞❛❞♦ ♣♦r e r = cos ϕ sin θe x + sin ϕ sin θe y + cos θe z . (1) (1)

        ✭✹✳✷✸✮ ❆ ♣r❡ssã♦ t♦t❛❧ p ❡ ❛ ✈❡❧♦❝✐❞❛❞❡ v ♥♦ ✢✉✐❞♦ ✐❞❡❛❧ sã♦ ❞❛❞♦s ♣❡❧❛ s♦♠❛ ❞♦s ❝❛♠♣♦s

        ✐♥❝✐❞❡♥t❡s ❡ ❡s♣❛❧❤❛❞♦s ❞❛ s❡❣✉✐♥t❡ ♠❛♥❡✐r❛✿ (1) ✐ωt

      (1) −

      ✐♥ s❝ ❡ ✭✹✳✷✹✮ ✐ωt v = (v + v ) ,

        ✐♥ s❝ ❡ ✭✹✳✷✺✮

        ♦♥❞❡ p ✐♥✭s❝✮ é ❛ ♣r❡ssã♦ ✐♥❝✐❞❡♥t❡ ✭❡s♣❛❧❤❛❞❛✮ ❡ v ✐♥✭s❝✮ é ❛ ✈❡❧♦❝✐❞❛❞❡ ✐♥❝✐❞❡♥t❡ ✭❡s♣❛❧❤❛❞❛✮✳ ❉❡st❛ ❢♦r♠❛✱ ❛ ❞❡♥s✐❞❛❞❡ ▲❛❣r❛♥❣✐❛♥❛ ❡ ♦ ✢✉①♦ ❞❡ ♠♦♠❡♥t♦ sã♦ ❡①♣r❡ss♦s ❡♠ t❡r♠♦s

        ❞❛s ♦♥❞❛s ✐♥❝✐❞❡♥t❡s ❡ ❡s♣❛❧❤❛❞❛s✱ 2 2 ρ v p ρ v p p p

        ✐♥ ✐♥ s❝ s❝ ✐♥ s❝

        · v · v

        ✐♥ s❝

      • ✐♥ s❝
      • (1) (1) 2 2ρ c 2 2ρ c ρ c v v v
      • ρ v L = − − · v − ✭✹✳✷✻✮
      • 2 2 2

          ρ = ρ (v + v + v + v ) ,

          ✐♥ ✐♥ s❝ s❝ ✐♥ s❝ s❝ ✐♥ ✭✹✳✷✼✮

          ♦♥❞❡ ♦ ♣r✐♠❡✐r♦ ❡ ♦ s❡❣✉♥❞♦ ♣❛rê♥t❡s❡s ❞❛ sã♦ ❛s ❝♦♥tr✐❜✉✐çõ❡s ❞❛s ♦♥❞❛s ✐♥❝✐✲ ❞❡♥t❡ ❡ ❡s♣❛❧❤❛❞❛s ♣❛r❛ ❛ ▲❛❣r❛♥❣✐❛♥❛✱ ❡♥q✉❛♥t♦ q✉❡ ♦ t❡r❝❡✐r♦ t❡r♠♦ é ❛ ▲❛❣r❛♥❣✐❛♥❛ ❞❡ ✐♥t❡r❛çã♦✳

          ❖❜s❡r✈❡ q✉❡ ❛s ❝♦♥tr✐❜✉✐çõ❡s s♦♠❡♥t❡ ❞❡ ❝❛♠♣♦s ✐♥❝✐❞❡♥t❡s ♥ã♦ ❝♦♥té♠ ✐♥❢♦r♠❛çã♦ s♦❜r❡ ♦ ❡s♣❛❧❤❛♠❡♥t♦✱ ♣♦rt❛♥t♦ ❞❡✈❡♠ s❡r ③❡r♦✳ ❆❧é♠ ❞✐ss♦✱ ❛ s✉♣❡r❢í❝✐❡ ❡s❢ér✐❝❛ ❞❡

          ❝

          ❝♦♥tr♦❧❡ ♣♦❞❡ s❡r ❝♦♥s✐❞❡r❛❞❛ s✉✜❝✐❡♥t❡♠❡♥t❡ ❞✐st❛♥t❡ ❞♦ ♦❜❥❡t♦ ✭R → ∞✮✱ ❞❡ ♠♦❞♦ q✉❡ ♦ s❡❣✉♥❞♦ t❡r♠♦ ❞❛ s❡❥❛ ③❡r♦✱ ✉♠❛ ✈❡③ q✉❡ ❛ ♦♥❞❛ ❡s♣❛❧❤❛❞❛ s❡rá ❧♦❝❛❧♠❡♥t❡ ✉♠❛ ♦♥❞❛ ♣❧❛♥❛ ♣r♦❣r❡ss✐✈❛✳ ❊♥tã♦✱ ✉t✐❧✐③❛♥❞♦ ❊q✳ ♣♦❞❡♠♦s ❡①♣r❡ss❛r ❛ ❢♦rç❛ ❞❡ r❛❞✐❛çã♦ ♥♦ ❝❛♠♣♦ ❞✐st❛♥t❡ ❝♦♠♦ Z p p

          ✐♥ s❝ r❛❞ 2 F v e v

          = lim R ρ r (v r )

          ✐♥ s❝ s❝ s❝ R →∞ ❝ · v − − ρ · e 2 ρ c

        • v (v ) + v (v )

          ✐♥ s❝ r s❝ ✐♥ r

          · e · e ❞Ω. ✭✹✳✷✽✮ ❯s❛♥❞♦ ❛ ❡q✉❛çã♦ ❞❛ ♠é❞✐❛ ❞♦ ♣r♦❞✉t♦ ❞❡ ❢✉♥çõ❡s✱ ❊q✳ ❡ ❡s❝r❡✈❡♥❞♦ ❛ ♣r❡ssã♦ ❝♦♠♦ p = ωφ

          

        ✐♥✭s❝✮ ✐ρ ✐♥✭s❝✮ ❡ ❛ ✈❡❧♦❝✐❞❛❞❡ ❞♦ ✢✉✐❞♦ v ✐♥✭s❝✮ ✐♥✭s❝✮ ♥❛ ❊q✳ ❡♥❝♦♥tr❛♠♦s

        Z = ∇φ

          ✐ ∗

          r❛❞ 2 2 F

          e lim R φ ∂ φ φ

          ✐♥ r ✐♥ s❝ r

          = −E ❝ ❘❡ − s❝ + |φ | ❞Ω, ✭✹✳✷✾✮ R →∞ ❝ k ♦♥❞❡ φ ✐♥ ❡ φ s❝ sã♦ ♦s ♣♦t❡♥❝✐❛✐s ❞❡ ✈❡❧♦❝✐❞❛❞❡ ✐♥❝✐❞❡♥t❡ ❡ ❡s♣❛❧❤❛❞♦✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✱ 2 2 E = p /(2ρ c )

          é ❛ ❞❡♥s✐❞❛❞❡ ❞❛ ❡♥❡r❣✐❛ ♣♦t❡♥❝✐❛❧✳ ❉❡st❛ ❢♦r♠❛✱ s✉❜st✐t✉✐♥❞♦ ❛s ❡①♣r❡ssõ❡s ❛ss✐♥tót✐❝❛s ♣❛r❛ ❝❛♠♣♦ ❞✐st❛♥t❡✱ ❊qs✳

          ❡ é ♣♦ssí✈❡❧ ♦❜t❡r Z " X X ′ ′∗ # E − ∗ ∗

          r❛❞ n n m m

          F (a + s a ) s a Y (θ, ϕ)Y ′ (θ, ϕ) e nm n nm r = − ❘❡ ✐ n nm n n ❞Ω. 2 k n,m n ,m ′ ′ ✭✹✳✸✵✮

          ❯t✐❧✐③❛♥❞♦ ♦ ✈❡t♦r ✉♥✐tár✐♦ ❊q✳ ❡ ❛s r❡❧❛çõ❡s ✐♥t❡❣r❛✐s ❞♦s ❤❛r♠ô♥✐❝♦s ❡s❢ér✐❝♦s

          ❞❛❞❛ ♥❛ ❘❡❢✳ ✱ ♣♦❞❡♠♦s ❡s❝r❡✈❡r ❛ ❢♦rç❛ ❞❡ r❛❞✐❛çã♦ ❡♠ ❝♦♦r❞❡♥❛❞❛s ❝❛rt❡s✐❛♥❛s ❝♦♠♦ s❡❣✉❡ 2

          r❛❞

          F = πa E (Y e + Y e + Y e ), x x y y z z ✭✹✳✸✶✮ x y z , e , e x y z

          ♦♥❞❡ e sã♦ ♦s ✈❡t♦r❡s ✉♥✐tár✐♦s ♥♦ ♣❧❛♥♦ ❝❛rt❡s✐❛♥♦ ❡ Y ✱ Y ✱ Y sã♦ ❛s ❝♦♠♣♦♥❡♥t❡s ❝❛rt❡s✐❛♥❛s ❞❛ ❢♦rç❛ ❞❡ r❛❞✐❛çã♦ ❛❞✐♠❡♥s✐♦♥❛❧ ❞❛❞❛s ♣♦r X s

          (n + m + 1)(n + m + 2) ✐

          Y = x y 2 ✐Y

        • 2π(ka) (2n + 1)(2n + 3) ∗ ∗ ∗ ∗ ∗ ∗ n,m

          (s n + s + 2s n s )a nm a + (s + s n+1 + 2s s n+1 )a a n+1,−m−1 , × n+1 n+1 n+1,m+1 n n n,−m X s

          ✭✹✳✸✷✮

          1 ∗ ∗ ∗ (n − m + 1)(n + m + 1)

          Y = z ■♠ n n nm 2 (s + s + 2s s )a a . n+1 n+1 n+1,m π(ka) (2n + 1)(2n + 3) n,m x , Y y z

          ✭✹✳✸✸✮ ◆♦t❡ q✉❡ Y ❡ Y sã♦ q✉❛♥t✐❞❛❞❡s r❡❛✐s q✉❡ ❞❡♣❡♥❞❡♠ ❞♦s ❝♦❡✜❝✐❡♥t❡s ❞❡ ❢♦r♠❛ nm n

          ❞♦ ❢❡✐①❡ a ❡ ❞♦s ❝♦❡✜❝✐❡♥t❡s ❞❡ ❡s♣❛❧❤❛♠❡♥t♦ ❡s❝❛❧❛r❡s s ✳ ❆ss✐♠✱ ❛ ❢♦rç❛ ❞❡ r❛❞✐❛çã♦ n ❞❡♣❡♥❞❡ ❞❛s ♣r♦♣r✐❡❞❛❞❡s ❛❝úst✐❝❛s ❞♦ ♦❜❥❡t♦ ❞❛❞❛s ♣❡❧♦s s ❡ ❛s ❝❛r❛❝t❡ríst✐❝❛s ❞❛ ♦♥❞❛ nm ✐♥❝✐❞❡♥t❡ ❞❡♥♦t❛❞❛s ♣♦r a ✳ 1

          ◆♦ r❡❣✐♠❡ ❞❡ ❡s♣❛❧❤❛♠❡♥t♦ ❘❛②❧❡✐❣❤✱ s♦♠❡♥t❡ s ❡ s sã♦ r❡❧❡✈❛♥t❡s ❝♦♠♦ ❞✐s❝✉t✐❞♦ ❛♥t❡r✐♦r♠❡♥t❡ ♥♦ ❝❛♣ít✉❧♦ P♦rt❛♥t♦✱ ♠❛♥t❡♥❞♦ ❛♣❡♥❛s ❡st❡s t❡r♠♦s✱ ✈❡r✐✜❝❛✲s❡ q✉❡ ♦s ❝♦♠♣♦♥❡♥t❡s ❝❛rt❡s✐❛♥❛s ❞❛ ❢♦rç❛ ❞❡ r❛❞✐❛çã♦ ❛❞✐♠❡♥s✐♦♥❛❧ sã♦ ❞❛❞❛s ♣♦r r 2

          ✐ ∗ ∗ ∗ ∗ ∗ ∗ x y 0,0 + Y = (s + s + 2s s )a a + (s + s + 2s s )a a 1 1 1,−1 ✐Y 2 1 1 1,1 0,0 ✭✹✳✸✹✮

          2π(ka) X 1 r

          3 (2 + m)(3 + m) ∗ ∗ ∗

          (s a a + s , + 1 1,m 2,−m−1 2,m+1 1 1,−m a a ) m=−1

          15 X 1 r r 1 1 ∗ ∗ ∗ ∗

          (2 − m)(2 + m) (s + s + 2s s + Y z = )a 0,0 a s 1 a 1,m a . 2 ■♠ 1 1 1,0 2,m

          π(ka)

          3 m=−1

          15 1 ✭✹✳✸✺✮ ❖❜s❡r✈❡ q✉❡✱ ❛♣❡s❛r ❞❡ s♦♠❡♥t❡ s ❡ s s❡r❡♠ r❡❧❡✈❛♥t❡s ♥♦ r❡❣✐♠❡ ❞❡ ❘❛②❧❡✐❣❤✱ é 2,m

          ♥❡❝❡ssár✐♦ ❝♦♠♣✉t❛r ♦ ♠♦♠❡♥t♦ ❞❡ q✉❛❞r✉♣♦❧♦ ❞❛ ♦♥❞❛ ✐♥❝✐❞❡♥t❡ a ✳ ➱ ♣♦ssí✈❡❧ ❞❡♠♦♥str❛r q✉❡ ❛ ❢♦rç❛ ❞❡ r❛❞✐❛çã♦ ❛❣✐♥❞♦ s♦❜r❡ ✉♠❛ ♣❛rtí❝✉❧❛ ❞❡ ❘❛②❧❡✐❣❤

          ♣♦❞❡ s❡r ❡①♣r❡ss❛ ❡♠ t❡r♠♦s ❞♦s ❝❛♠♣♦s ❛❝úst✐❝♦s ✐♥❝✐❞❡♥t❡s ❡ ❞♦ ♠♦♥♦♣♦❧♦ s ❡ ❞✐♣♦❧♦ s 1 ❞♦s ❝♦❡✜❝✐❡♥t❡s ❞❡ ❡s♣❛❧❤❛♠❡♥t♦ ❡s❝❛❧❛r❡s✳ P❛r❛ ✐ss♦✱ ✐r❡♠♦s ❡①♣❛♥❞✐r ❛ ♣r❡ssã♦ ❡♠ t♦r♥♦ ❞❛ ♦r✐❣❡♠ ❛té ❛ ❛♣r♦①✐♠❛çã♦ ❞❡ s❡❣✉♥❞❛ ♦r❞❡♠✱ c

          ✐ρ kr · [r · ∇v(0)] p(r) = p(0) + c

          ✐ρ ✭✹✳✸✻✮ kr · v(0) +

          2 e e e x x y y z z + v + v ♦♥❞❡✱ v = v ✳ ❙✉❜st✐t✉✐♥❞♦ ❛ ❊q✳ ♥❛ ❡q✉❛çã♦ ❞♦s ❝♦❡✜❝✐❡♥t❡s ❞❡

          (cos ϕ sin θe + sin ϕ sin θe + cos θe ) x y z ❢♦r♠❛ ❞♦ ❢❡✐①❡ ❊q✳ ❥✉♥t❛♠❡♥t❡ ❝♦♠ r = R ❝ ❝♦♠ ♦ r❛✐♦ ❞❡ ❝♦♥tr♦❧❡ R ❝ → 0✱ ♦❜t❡♠♦s ♦s ❝♦❡✜❝✐❡♥t❡s ❞❡ ❢♦r♠❛ ❞♦ ❢❡✐①❡ ❛té ❛♣r♦①✐♠❛çã♦ ❞❡ q✉❛❞r✉♣♦❧♦ ❝♦♠♦

          √ a = 4πp(0), 0,0 √ a = ρ c (0) + v (0)] , 1,∓1 x y

          6π [±✐v √ a 1,0 = 2 c 3πv z (0),

          ✐ρ ρ c r 15π a = [( v v v (0)] , 2,∓2 x x y y y x

          ✐∂ (0) − ∂ (0)) ± 2∂ ✭✹✳✸✼✮ k

          2 √

          ρ c a = v (0) + ∂ v (0)] , 2,∓1 z x z y 30π [±✐∂ k c √

          ✐ρ a 5π [∂ v (0) + ∂ v v (0)] . 2,0 x x y y z z = − (0) − 2∂ k

          ❆❣♦r❛✱ s✉❜st✐t✉✐♥❞♦ ❛ ❊q✳ ❡♥❝♦♥tr❛♠♦s ❛ s❡❣✉✐♥t❡ ❡①♣r❡ssã♦ ♣❛r❛ ❛ ❢♦rç❛ ❞❡ r❛❞✐❛çã♦ ❛❝úst✐❝❛✱ 2

          2πa 3 c s 1 ✐ρ ∗ ∗ ∗ 8 7

          r❛❞ F v + (s + 2s s )p v .

          ✐♥ ✐♥

          = − ❘❡ · ∇v 1 ❖ ε ✐ε 2 ✐♥ ✐♥ ε c k

          ✭✹✳✸✽✮

          ❯t✐❧✐③❛♥❞♦ ❛ ✐❞❡♥t✐❞❛❞❡ ∗ ∗ ∗ ∗ v = v + v ),

          ✐♥ ✐♥

          ∇ · v ✐♥ · ∇v ✐♥ ✐♥ (∇ · v ✐♥ ✭✹✳✸✾✮ ❡ ❝♦♥s✐❞❡r❛♥❞♦ ❛ r❡❧❛çã♦

          ✐kp ✐♥ = ,

          ∇ · v ✐♥ ✭✹✳✹✵✮

          ρ c q✉❡ é ❞❡r✐✈❛❞❛ ❞❛ ❊q✳ t♦r♥❛✲s❡ 2

          3 2πa c s 1

          ✐ρ ∗ ∗ ∗

          r❛❞ 8 7 F v v + (s + 3s + 2s s )p .

          ✐♥ 1 ✐♥ ❖ ε ✐ε ✭✹✳✹✶✮

          = − ∇ · v ✐♥ 2 1 ✐♥ ε c k

        • ❘❡

          ❖❜s❡r✈❡ q✉❡✱ ♥❛ ❛♣r♦①✐♠❛çã♦ ♠♦♥♦♣♦❧♦✲❞✐♣♦❧♦✱ ❛ ❢♦rç❛ ❞❡ r❛❞✐❛çã♦ ❡①❡r❝✐❞❛ s♦❜r❡ ✉♠❛ 1 ♣❛rtí❝✉❧❛ ♣❡q✉❡♥❛ é ❞❛❞❛ ❡♠ t❡r♠♦s ❞❡ s ✱ s ✱ ♣r❡ssã♦ ✐♥❝✐❞❡♥t❡ p ✐♥ ❡ ✈❡❧♦❝✐❞❛❞❡ ❞♦ ✢✉✐❞♦ v

          ✐♥

          ❛♠❜♦s ❡✈♦❧✉í❞♦s ♥♦ ❝❡♥tr♦ ❞❛ ♣❛rtí❝✉❧❛✳ ❊st❛ ❢ór♠✉❧❛ ❡①♣r❡ss❛ ❛ s♦❧✉çã♦ ❝♦♠♣❧❡t❛ ♣❛r❛ ❛ ❛♣r♦①✐♠❛çã♦ ♠♦♥♦♣♦❧♦✲❞✐♣♦❧♦✳ ❊♥tr❡t❛♥t♦✱ ❛ ❛♥á❧✐s❡ ❞❛ ❢♦rç❛ ❞❡ r❛❞✐❛çã♦ ❞✐r❡t❛ 1

          ❡♠ t❡r♠♦s ❞♦s ❝♦❡✜❝✐❡♥t❡s s ❡ s ♥ã♦ é tr✐✈✐❛❧✳ P♦r ✐ss♦✱ ✐r❡♠♦s s✐♠♣❧✐✜❝❛r ❛ ❊q✳ ❉❡♣♦✐s ❞❡ s✉❜st✐t✉✐r ❛s ❊qs✳ ❡ ✉t✐❧✐③❛r ❛ ❡①♣r❡ssã♦ ❞♦

          ✢✉①♦ ❞♦ ♠♦♠❡♥t♦ ❞❛❞❛ ♥❛ ❘❡❢✳ 2 2 ∗ |v ✐♥ | |p ✐♥ | ρ v ,

          ❘❡ [ρ ✐♥ ✭✹✳✹✷✮

          ∇ · v ✐♥ ] = ∇ − 2 2 2ρ c ♦❜t❡♠♦s ❛ ❢♦rç❛ ❞❡ r❛❞✐❛çã♦ ❝♦♠♦ ❛ s♦♠❛ ❞❡ três ❝♦♥tr✐❜✉✐çõ❡s✱ ❞❡♥♦♠✐♥❛❞❛s ❝♦♠♦ ❢♦rç❛

          ❣r❛❞✐❡♥t❡ F ❣r❛❞ ✱ ❢♦rç❛ ❞❡ ❡s♣❛❧❤❛♠❡♥t♦ F s❝❛ ✱ ❡ ❢♦rç❛ ❞❡ ❛❜s♦rçã♦ F ❛❜s ✱ ❞❛❞❛s ♣♦r 2 2 ρ

          ✭❘✮ |p ✐♥ | ✭❘✮ |v ✐♥ |

        • F ε ˜ f ),

          ❣r❛❞ 1 ❖(ε ✭✹✳✹✸✮

          = −πa ∇ 2 − f 3ρ c

          2 2 2 2 ✭❘✮ ✭❘✮ 4 3f f 2 4 ✭❘✮ ✭❘✮ ✭❘✮

        1

        1 ∗ 5

          I v v = πa ε f + f f

        • s❝❛ 1 − ■♠[ ˜ ∇ · ρ

          ] ),

        • F

          ✐♥ ✐♥ ✐♥ ❖(ε

          9c

          4

          6 ✭✹✳✹✹✮

          4 I

          2I 2 ✭■✮ ✐♥ ✭■✮ ✐♥ ∗ 2 4 F v v

          ε f

        • g ] ),

          ❛❜s 1 ✐♥ ❖(ε ✭✹✳✹✺✮

          = −πa − 6ε − ■♠[ ˜ ∇ · ρ ✐♥ 1 3 c c = v ]/2

          ✐♥ ✐♥

          ♦♥❞❡ ˜ ∇ = k ∇ é ♦ ♦♣❡r❛❞♦r ❣r❛❞✐❡♥t❡ ❛❞✐♠❡♥s✐♦♥❛❧ ❝♦♥✈❡♥✐❡♥t❡ ❡ I ❘❡[p ✐♥

          ✭❘✮ ✭■✮

          é ❛ ✐♥t❡♥s✐❞❛❞❡ ♠é❞✐❛ ✐♥❝✐❞❡♥t❡✳ ❖s s✉♣❡r í♥❞✐❝❡s ❡ ❝♦rr❡s♣♦♥❞❡♠ ❛ ♣❛rt❡ r❡❛❧ ❡ ✐♠❛❣✐♥ár✐❛ ❞♦s ❝♦❡✜❝✐❡♥t❡s f ✱ f 1 ✱ ❡ g 1 ✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✳ ❘❡ss❛❧t❛♠♦s q✉❡ ♦s ❝❛♠♣♦s ❛❝úst✐❝♦s ✐♥❝✐❞❡♥t❡s p ✐♥ ❡ v ✐♥ ❞❡✈❡♠ s❡r ❛✈❛❧✐❛❞♦s ♥♦ ❝❡♥tr♦ ❞❛ ♣❛rtí❝✉❧❛✳

          ❆ ❢♦rç❛ ❞❡ r❛❞✐❛çã♦ ❣r❛❞✐❡♥t❡ F ❣r❛❞ ❢♦✐ ♣r✐♠❡✐r❛♠❡♥t❡ ❞❡r✐✈❛❞♦ ♣♦r ●♦r❦♦✈ ❝♦♥✲ s✐❞❡r❛♥❞♦ ✉♠❛ ♣❛rtí❝✉❧❛ ✢✉✐❞❛ ❝♦♠♣r❡ssí✈❡❧ ♥ã♦ ❛❜s♦r✈❡❞♦r❛✳ ❊ss❛ ❝♦♠♣♦♥❡♥t❡ ❞❛ ❢♦rç❛ ❝♦rr❡s♣♦♥❞❡ ❛ ♣❛rt❡ ❝♦♥s❡r✈❛t✐✈❛ ❞❛ ❢♦rç❛ ❞❡ r❛❞✐❛çã♦ ❛❝úst✐❝❛✳ ❖ s❡✉ ♣❛♣❡❧ é ❞♦♠✐♥❛♥t❡ q✉❛♥❞♦ ❛ ♦♥❞❛ ✐♥❝✐❞❡♥t❡ t❡♠ ✉♠❛ ✈❛r✐❛çã♦ ❡s♣❛❝✐❛❧ ♥❛ ❞❡♥s✐❞❛❞❡ ❞❡ ❡♥❡r❣✐❛✳

          

        s❝❛

          ❆ ❢♦rç❛ ❞❡ r❛❞✐❛çã♦ ❞❡ ❡s♣❛❧❤❛♠❡♥t♦ F ❡①❡r❝✐❞❛ s♦❜r❡ ✉♠❛ ♣❛rtí❝✉❧❛ ❝♦♠♣r❡ssí✈❡❧ ❛❜s♦r✈❡❞♦r❛ ❢♦✐ ❞❡r✐✈❛❞❛ ♣♦r ❙✐❧✈❛ ✳ ❖ t❡r♠♦ ✏❢♦rç❛ ❞❡ ❡s♣❛❧❤❛♠❡♥t♦✑ ❡stá r❡❧❛❝✐♦♥❛❞♦ ❛♦ ❢❛t♦ ❞❡ q✉❡ ❛ ❢♦rç❛ é ♣r♦♣♦r❝✐♦♥❛❧ à s❡çã♦ tr❛♥s✈❡rs❛❧ ❞❡ ❡s♣❛❧❤❛♠❡♥t♦ ♣❛r❛ ✉♠❛ ♦♥❞❛ ♣❧❛♥❛

          ❆ ❝♦♥tr✐❜✉✐çã♦ ❞❛ ❢♦rç❛ ❞❡ r❛❞✐❛çã♦ ❞❡ ❛❜s♦rçã♦ F ❛❜s ❡st❛ r❡❧❛❝✐♦♥❛❞❛ ❝♦♠ ❛s ♣r♦♣r✐❡✲ ❞❛❞❡s ❞❡ ❛❜s♦rçã♦ ❞❛ ♣❛rtí❝✉❧❛✳ ❊ss❛ ❝♦♠♣♦♥❡♥t❡ ❢♦✐ ❛♥t❡r✐♦r♠❡♥t❡ ❞❡r✐✈❛❞❛ ❝♦♥s✐❞❡r❛♥❞♦ ✉♠❛ ❡s❢❡r❛ ✢✉✐❞❛ ❛♣❡♥❛s ❝♦♠ ❛❜s♦rçã♦ ❞❡✈✐❞♦ ❛ ❞✐❧❛t❛çã♦ ❣❡♥❡r❛❧✐③❛ ❡ss❡s r❡s✉❧t❛❞♦s ✐♥❝❧✉✐♥❞♦ ❛ ❛❜s♦rçã♦ ❞❡ ❝✐s❛❧❤❛♠❡♥t♦✳

          

        ✹✳✸ ❚♦rq✉❡ ❞❡ r❛❞✐❛çã♦ ❛❝úst✐❝♦ ♥♦ ❧✐♠✐t❡ ❞❡ ❘❛②❧❡✐❣❤

          ❆ ✐♥t❡r❛çã♦ ❞❡ ✉♠❛ ♣❛rtí❝✉❧❛ s✉s♣❡♥s❛ ❞❡ ✈♦❧✉♠❡ V ❡ s✉♣❡r❢í❝✐❡ S ❝♦♠ ✉♠❛ ♦♥❞❛ ❛❝úst✐❝❛ ♣♦❞❡ ❞❛r ♦r✐❣❡♠ ❛ ✉♠ t♦rq✉❡ ❞❡ r❛❞✐❛çã♦ ❛❝úst✐❝♦✳ ■ss♦ ❛❝♦♥t❡❝❡ ❞❡✈✐❞♦ ❛ tr❛♥s❢❡rê♥❝✐❛ ❞♦ ✢✉①♦ ❞♦ ♠♦♠❡♥t♦ ❛♥❣✉❧❛r ♣❛r❛ ❛ ♣❛rtí❝✉❧❛✳ ❆❧é♠ ❞✐ss♦✱ ❛ ♣❛rtí❝✉❧❛ ♣♦❞❡ s❡r ❝♦♥✜❣✉r❛❞❛ ♣❛r❛ ❣✐r❛r ❡♠ t♦r♥♦ ❞❡ ✉♠ ❞❡t❡r♠✐♥❛❞♦ ❡✐①♦ ❞❡ r♦t❛çã♦ ❝♦♠♦ r❡s✉❧t❛❞♦ ❞♦ t♦rq✉❡ ❞❡ r❛❞✐❛çã♦ ❛❝úst✐❝♦ ❈♦♥s❡q✉❡♥t❡♠❡♥t❡✱ ✉♠ ❣r❛✉ ❞❡ ❧✐❜❡r❞❛❞❡ r♦t❛❝✐♦♥❛❧ t❛♠❜é♠ ❡st❛ ❞✐s♣♦♥í✈❡❧ ♥❛ ♠❛♥✐♣✉❧❛çã♦ ❞❡ ♣❛rtí❝✉❧❛s ❝♦♠ ❜❛s❡ ❡♠ ♠ét♦❞♦s ❛❝úst✐❝♦s✳ ◆❡st❛ s❡çã♦✱ ❝♦♥s✐❞❡r❛♠♦s ♦ t♦rq✉❡ ❞❡ r❛❞✐❛çã♦ ❛❝úst✐❝♦ ❡①❡r❝✐❞♦ ♣♦r ✉♠❛ ♦♥❞❛ ❛r❜✐trár✐❛ ❡♠ ✉♠❛ ♣❛rtí❝✉❧❛ ✈✐s❝♦❡❧ást✐❝❛ s✉s♣❡♥s❛ ❡♠ ✉♠ ✢✉✐❞♦ ✐❞❡❛❧✳

          ✹✳✸✳✶ ❋❧✉①♦ ❞♦ ♠♦♠❡♥t♦ ❛♥❣✉❧❛r

          ❖ t♦rq✉❡ ❞❡ r❛❞✐❛çã♦ ❛❝úst✐❝♦ é ✉♠ ❡❢❡✐t♦ ♥ã♦ ❧✐♥❡❛r ❞❡ s❡❣✉♥❞❛ ♦r❞❡♠ ♣r♦❞✉③✐❞♦ ♣❡❧❛ tr❛♥s❢❡rê♥❝✐❛ ❞♦ ♠♦♠❡♥t♦ ❛♥❣✉❧❛r ❞♦ ❝❛♠♣♦ ❛❝úst✐❝♦ ♣❛r❛ ❛ ♣❛rtí❝✉❧❛✳ ❙❡♠❡❧❤❛♥t❡ à ❢♦rç❛ ❞❡ r❛❞✐❛çã♦ ❛❝úst✐❝❛✱ ♦ t♦rq✉❡ ❞❡ r❛❞✐❛çã♦ t❛♠❜é♠ é ✉♠❛ q✉❛♥t✐❞❛❞❡ ♠é❞✐❛✳

          ❉❡st❛ ❢♦r♠❛✱ ♦ ✢✉①♦ ❞❡ ♠♦♠❡♥t♦ ❛♥❣✉❧❛r ♠é❞✐♦ ❡♠ r❡❧❛çã♦ ❛ ♦r✐❣❡♠ ❞❡ ✉♠❛ s✐st❡♠❛ ❞❡ ❝♦♦r❞❡♥❛❞❛s✱ é ❞❡✜♥✐❞♦ ❞❛ s❡❣✉✐♥t❡ ❢♦r♠❛✿ M (2) (2)

          , (2) = r × S ✭✹✳✹✻✮ ♦♥❞❡ M é ❝♦♥❤❡❝✐❞♦ ❝♦♠♦ t❡♥s♦r t❡♥sã♦ ❞❡ r❛❞✐❛çã♦ ❞♦ ♠♦♠❡♥t♦ ❛♥❣✉❧❛r✱ r é ♦ ✈❡t♦r (2) ♣♦s✐çã♦ ❡ S é ♦ t❡♥s♦r ❞❡ t❡♥sã♦ ❞❡ r❛❞✐❛çã♦✳

          ✹✳✸✳✷ ▼ét♦❞♦ ❞❡ ❝❛♠♣♦ ❞✐st❛♥t❡ r❛❞

          ❖ t♦rq✉❡ ❞❡ r❛❞✐❛çã♦ ❛❝úst✐❝♦ N s♦❜r❡ ❛ s✉♣❡r❢í❝✐❡ S é ❞❡✜♥✐❞♦ ❝♦♠♦ ❛ ✐♥t❡❣r❛❧ ❞♦ ✢✉①♦ ❞❡ ♠♦♠❡♥t♦ ♠é❞✐♦✱ Z Z h i

          r❛❞ M (2) (2) 2 2 N r r r = = . S S · n❞ × S · n❞ ✭✹✳✹✼✮

          ❉❡ ❢♦r♠❛ ❛♥á❧♦❣❛ à ❞❡♠♦♥str❛çã♦ ❞❛ ❢♦rç❛ ❞❡ r❛❞✐❛çã♦✱ é ♣♦ssí✈❡❧ ❡①♣r❡ss❛r ♦ t♦rq✉❡

          ❝

          ❞❡ r❛❞✐❛çã♦ ❛❝úst✐❝♦ ❝♦♠♦ ❛ ✐♥t❡❣r❛❧ s♦❜r❡ ✉♠❛ s✉♣❡r❢í❝✐❡ ❞❡ ❝♦♥tr♦❧❡ S q✉❡ ❡♥❣❧♦❜❡ ❛ ♣❛rtí❝✉❧❛ ✉t✐❧✐③❛♥❞♦ ❛ ❊q✳ ❞❛ s❡❣✉✐♥t❡ ❢♦r♠❛✿ Z h i 2

          r❛❞ (2) N r r .

          = − × S · n❞ ✭✹✳✹✽✮ S ❊ss❛ ❡q✉❛çã♦ ♥♦s ♣❡r♠✐t❡ ✉t✐❧✐③❛r ❛s ❡①♣❛♥sõ❡s ❞❡ ❝❛♠♣♦ ❞✐st❛♥t❡ ♣❛r❛ ❝❛❧❝✉❧❛r ♦ t♦rq✉❡ ❞❡ r❛❞✐❛çã♦ ❛❝úst✐❝♦✳ ❯t✐❧✐③❛♥❞♦ ❛ t❡♥sã♦ ❞❡ s❡❣✉♥❞❛ ♦r❞❡♠✱ ❊q✳ ♦❜t❡♠♦s Z " ! # (1)2 (1)2

          ρ v p 2

          r❛❞ I (1) (1)

          N = r v v r , × − − ρ · n❞ 2 S Z h i 2 2ρ c 2

          r❛❞ (1) (1)

          N r v r .

          ✭✹✳✹✾✮ = −ρ × v · n❞ S

          P♦❞❡♠♦s ❡s❝r❡✈❡r ♦ t♦rq✉❡ ❞❡ r❛❞✐❛çã♦ ❞❛ ❊q✳ ❝♦♠♦ Z ρ ∗ ∗ ∗ ∗

          r❛❞ 2 N v v v v r + v + v + v .

          ❘❡ [r × (v ✐♥ ✐♥ s❝ s❝ ✭✹✳✺✵✮ = − ✐♥ s❝ ✐♥ s❝ )] · n❞

          2 S ◆♦t❡ q✉❡✱ ❛♣❡♥❛s ❛ ❝♦♥tr✐❜✉✐çã♦ ❞❛ ✈❡❧♦❝✐❞❛❞❡ ✐♥❝✐❞❡♥t❡ ♥ã♦ ❝♦♥té♠ ✐♥❢♦r♠❛çã♦ s♦❜r❡ ♦ ❡s♣❛❧❤❛♠❡♥t♦✱ ♣♦rt❛♥t♦ ❞❡✈❡ s❡r ③❡r♦✳ ❈♦♥s✐❞❡r❛♥❞♦ ❛ s✉♣❡r❢í❝✐❡ ❞❡ ❝♦♥tr♦❧❡ ❡s❢ér✐❝❛ ❞✐st❛♥t❡ ❞❛ ♣❛rtí❝✉❧❛ ✭R ❝

          → ∞✮✱ ♣♦❞❡♠♦s ❡①♣r❡ss❛r ♦ t♦rq✉❡ ❞❡ r❛❞✐❛çã♦ ♥♦ ❝❛♠♣♦ ❞✐st❛♥t❡ ❞❛ s❡❣✉✐♥t❡ ❢♦r♠❛✿ Z

          ρ 2 ∗ ∗ ∗

          r❛❞

          N lim R (v ) + v (v ) + v (v r r r = − ❘❡{r × [v ✐♥ · e s❝ · e s❝ · e )]}❞Ω. ✭✹✳✺✶✮ R →∞ ❝ s❝ ✐♥ s❝

          2 ❊♠ t❡r♠♦s ❞♦ ♣♦t❡♥❝✐❛❧ ❞❡ ✈❡❧♦❝✐❞❛❞❡ φ ✐♥(s❝) ✱ ♦ t♦rq✉❡ ♣♦❞❡ s❡r ❡①♣r❡ss♦ ❝♦♠♦

          Z h ρ ∗ ∗ ∗

          r❛❞ 2 L L L

          = lim R (ˆ φ )(∂ φ ) + (ˆ φ )(∂ φ ) + (ˆ φ )(∂ φ ) ■♠ ✐♥ r s❝ r s❝ r i ❞Ω, ✭✹✳✺✷✮ R →∞ ❝ s❝ ✐♥ s❝

          2 ❝ ♦♥❞❡ ✏■♠✑ s✐❣♥✐✜❝❛ ♣❛rt❡ ✐♠❛❣✐♥ár✐❛ ❡ ˆL ≡ −✐ (r × ∇) é ♦ ♦♣❡r❛❞♦r ♠♦♠❡♥t♦ ❛♥❣✉❧❛r✳

          ❆❣♦r❛✱ s✉❜st✐t✉✐♥❞♦ ❛s ❡①♣r❡ssõ❡s ❛ss✐♥tót✐❝❛s ♣❛r❛ ❝❛♠♣♦ ❞✐st❛♥t❡ ❊qs✳ ❞❡♥tr♦ ❞❛ ❊q✳ ❡♥❝♦♥tr❛♠♦s X X ′ ∗ ′ Z

          V E n−n ∗ ∗ ∗ m m

          r❛❞ ′ ′ ′ L ˆ

          N (a + s a )s a Y Y ′ n n m = − ❘❡ ✐ nm n nm n n ❞Ω. ✭✹✳✺✸✮ 3

          π(ka) n,m n ,m ′ ′ P❛r❛ ♦❜t❡r ♦ t♦rq✉❡ ❞❡ r❛❞✐❛çã♦ ❛❝úst✐❝♦ ❡♠ ❝♦♦r❞❡♥❛❞❛s ❝❛rt❡s✐❛♥❛s✱ ✐♥tr♦❞✉③✐♠♦s ♦s

          ♦♣❡r❛❞♦r❡s ♠♦♠❡♥t♦ ❛♥❣✉❧❛r ±

          ✐ϕ

          ˆ L ± = ˆ L x L y (∂ θ ϕ ) ,

          ✭✹✳✺✹✮ ± ✐ ˆ = ±❡ ± ✐ cot θ ∂

          ˆ L z ϕ ,

          ✭✹✳✺✺✮ = −✐∂

        • + i

          (i = x, y, z) ♦♥❞❡ ˆL sã♦ ❛s ❝♦♠♣♦♥❡♥t❡s ❝❛rt❡s✐❛♥❛s ❞♦ ♦♣❡r❛❞♦r ˆL✱ ❡ ˆL ❡ ˆL sã♦ ❝❤❛♠❛❞♦s ❞❡ ♦♣❡r❛❞♦r❡s ❡s❝❛❞❛✳

          ➱ ♣♦ssí✈❡❧ ♠♦str❛r q✉❡✱ q✉❛♥❞♦ ♦s ♦♣❡r❛❞♦r❡s ❡s❝❛❞❛s ❡ ♦ ♦♣❡r❛❞♦r ❞❛ ❝♦♠♣♦♥❡♥t❡ z ❞♦ ♠♦♠❡♥t♦ ❛♥❣✉❧❛r ❛t✉❛♠ ♥❛ ❢✉♥çã♦ ❤❛r♠ô♥✐❝♦ ❡s❢ér✐❝♦✱ ♦❜t❡♠♦s m m±1

          ˆ L ± Y = b n,±m Y , n n m m ✭✹✳✺✻✮

          ˆ L Y = mY , z n n ✭✹✳✺✼✮

          = ♦♥❞❡ b n,±m p(n ∓ m)(n ± m + 1)✳ ❙✉❜st✐t✉✐♥❞♦ ❡ss❛s ❡q✉❛çõ❡s ❞❡♥tr♦ ❊q✳ ♦❜✲ t❡♠♦s ❛s ❝♦♠♣♦♥❡♥t❡s ❝❛rt❡s✐❛♥❛s ❛❞✐♠❡♥s✐♦♥❛✐s ❞♦ t♦rq✉❡ ❞❡ r❛❞✐❛çã♦ ❝♦♠♦ X

          1 ∗ ∗ x y + N n nm )s a a ✐N = − p(n − m)(n + m + 1) (1 + s n n,m+1 3

          2π(ka) ∗ ∗ n,m ,

        • (1 + s )s n a a n,−m−1 n n,−m
        • X ✭✹✳✺✽✮

            1 2 N m[(1 + s )s . z n nm = − ❘❡ n ]|a | ✭✹✳✺✾✮ 3

            π(ka) n,m ◆♦t❡ q✉❡ ❛s ❊qs✳ sã♦ ❡①♣r❡ssõ❡s ❣❡r❛✐s ❞♦ t♦rq✉❡ ❞❡ r❛❞✐❛çã♦ ❛❝úst✐❝♦✱

            ♦✉ s❡❥❛✱ sã♦ ✈á❧✐❞❛s ♣❛r❛ q✉❛❧q✉❡r ❝❛♠♣♦ ❛❝úst✐❝♦ ✐♥❝✐❞❡♥t❡✳ ❆❧é♠ ❞✐ss♦✱ s✐♠✐❧❛r♠❡♥t❡ à ❢♦rç❛ ❞❡ r❛❞✐❛çã♦ ❛♥❛❧✐s❛❞❛ ❛♥t❡r✐♦r♠❡♥t❡✱ ♦ t♦rq✉❡ ❞❡ r❛❞✐❛çã♦ ❛❝úst✐❝♦ ❞❡♣❡♥❞❡ ❞❛s n ♣r♦♣r✐❡❞❛❞❡s ❢ís✐❝❛s ❞❛ ♣❛rtí❝✉❧❛ ❡s♣❛❧❤❛❞♦r❛ s ❡ ❞❛s ❝❛r❛❝t❡ríst✐❝❛s ❞♦ ❢❡✐①❡ ✐♥❝✐❞❡♥t❡ nm ❛tr❛✈és ❞♦s a ✳

            ◆♦ ❧✐♠✐t❡ ❞❡ ❡s♣❛❧❤❛♠❡♥t♦ ❞❡ ❘❛②❧❡✐❣❤✱ ❛s ❝♦♠♣♦♥❡♥t❡s ❝❛rt❡s✐❛♥❛s ❛❞✐♠❡♥s✐♦♥❛✐s ❞♦ t♦rq✉❡ ♣♦❞❡♠ s❡r ❝❛❧❝✉❧❛❞❛s ❝♦♥s✐❞❡r❛♥❞♦ s♦♠❡♥t❡ ♦s ❝♦❡✜❝✐❡♥t❡s ❞❡ ❡s♣❛❧❤❛♠❡♥t♦ ❞❡

            1

            ♠♦♥♦♣♦❧♦ s ❡ ❞✐♣♦❧♦ s ✳ P♦r ❝♦♥s❡❣✉✐♥t❡✱ ♦❜t❡♠♦s √ s

            2 + s 1 ∗ ∗ x ✐N y + N 1 1,−1 1,0 ✭✹✳✻✵✮ (a a + a a ) , = − + |s | 1,0 1,1 3

            π(ka)

            2 1 s + s 1 1 2 2 2 N z 1 1,1 1,−1 ) .

            ✭✹✳✻✶✮ = − + |s | (|a | − |a | 3

            π(ka)

            2 nm ❙✉❜st✐t✉✐♥❞♦ ♦s ❝♦❡✜❝✐❡♥t❡s ❞❡ ❢♦r♠❛ ❞♦ ❢❡✐①❡ a ❞❛❞♦s ♥❛ ❊q✳ ❡ ❡♥❝♦♥tr❛♠♦s ♦ t♦rq✉❡ ❞❡ r❛❞✐❛çã♦ ❛❝úst✐❝♦ ♥❛ ❛♣r♦①✐♠❛çã♦ ♠♦♥♦♣♦❧♦✲❞✐♣♦❧♦ ❝♦♠♦ s + s 1

            ✐6πρ 1 2 ∗ 8 7

            r❛❞

          • N (v ) + .
          • 1 = − + |s | ✐♥ × v ❖ ε ✐ε ✭✹✳✻✷✮ 3 ✐♥ k

              2 ❯s❛♥❞♦ ❛ ❊q✳ ❞❡♥tr♦ ❞❡st❛ ❡q✉❛çã♦✱ ❡♥❝♦♥tr❛♠♦s q✉❡ ♦ t♦rq✉❡ ❞❡ r❛❞✐❛çã♦ ❛❝úst✐❝♦ ❡♠ ✉♠❛ ♣❛rtí❝✉❧❛ ✈✐s❝♦❡❧ást✐❝❛ é t♦t❛❧♠❡♥t❡ ❝❛✉s❛❞♦ ❞❡✈✐❞♦ ❛ ❛❜s♦rçã♦ ❞❛ ♣❛rtí❝✉❧❛✱ 3 2 ✭■✮ ∗ 4 N = ε ρ g (v ) + ).

              ❛❜s ✐6πa ✐♥ ❖(ε ✭✹✳✻✸✮ 1 × v ✐♥

              ❆ ✈❡❧♦❝✐❞❛❞❡ ✐♥❝✐❞❡♥t❡ v ✐♥ é ❡✈♦❧✉í❞❛ ♥♦ ❝❡♥tr♦ ❞❛ ♣❛rtí❝✉❧❛✳ P❛r❛ ✉♠❛ ♣❛rtí❝✉❧❛ ♥ã♦

              ✭■✮

              = 0 ❛❜s♦r✈❡❞♦r❛ g ✱ ❝♦♥s❡q✉❡♥t❡♠❡♥t❡ ♥❡♥❤✉♠ t♦rq✉❡ ❞❡ r❛❞✐❛çã♦ ❛❝úst✐❝♦ é ♣r♦❞✉③✐❞♦ 1

              ♣❡❧❛ ♦♥❞❛ ✐♥❝✐❞❡♥t❡✳

              

            ✹✳✹ ❊①♣❛♥sã♦ ❛ss✐♥tót✐❝❛ ❞♦ í♥❞✐❝❡ ❞❡ r❡❢r❛çã♦ ❝♦♠✲

            ♣❧❡①♦

              ❆♣❡s❛r ❞❡ t♦❞❛s ❛s s✐♠♣❧✐✜❝❛çõ❡s ♥❛ ❛♥á❧✐s❡ ❞❡ ❢♦rç❛ ❡ t♦rq✉❡ ❞❡ r❛❞✐❛çã♦ ❛❝úst✐❝♦ ♦❜t✐❞❛s ♥❛s ❊qs✳ P❛r❛ ♦❜t❡r r❡s✉❧t❛❞♦s ♠❛✐s s✐♠♣❧❡s✱ é ♥❡❝❡ssár✐♦ ❡❢❡t✉❛r ✉♠❛ ❡①♣❛♥sã♦ ❛ss✐♥tót✐❝❛ ❞♦ í♥❞✐❝❡ ❞❡ r❡❢r❛çã♦ ❝♦♠✲ ν j ) ♣❧❡①♦ ❝♦♠ ♦s t❡♠♣♦s ❞❡ r❡❧❛①❛çã♦ ❛❞✐♠❡♥s✐♦♥❛✐s ♥♦s s❡❣✉✐♥t❡s r❡❣✐♠❡s✿ (ωτ ≪ 1 ❡ ν (ωτ ) j

              ≫ 1✳ ❊st❡s r❡❣✐♠❡s sã♦ r❡❢❡r✐❞♦s ❝♦♠♦ ❛♣r♦①✐♠❛çã♦ ❞❡ ❜❛✐①❛ ❡ ❛❧t❛ ❢r❡q✉ê♥❝✐❛ ♣❛r❛ ♦♥❞❛s ❧♦♥❣✐t✉❞✐♥❛✐s j = ℓ ❡ ♦♥❞❛s ❞❡ ❝✐s❛❧❤❛♠❡♥t♦ j = s✳ ◆♦t❡ q✉❡ ❡ss❛s ❡①♣❛♥sõ❡s ❛ss✐♥tót✐❝❛s ❢♦r❛♠ ❛♥❛❧✐s❛❞❛s ❡♠ ❞❡t❛❧❤❡s ♥❛ ❘❡❢✳ ■r❡♠♦s r❡❝✉♣❡r❛r ♣❛r❝✐❛❧♠❡♥t❡ ❛❧❣✉♥s r❡s✉❧t❛❞♦s r❡❧❡✈❛♥t❡s ♣❛r❛ ♦s ♣r♦❜❧❡♠❛s ❞❡ ❢♦rç❛ ❡ t♦rq✉❡ ❞❡ r❛❞✐❛çã♦ ❛❝úst✐❝♦ ♣r❡s❡♥t❡s ♥❡ss❛ r❡❢❡rê♥❝✐❛✳

              ❖s t❡♠♣♦s ❞❡ r❡❧❛①❛çã♦ ❛❞✐♠❡♥s✐♦♥❛✐s ✐♥tr♦❞✉③❡♠ ♥♦✈❛s ❡s❝❛❧❛s ❡♠ ♥♦ss❛ ❛♥á❧✐s❡✳ ν j j = (ωτ ) P♦r ✐ss♦✱ s❡rá út✐❧ ❞❡✜♥✐r ♦ s❡❣✉✐♥t❡ ♣❛râ♠❡tr♦ ❞❡ ❡s❝❛❧❛ ε ✳ ❆ss✉♠✐♠♦s q✉❡ ♦ ❝♦♠♣♦rt❛♠❡♥t♦ ❧✐♠✐t❡ ❞❡st❡ ♣❛râ♠❡tr♦ ❡st❛ r❡❧❛❝✐♦♥❛❞♦ ❝♦♠ ♦ ❢❛t♦r t❛♠❛♥❤♦ ❞❛ ♣❛rtí❝✉❧❛ 1

              ε = ka = j =

              ✳ P♦rt❛♥t♦✱ t❡♠♦s ε ❖(ε) ♣❛r❛ ♦ r❡❣✐♠❡ ❞❡ ❜❛✐①❛ ❢r❡q✉ê♥❝✐❛✱ ❡ ε j ❖(ε) ♥♦ r❡❣✐♠❡ ❞❡ ❛❧t❛ ❢r❡q✉ê♥❝✐❛✳

              ✹✳✹✳✶ ❘❡❣✐♠❡ ❞❡ ❜❛✐①❛ ❢r❡q✉ê♥❝✐❛ ℓ s , ε

              ❆ ❡①♣❛♥sã♦ ❛ss✐♥tót✐❝❛ ❞♦ í♥❞✐❝❡ ❝♦♠♣❧❡①♦ ❞❛ ❊q✳ q✉❛♥❞♦ ε ≪ 1✱ ❝♦rr❡s✲ ♣♦♥❞❡ ❛♦ r❡❣✐♠❡ ❡♠ q✉❡ ❛ ❡❧❛st✐❝✐❞❛❞❡ ❞♦♠✐♥❛ s♦❜r❡ ❛ ✈✐s❝♦s✐❞❛❞❡✳ ❊st❛ ❛♣r♦①✐♠❛çã♦ ♣♦❞❡ t❛♠❜é♠ s❡r ✈✐st❛ ♥❛ ❞❡s❝r✐çã♦ ❞❛ ♣r♦♣❛❣❛çã♦ ❞❡ ♦♥❞❛s ❡♠ ✢✉✐❞♦s ✈✐s❝♦s♦s ◆♦ r❡❣✐♠❡ ❞❡ ❜❛✐①❛ ❢r❡q✉ê♥❝✐❛✱ ♦ í♥❞✐❝❡ ❞❡ r❡❢r❛çã♦ ❝♦♠♣❧❡①♦ t♦r♥❛✲s❡ ν

              (−✐) 2 n + (ε ε ,

              ❝ j j

              ) = 1 − ❖ ε j ∈ {ℓ, s}. ✭✹✳✻✹✮

              2 ❋❛③❡♥❞♦ r❡❢❡rê♥❝✐❛ ❛ ❊q✳ r❡❝♦♥❤❡❝❡♠♦s q✉❡ ❛ ❛❜s♦rçã♦ ❛❞✐♠❡♥s✐♦♥❛❧ ❡ í♥❞✐❝❡ ❞❡ r❡❢r❛çã♦ é ❞❛❞♦✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✱ ♣♦r

              ε j πν α(ε ˜ j ) = sin ,

              ✭✹✳✻✺✮

              2

              2 ε j πν n (ε j cos .

              ❘

              ✭✹✳✻✻✮ ) = 1 −

              2

              2 j ) > 0 ❯♠❛ ✈❡③ q✉❡ ❛ ❛❜s♦rçã♦ ❛❞✐♠❡♥s✐♦♥❛❧ ❞❡✈❡ s❡r s❡♠♣r❡ ♣♦s✐t✐✈❛✱ ˜α(ε ✱ ❡♥tã♦ ν ∈ (0, 2)✳ ❖s ✈❛❧♦r❡s ♦r❞❡♠ ❞❛ ❞❡r✐✈❛❞❛ ❢r❛❝✐♦♥ár✐❛ ν = 0 ❡ 2 sã♦ ❡①❝❧✉í❞♦s ♣♦rq✉❡ ❡❧❡s r❡♣r❡s❡♥t❛♠ ✉♠ ♠❛t❡r✐❛❧ s❡♠ ❛❜s♦rçã♦✳ ❱❛❧❡ r❡ss❛❧t❛r q✉❡✱ ♥♦ ❧✐♠✐t❡ ❞❡ ❜❛✐①❛ ❢r❡q✉ê♥❝✐❛✱ ♦ í♥❞✐❝❡ ❞❡

              (ε j ) r❡❢r❛çã♦ ♠❛t❡r✐❛❧ é ♠❡♥♦r q✉❡ ❛ ✉♥✐❞❛❞❡✳ ❆ss✐♠✱ ❛ ✈❡❧♦❝✐❞❛❞❡ ❞❡ ❢❛s❡ c ♣ é ♠❛✐♦r q✉❡ j ❛ ✈❡❧♦❝✐❞❛❞❡ ❞♦ s♦♠ c ✳ ❆❧é♠ ❞✐ss♦✱ ✉t✐❧✐③❛♥❞♦ ❛s ❊qs✳ ❡ ❝♦♠♣❛r❛♥❞♦ ❝♦♠ ❛ ❊q✳ ❡♥❝♦♥tr❛♠♦s q✉❡ ❛ ♦r❞❡♠ ❞❛ ❞❡r✐✈❛❞❛ ❢r❛❝✐♦♥ár✐❛ ν ❡stá r❡❧❛❝✐♦♥❛❞❛ ❝♦♠ ♦ ❡①♣♦❡♥t❡ ❞❡ ❛❜s♦rçã♦ ❞❛ ❧❡✐ ❞❡ ♣♦tê♥❝✐❛ y ♣♦r y = 1 + ν.

              ✭✹✳✻✼✮ P❛r❛ ❝❛❧❝✉❧❛r ♦s ❝♦❡✜❝✐❡♥t❡s ❞❛s ❊qs✳ r❡❛❧✐③❛♠♦s ✉♠❛ ❡①♣❛♥sã♦ ❡♠ sér✐❡

              (ε, ε , ε ) ℓ s ❞❡ ❚❛②❧♦r ♥♦s ❝♦❡✜❝✐❡♥t❡s ❞❡ ❡s♣❛❧❤❛♠❡♥t♦ ❡s❝❛❧❛r❡s ❞❡ ♠♦♥♦♣♦❧♦ s ❡ ❞✐♣♦❧♦ s (ε, ε , ε ) 1 ℓ s

              ❞❛❞♦s ♥❛ ❊q✳ ❯♠❛ ℓ s ✈❡③ q✉❡ ♦s t❡♠♣♦s ❞❡ r❡❧❛①❛çã♦ sã♦ ❝♦♥st❛♥t❡s✱ ♦s ♣❛râ♠❡tr♦s ❡s❝❛❧❛r❡s ε ❡ ε sã♦ ♣❛✲ ν ℓ ℓ s s = (τ /τ ) ε r❛♠❡tr✐③❛❞♦s ♣♦r ✉♠❛ ❢r❡q✉ê♥❝✐❛ ❛♥❣✉❧❛r ω✳ P♦rt❛♥t♦✱ ✉s❛♥❞♦ ❛ r❡❧❛çã♦ ε ✱

              = 0 ♥ós ❡①❡❝✉t❛♠♦s ✉♠❛ ♦✉tr❛ ❡①♣❛♥sã♦ ❡♠ sér✐❡ ❞❡ ❚❛②❧♦r ♥❛s ✐♠❡❞✐❛çõ❡s ❞❡ ε s ✳ ❆s ❡①♣❛♥sõ❡s ❢♦r❛♠ r❡❛❧✐③❛❞❛s ♥♦ s♦❢t✇❛r❡ ▼❛t❤❡♠❛t✐❝❛ ❋✐♥❛❧♠❡♥t❡✱ ♦❜t❡♠♦s ♦s ❝♦❡✜❝✐❡♥t❡s ♥♦ ❧✐♠✐t❡ ❞❡ ❜❛✐①❛ ❢r❡q✉ê♥❝✐❛ ❝♦♠♦ 2 2

              ρ c [ε ℓ /c ℓ ) ε ] πν

              ✭❘✮ ✭❊✮ s s

              − (4/3)(c 2 f (ε , ε + ℓ ) = f cos ), 2 + 2 2

              s

              ❖(ε ✭✹✳✻✽✮ 2 ρ c /c ) ] 1 s ℓ ℓ [1 − (4/3)(c 2

              2 ρ c [ε ℓ /c ℓ ) ε ] πν

              ✭■✮ s s

              − (4/3)(c 2 f

            • (ε ℓ , ε sin ),

              s

              ❖(ε ✭✹✳✻✾✮ ) = − 2 2 2

              ρ c /c ) ] 1 s ℓ ℓ [1 − (4/3)(c

              2 2(ρ ) 1

              ✭❘✮ − ρ

              f = , 1 ✭✹✳✼✵✮

              2ρ + ρ 1

              ✭■✮

              f (ε , ε )

              ✭■✮ ℓ s 2 g (ε + , ε ). 1 ) = − ❖(ε ✭✹✳✼✶✮ ℓ s 2

              5(ρ /ρ + 2) 1

              ✭❊✮

              ❖ ❝♦❡✜❝✐❡♥t❡ f é ❞❛❞♦ ♣♦r 2 ρ c

              ✭❊✮ f .

              = 1 − 2 2 ✭✹✳✼✷✮ ρ c /c ) ] 1 ℓ [1 − (4/3)(c s

              ✹✳✹✳✷ ❘❡❣✐♠❡ ❞❡ ❛❧t❛ ❢r❡q✉ê♥❝✐❛ ℓ s , ε

              ◆♦ ♦✉tr♦ ❡①tr❡♠♦✱ t❡♠♦s ♦ ❧✐♠✐t❡ ❞❡ ❛❧t❛ ❢r❡q✉ê♥❝✐❛ ε ≫ 1✱ ♥♦ q✉❛❧ ❛ ✈✐s❝♦s✐❞❛❞❡ s✉♣❡r❛ ❛ ❡❧❛st✐❝✐❞❛❞❡✳ ◆❡st❡ ❝❛s♦✱ ♦ í♥❞✐❝❡ ❞❡ r❡❢r❛çã♦ ❝♦♠♣❧❡①♦ ❞❛ ❊q✳ é ❡①♣r❡ss♦ ♣♦r − − ν/2 1/2 3/2

              , n (ε + j ε

              ❝ ❖ ε ✭✹✳✼✸✮ ) = (−✐) j j ∈ {ℓ, s}.

              ❆ ♣❛rt✐r ❞❡st❛ ❡q✉❛çã♦✱ ♦❜s❡r✈❛♠♦s q✉❡ ❛ ❛❜s♦rçã♦ ❛❞✐♠❡♥s✐♦♥❛❧ ❡ í♥❞✐❝❡ ❞❡ r❡❢r❛çã♦ é ❞❛❞♦ ♣♦r − πν 1/2

              α(ε ˜ ) = ε sin , j j ✭✹✳✼✹✮ − πν 1/2

              4 n (ε ) = ε cos .

              ❘ j j

              ✭✹✳✼✺✮

              4 1/2 (ε ) = )

              

            ❘ j

              ◆♦t❡ q✉❡ ♦ í♥❞✐❝❡ ❞❡ r❡❢r❛çã♦ t❡♠ ♦r❞❡♠ n ❖(ε j ✳ ❆ss✐♠✱ ❛ ✈❡❧♦❝✐❞❛❞❡ ❞❡ ❢❛s❡ ❞❛ ♦♥❞❛ é ♠✉✐t♦ ♠❛✐♦r q✉❡ ❛ ✈❡❧♦❝✐❞❛❞❡ ❞♦ s♦♠ ❧♦♥❣✐t✉❞✐♥❛❧ ❡ ❞❡ ❝✐s❛❧❤❛♠❡♥t♦✳

              P❛r❛ ❞❡t❡r♠✐♥❛r ♦ ✐♥t❡r✈❛❧♦ ❞❡ ✈❛r✐❛çã♦ ❞♦ ♣❛râ♠❡tr♦ ν✱ ♥♦t❛♠♦s q✉❡ t❛♥t♦ ❛ ❛❜s♦rçã♦ )

              ❛❞✐♠❡♥s✐♦♥❛❧ ˜α(ε j ❝♦♠♦ ♦ í♥❞✐❝❡ ❞❡ r❡❢r❛çã♦ n ❘ ❞❡✈❡rã♦ s❡r❡♠ ♣♦s✐t✐✈♦s✱ ♣❡❧♦ ♠❡♥♦s ♣❛r❛ ♦ ♠❛t❡r✐❛❧ ✈✐s❝♦❡❧ást✐❝♦✳ ❆ss✐♠✱ ♦ ✐♥t❡r✈❛❧♦ ♠á①✐♠♦ ❞❡ss❡ ♣❛râ♠❡tr♦ é ν ∈ (0, 2)✳ ❇❛s❡❛❞♦s ♥❛s ❛♥á❧✐s❡s ❞❛s ❛♣r♦①✐♠❛çõ❡s ❞❡ ❜❛✐①❛ ❡ ❛❧t❛ ❢r❡q✉ê♥❝✐❛✱ ❝♦♥❝❧✉í♠♦s q✉❡ ❛ ♦r❞❡♠ ❞❛ ❞❡r✐✈❛❞❛ ❢r❛❝✐♦♥ár✐❛ ν ❞❡✜♥✐❞❛ ♥❛ ❊q✳ ❞❡✈❡rá ✈❛r✐❛r ♥♦ ✐♥t❡r✈❛❧♦ ν ∈ (0, 2)✳ ❯t✐❧✐③❛♥❞♦ ❛s ❊qs✳ ❡♥❝♦♥tr❛♠♦s ❛ r❡❧❛çã♦ ❡♥tr❡ ❛ ♦r❞❡♠ ❞❛ ❞❡r✐✈❛❞❛ ❢r❛❝✐♦♥ár✐❛ ν ❡ ♦ ❡①♣♦❡♥t❡ ❞❡ ❛❜s♦rçã♦ ❞❛ ❧❡✐ ❞❡ ♣♦tê♥❝✐❛ y ♥❛ ❛♣r♦①✐♠❛çã♦ ❞❡ ❛❧t❛ ❢r❡q✉ê♥❝✐❛ é

              ν y . = 1 − ✭✹✳✼✻✮

              2 P❛r❛ ❝❛❧❝✉❧❛r ♦s ❝♦❡✜❝✐❡♥t❡s ❞❡ ♠♦♥♦♣♦❧♦ ❡ ❞✐♣♦❧♦✱ ♥❛s ❊qs✳ ❞❡s❡♥✈♦❧✲ ✈❡♠♦s ✉♠ ♣r♦❝❡❞✐♠❡♥t♦ s❡♠❡❧❤❛♥t❡ ❛♦ r❡❣✐♠❡ ❞❡ ❜❛✐①❛ ❢r❡q✉ê♥❝✐❛ ❞❡s❝r✐t♦ ♥❛ ❙❡çã♦

              s

              ◆♦ ❡♥t❛♥t♦✱ ❛ ú❧t✐♠❛ ❡①♣❛♥sã♦ ❡♠ sér✐❡ ❞❡ ❚❛②❧♦r é r❡❛❧✐③❛❞❛ ❝♦♠ ε → ∞✳ ❈♦♥s❡q✉❡♥✲ t❡♠❡♥t❡✱ ♦s ❝♦❡✜❝✐❡♥t❡s ♥♦ ❧✐♠✐t❡ ❞❡ ❛❧t❛ ❢r❡q✉ê♥❝✐❛ sã♦ 2 ρ c cos(πν/2)

              ✭❘✮ 2 ℓ s + f (ε , ε

              ), ) = 1 − ❖(ε ✭✹✳✼✼✮ 2 2

              ρ c [ε /c ) ε ] 1 ℓ s ℓ s 2 − (4/3)(c ρ c sin(πν/2)

              ✭■✮ 2

              f (ε , ε ℓ s ), ) = − ❖(ε ✭✹✳✼✽✮ 2 2

            • ρ c [ε /c ) ε ] 1 ℓ ℓ

              − (4/3)(c s s

              ✭■✮

              f (ε , ε )

              ✭■✮ ℓ s 2 g (ε , ε ). 1 ) = − ℓ s ❖(ε ✭✹✳✼✾✮ 2 +

              5(ρ /ρ + 2) 1

              ✭❘✮

              ❖ ❝♦❡✜❝✐❡♥t❡ f 1 é ♦ ♠❡s♠♦ ❞♦ r❡❣✐♠❡ ❞❡ ❜❛✐①❛ ❢r❡q✉ê♥❝✐❛ ❞❛❞♦ ♥❛ ❊q✳ ◆❡st❡ ❝❛♣ít✉❧♦ ❢♦r❛♠ ❞❡♠♦♥str❛❞❛s ❛s ❡①♣r❡ssõ❡s ❛♥❛❧ít✐❝❛s ♣❛r❛ ❢♦rç❛ ❡ t♦rq✉❡ ❞❡ r❛❞✐❛çã♦ ❛❝úst✐❝♦ s♦❜r❡ ✉♠❛ ♣❛rtí❝✉❧❛ ❡s❢ér✐❝❛ ✉t✐❧✐③❛♥❞♦ ♦ ♠ét♦❞♦ ❞❡ ❝❛♠♣♦ ❞✐st❛♥t❡✱ ❝♦♠ ê♥❢❛s❡ ♥❛ ❛♣r♦①✐♠❛çã♦ ♠♦♥♦♣♦❧♦✲❞✐♣♦❧♦ ❞♦s ❝♦❡✜❝✐❡♥t❡s ❞❡ ❡s♣❛❧❤❛♠❡♥t♦ ❘❛②❧❡✐❣❤✳ ❆❧é♠ ❞✐ss♦✱ ❡❢❡t✉❛♠♦s ✉♠❛ ❡①♣❛♥sã♦ ❛ss✐♥tót✐❝❛ ❞♦ í♥❞✐❝❡ ❞❡ r❡❢r❛çã♦ ❝♦♠♣❧❡①♦ ❡♠ ❞♦✐s r❡❣✐♠❡s ✭❜❛✐①❛ ❡ ❛❧t❛ ❢r❡q✉ê♥❝✐❛✮ ♣❛r❛ ♦❜t❡r r❡s✉❧t❛❞♦s ♠❛✐s s✐♠♣❧❡s ❞❡ s❡r❡♠ ❛♥❛❧✐s❛❞♦s✳ ❊ss❡ ❡st✉❞♦ s❡rá ❛ ❜❛s❡ ♣❛r❛ ♥♦ss❛s ❞✐s❝✉ssõ❡s ♥♦ ♣ró①✐♠♦ ❝❛♣ít✉❧♦✱ ♦♥❞❡ ✐♥✈❡st✐❣❛r❡♠♦s ♦ ❡❢❡✐t♦ ❞❛ ❢♦rç❛ ❡ t♦rq✉❡ ❞❡ r❛❞✐❛çã♦ ❛❝úst✐❝♦ s♦❜r❡ ✉♠❛ ♣❛rtí❝✉❧❛ ✈✐s❝♦❡❧ást✐❝❛ ♣❛r❛ três ❞✐❢❡r❡♥t❡s ❢r❡♥t❡s ❞❡ ♦♥❞❛✳

              5 ❘❡s✉❧t❛❞♦s ❡ ❞✐s❝✉ssõ❡s

              P❛rtí❝✉❧❛s s✉s♣❡♥s❛s ❡♠ ✉♠ ✢✉✐❞♦ ❡ ❡①♣♦st❛s ❛ ✉♠ ❝❛♠♣♦ ❛❝úst✐❝♦ s❡rã♦ ❛❢❡t❛❞❛s ♣♦r ✉♠❛ ❢♦rç❛ ❞❡ r❛❞✐❛çã♦ ❛❝úst✐❝❛✳ ❆♣❧✐❝❛çõ❡s q✉❡ ✉t✐❧✐③❛♠ ♦ ♠♦✈✐♠❡♥t♦ ❞❡ ✉♠ ♦❜❥❡t♦ ❞❡✈✐❞♦ ❛♦ ❢❡♥ô♠❡♥♦ ❞❡ ❢♦rç❛ ❞❡ r❛❞✐❛çã♦ t♦r♥❛r❛♠✲s❡ ✉♠ ♠ét♦❞♦ ♣r♦♠✐ss♦r ❡♠ ❜✐♦t❡❝✲ ♥♦❧♦❣✐❛ ✳ ❊♥tr❡ ❡ss❛s ❛♣❧✐❝❛çõ❡s✱ ♣♦❞❡♠♦s ❝✐t❛r ♠❛♥✐♣✉❧❛çã♦ ❞❡ ♣❛rtí❝✉❧❛s s❡♠ ❝♦♥t❛t♦✱ ✐♥❝❧✉✐♥❞♦ ❝é❧✉❧❛s ❜✐♦❧ó❣✐❝❛s ❡ ♦✉tr♦s ♠✐❝r♦r❣❛♥✐s♠♦s✱ ❡ ❧❡✈✐t❛çã♦ ❛❝úst✐❝❛ P♦rt❛♥t♦✱ ✉♠❛ ✐♥✈❡st✐✲ ❣❛çã♦ ♠❛✐s ❛♠♣❧❛ s♦❜r❡ ❝♦♠♦ ❛ ✈✐s❝♦❡❧❛st✐❝✐❞❛❞❡ ❞❛ ♣❛rtí❝✉❧❛ ❛❢❡t❛ ❛ ❢♦rç❛ ❡ ♦ t♦rq✉❡ ❞❡ r❛❞✐❛çã♦ ❛❝úst✐❝♦ ♥❛ ❛♣r♦①✐♠❛çã♦ ❞❡ ❘❛②❧❡✐❣❤ s❡rá ❛♣r❡s❡♥t❛❞❛ ♥❡ss❡ ❝❛♣ít✉❧♦✳ P❛r❛ ✐❧✉s✲ tr❛r✱ ✐r❡♠♦s ❛♥❛❧✐s❛r ❡ss❡s ❢❡♥ô♠❡♥♦s ❛t✉❛♥❞♦ s♦❜r❡ ✉♠❛ ♣❡q✉❡♥❛ ♣❛rtí❝✉❧❛ s✉s♣❡♥s❛ ❡♠ ✉♠ ✢✉✐❞♦ ✐❞❡❛❧ ❞❡✈✐❞♦ ❛ ✐♥t❡r❛çã♦ ❝♦♠ ✉♠❛ ♦♥❞❛ ❛❝úst✐❝❛ ❞❡ três t✐♣♦s✿ ✉♠❛ ♦♥❞❛ ♣❧❛♥❛ ♣r♦❣r❡ss✐✈❛✱ ♦♥❞❛ ♣❧❛♥❛ ❡st❛❝✐♦♥ár✐❛ ❡ ❢❡✐①❡ ❞❡ ❇❡ss❡❧✳ P❛r❛ t❛❧ ❛♥á❧✐s❡✱ ✐r❡♠♦s ❝♦♥s✐❞❡r❛r ✉♠❛ ♣❛rtí❝✉❧❛ ❞❡ ♣♦❧✐❡t✐❧❡♥♦ ❞❡ ❜❛✐①❛ ❞❡♥s✐❞❛❞❡ ❡ ♦✉tr❛ ❞❡ ❛❧t❛ ❞❡♥s✐❞❛❞❡✳

              ✺✳✶ ❊①❡♠♣❧♦s ❞❡ ♦♥❞❛s ❛❝úst✐❝❛s

              ❈♦♥s✐❞❡r❡ ✉♠ ❢❡✐①❡ ❛❝úst✐❝♦ q✉❡ s❡ ♣r♦♣❛❣❛ ♥❛ ❞✐r❡çã♦ ❛①✐❛❧✱ ✐st♦ é✱ ♥♦ ❡✐①♦ z✳ ❯♠❛ ♣❛rtí❝✉❧❛ ✈✐s❝♦❡❧ást✐❝❛ ❞❡ r❛✐♦ a é ❝♦❧♦❝❛❞❛ ♥♦ tr❛❥❡t♦ ❞❛ ♦♥❞❛✳ ◆❡st❡ ❝❡♥ár✐♦✱ ❞✉❛s ❢✉♥çõ❡s r❡❧❡✈❛♥t❡s s❡rã♦ ✉t✐❧✐③❛❞❛s ♠❛✐s ❛❞✐❛♥t❡✱ ❛ ✐♥t❡♥s✐❞❛❞❡ ♠é❞✐❛ ✐♥❝✐❞❡♥t❡ I ✐♥ ❡ ❛ ❝♦♠♣♦♥❡♥t❡ ❛①✐❛❧ ❞♦ ✢✉①♦ ❞❡ ♠♦♠❡♥t♦✳ ❊st❛s ❢✉♥çõ❡s sã♦ ❞❛❞❛s ❡♠ t❡r♠♦s ❞❛ ♣r❡ssã♦ ✐♥❝✐❞❡♥t❡ p ✐♥ ❡ ❞❛ ✈❡❧♦❝✐❞❛❞❡ ❞♦ ✢✉✐❞♦ v ✐♥ ✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✱ ♣♦r

              1

              I = v ],

              ✐♥ ✐♥

              ❘❡[p ✐♥ ✭✺✳✶✮ ∗ ∗ ∗

              2 ρ v = ρ ∂ (v v ) + ∂ (v v ) ,

              

            ✐♥ z x ✐♥,x y ✐♥,y

              ∇ · v | ✭✺✳✷✮

              ✐♥ ✐♥,z ✐♥,z x y = ∂/∂x = ∂/∂y

              ♦♥❞❡ ρ é ❞❡♥s✐❞❛❞❡ ❞♦ ✢✉✐❞♦✱ ∂ ❡ ∂ ✳ ❆❧é♠ ❞✐ss♦✱ é út✐❧ ❞❡✜♥✐r ❛ ♠❛❣♥✐t✉❞❡ ❞❛ ❢♦rç❛ ❞❡ r❛❞✐❛çã♦ ❛❝úst✐❝❛ ❝♦♠♦ 2

              πa

              I F = , ✭✺✳✸✮ c

              2

              = ρ c v /2 ♦♥❞❡ I é ❛ ♠❛❣♥✐t✉❞❡ ❞❛ ✐♥t❡♥s✐❞❛❞❡ ♠é❞✐❛ ✐♥❝✐❞❡♥t❡ ❝♦♠ v s❡♥❞♦ ❛ ♠❛❣♥✐✲ t✉❞❡ ❞❛ ✈❡❧♦❝✐❞❛❞❡ ❞♦ ✢✉✐❞♦ ♥❛ ❢♦♥t❡ ❛❝úst✐❝❛ ❡ c ❛ ✈❡❧♦❝✐❞❛❞❡ ❞❡ ♣r♦♣❛❣❛çã♦ ❞♦ s♦♠ ♥♦ ✢✉✐❞♦ ✐❞❡❛❧✳

              ✺✳✶✳✶ ❖♥❞❛ ♣❧❛♥❛ ♣r♦❣r❡ss✐✈❛

              (z)

              ✐♥

              ❈♦♥s✐❞❡r❡ ❛ ❛♠♣❧✐t✉❞❡ ❞♦ ♣♦t❡♥❝✐❛❧ ❞❡ ✈❡❧♦❝✐❞❛❞❡ φ ❞❡ ✉♠❛ ♦♥❞❛ ♣❧❛♥❛ ♣r♦❣r❡ss✐✈❛ ✭❖PP✮ ❝♦♠♦ v

              ✐kz

              φ (z) = ,

              ✐♥ ❡ ✭✺✳✹✮

              k ♦♥❞❡ k é ♦ ♥ú♠❡r♦ ❞❡ ♦♥❞❛✳

              ✐kz

              = c v

              ✐♥

              ❆s ❛♠♣❧✐t✉❞❡s ❞❛ ♣r❡ssã♦ ✐♥❝✐❞❡♥t❡ ❡ ❞❛ ✈❡❧♦❝✐❞❛❞❡ ❞♦ ✢✉✐❞♦ sã♦ p ✐ρ ❡ ❡

              ✐kz

              v =

              

            ✐♥,z ✐v ❡ ✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✳ ❆ ♣❛rt✐r ❞❛ ❊q✳ ❡♥❝♦♥tr❛♠♦s q✉❡ ❛ ✐♥t❡♥s✐❞❛❞❡ ♠é❞✐❛

            z = I

              ✐♥❝✐❞❡♥t❡ é I ❡ ♦ ❞✐✈❡r❣❡♥t❡ ❞♦ ✢✉①♦ ❞❡ ♠♦♠❡♥t♦ ❞❛❞♦ ♥❛ ❊q✳ é ③❡r♦✳ ◆♦t❡ q✉❡ ❛ ♣❛rt❡ ❝♦♥s❡r✈❛t✐✈❛ ❞❛ ❢♦rç❛ ❞❡ r❛❞✐❛çã♦ ❛❝úst✐❝❛✱ ❢♦rç❛ ❣r❛❞✐❡♥t❡ ❊q✳ t❛♠❜é♠ é ③❡r♦✳ ❉❡st❛ ❢♦r♠❛✱ ❛ ❢♦rç❛ ❞❡ r❛❞✐❛çã♦ ❛❝úst✐❝❛ ❡①❡r❝✐❞❛ ❡♠ ✉♠❛ ♣❛rtí❝✉❧❛ ✈✐s❝♦❡❧ást✐❝❛ ❞❡✈✐❞♦ ❛ ✉♠❛ ♦♥❞❛ ♣❧❛♥❛ ♣r♦❣r❡ss✐✈❛ é ❞♦♠✐♥❛♥t❡♠❡♥t❡ ❞❡ ❛❜s♦rçã♦✳ P♦rt❛♥t♦✱ ❞❛ ❊q✳ t❡♠♦s q✉❡ ❛ ❢♦rç❛ ❞❡ r❛❞✐❛çã♦ ❛❝úst✐❝❛ ❛①✐❛❧ é ❞❛❞❛ ♣♦r

              4F

              ✭■✮ ❖PP ❖PP 3 F = F (ε , ε εf (ε , ε ) + ). ℓ s ℓ s r❛❞,z ❛❜s,z ) = − ❖(ε ✭✺✳✺✮

              3

              ✭■✮

              (ε , ε ) ❆ ❢✉♥çã♦ f s é ❞❛❞❛ ♣❡❧❛s ❊qs✳ ♣❛r❛ ❛♣r♦①✐♠❛çã♦ ❞❡ ❜❛✐①❛ ❡ ❛❧t❛ ❢r❡q✉ê♥❝✐❛ ❞❛ ♣❛rtí❝✉❧❛ ✈✐s❝♦❡❧ást✐❝❛✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✳ ❊♥tã♦ ❛ ❢♦rç❛ ❞❡ r❛❞✐❛çã♦ ❛❝úst✐❝❛

              ❖PP 2

              = ) s❡ ❝♦♠♣♦rt❛ ❝♦♠♦ F z ❖(ε ✳ ◆♦t❡ q✉❡✱ ❞❡ ❛❝♦r❞♦ ❝♦♠ ❛ ❊q✳ ❛ ❢♦rç❛ ❞❡ r❛❞✐❛✲ 3 çã♦ ❞❡ ❡s♣❛❧❤❛♠❡♥t♦ é ε ♠❛✐s ❢r❛❝❛ ❞♦ q✉❡ ❛ ❢♦rç❛ ❞❡ r❛❞✐❛çã♦ ❞❡ ❛❜s♦rçã♦ ❛♣r❡s❡♥t❛❞❛ ♥❛ ❊q✳

              ➱ ✐♠♣♦rt❛♥t❡ r❡ss❛❧t❛r q✉❡ ❛ ❢♦rç❛ ❞❡ r❛❞✐❛çã♦ ❛❝úst✐❝❛ ♣♦❞❡ s❡r ♥❡❣❛t✐✈❛ ✭✐st♦ é✱

              ✭■✮

              (ε , ε ) > 0 ❛ ❢♦rç❛ é ♦♣♦st❛ ❛ ❞✐r❡çã♦ ❞❡ ♣r♦♣❛❣❛çã♦ ❞♦ ❢❡✐①❡✮✱ s❡ f ℓ s ✱ ♦ q✉❡ ❝♦♥❞✉③ à s❡❣✉✐♥t❡ ❝♦♥❞✐çã♦ " # 1/ν 2 c

              τ ℓ

              4

              s

              < , ✭✺✳✻✮

              τ 3 c ℓ s s ℓ ♦♥❞❡ τ ❡ τ sã♦ ♦ t❡♠♣♦ ❞❡ r❡❧❛①❛çã♦ ❧♦♥❣✐t✉❞✐♥❛❧ ❡ ❞❡ ❝✐s❛❧❤❛♠❡♥t♦✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✱ c ℓ s

              é ✈❡❧♦❝✐❞❛❞❡ ❧♦♥❣✐t✉❞✐♥❛❧✱ c é ❛ ✈❡❧♦❝✐❞❛❞❡ ❞❡ ❝✐s❛❧❤❛♠❡♥t♦ ❡ ν é ❛ ♦r❞❡♠ ❢r❛❝✐♦♥ár✐❛ ✈✐s❝♦❡❧ást✐❝❛✳ ❖❜s❡r✈❡ q✉❡ ❡ss❛ ❝♦♥❞✐çã♦ é ✈á❧✐❞❛ t❛♥t♦ ♣❛r❛ ♦ r❡❣✐♠❡ ❞❡ ❜❛✐①❛ ❢r❡q✉ê♥❝✐❛ ❝♦♠♦ ♣❛r❛ ♦ r❡❣✐♠❡ ❞❡ ❛❧t❛ ❢r❡q✉ê♥❝✐❛✳ ❋♦rç❛ ❞❡ r❛❞✐❛çã♦ ❛❝úst✐❝❛ ♥❡❣❛t✐✈❛ ❞❡✈✐❞♦ ❛ ✉♠❛ ♦♥❞❛ ♣❧❛♥❛ t❛♠❜é♠ ❢♦✐ ♥♦t❛❞❛ ❡♠ ✉♠❛ ♣❛rtí❝✉❧❛ ❞❡ ♦✉r♦ r❡✈❡st✐❞❛ ❝♦♠ ✉♠❛ ❝❛♠❛❞❛ ❞❡ ✉♠ ♣♦❧í♠❡r♦ ✭✈✐s❝♦❡❧ást✐❝♦✮

              ❆ ❢♦rç❛ ❞❡ r❛❞✐❛çã♦ ❛❝úst✐❝❛ s♦❜r❡ ✉♠❛ ♣❛rtí❝✉❧❛ ✢✉✐❞❛ ❝♦♠ ❛❜s♦rçã♦ ❧♦♥❣✐t✉❞✐♥❛❧ s s = 0 = 0 t❛♠❜é♠ é ❞❛❞❛ ♣❡❧❛ ❊q✳ ❖ r❡s✉❧t❛❞♦ ♦❜t✐❞♦ ❡stá ❡♠ ❛❝♦r❞♦ ❝♦♠ ❛ ❘❡❢✳ ❆❧é♠ ❞✐ss♦✱ ❛ ❢♦rç❛ ❞❡ r❛❞✐❛çã♦ ❛❝úst✐❝❛ ❡①❡r❝✐❞❛ s♦❜r❡ ✉♠❛ ♣❛rtí❝✉❧❛ só❧✐❞❛ ❡❧ást✐❝❛ ♣♦❞❡ s❡r ♦❜t✐❞❛ ❞❛ ❊q✳ ❢❛③❡♥❞♦ τ = τ = 0 ℓ s

              ♥❛ ❖ r❡s✉❧t❛❞♦ ❡stá ❞❡ ❛❝♦r❞♦ ❝♦♠ ♦ q✉❡ ❢♦✐ ♣r❡✈✐❛♠❡♥t❡ ❞❡r✐✈❛❞♦ ♥❛ ❘❡❢✳

              ✺✳✶✳✷ ❖♥❞❛ ♣❧❛♥❛ ❡st❛❝✐♦♥ár✐❛

              ❈♦♥s✐❞❡r❡ ✉♠❛ ♦♥❞❛ ♣❧❛♥❛ ❡st❛❝✐♦♥ár✐❛ ✭❖P❊✮ ❞✐str✐❜✉í❞❛ ❛♦ ❧♦♥❣♦ ❞♦ ❡✐①♦ z✳ ❊ss❡ t✐♣♦ ❞❡ ♦♥❞❛ ❛❝úst✐❝❛ é ♠✉✐t♦ ✐♠♣♦rt❛♥t❡ ♣❛r❛ ❛♣❧✐❝❛çõ❡s q✉❡ ❡♥✈♦❧✈❡♠ ❢♦rç❛ ❞❡ r❛❞✐❛çã♦ ❛❝úst✐❝❛✳ ❆ ❛♠♣❧✐t✉❞❡ ❞♦ ♣♦t❡♥❝✐❛❧ ❞❡ ✈❡❧♦❝✐❞❛❞❡ ❞❛ ♦♥❞❛ ❡st❛❝✐♦♥ár✐❛ é v

              φ (z) =

              ✐♥

              sin[k(z − h)], ✭✺✳✼✮ k ♦♥❞❡ h é ♦ ♣❛râ♠❡tr♦ ❞❡ ❞❡s❧♦❝❛♠❡♥t♦✱ q✉❡ r❡♣r❡s❡♥t❛ ❛ ❞✐stâ♥❝✐❛ ❞♦ ♣r✐♠❡✐r♦ ♥ó ♣❛r❛ ❛ ♦r✐❣❡♠ ❞♦ s✐st❡♠❛ ❞❡ ❝♦♦r❞❡♥❛❞❛s✳

              = c v = v ❖s ❝❛♠♣♦s ❛❝úst✐❝♦s ✐♥❝✐❞❡♥t❡s sã♦ p ✐♥ ✐ρ sin[k(z − h)] ❡ v ✐♥ cos[k(z − h)]✳

              ◆❡st❡ ❝❛s♦✱ ♦❜s❡r✈❛♠♦s ❞❛s ❊qs✳ q✉❡ t❛♥t♦ ❛ ✐♥t❡♥s✐❞❛❞❡ ✐♥❝✐❞❡♥t❡ ❝♦♠♦ ♦ ✢✉①♦ ❞❡ ♠♦♠❡♥t♦ ❞❡s❛♣❛r❡❝❡♠✳ P♦rt❛♥t♦✱ ❛♣❡♥❛s ❛ ♣❛rt❡ ❝♦♥s❡r✈❛t✐✈❛ ❞❛ ❢♦rç❛ ❞❡ r❛❞✐❛çã♦ ❛❝úst✐❝❛✱ q✉❡ ❝♦rr❡s♣♦♥❞❡ ❛ ❢♦rç❛ ❣r❛❞✐❡♥t❡ ❞❛ ❊q✳ ♣❡r♠❛♥❡❝❡✳ ❈♦♥s❡✲ q✉❡♥t❡♠❡♥t❡✱ ❡♥❝♦♥tr❛♠♦s h i

              F ( (

              

            ❘) ❘)

            ❖P❊ 2 .

              F (ε ℓ , ε ε 2f (ε ℓ , ε ) + 3f

              s s

              ✭✺✳✽✮

              ❣r❛❞,z ; z) = − 1 sin[2k(z − h)] + ❖ ε

              3 ❊st❡ r❡s✉❧t❛❞♦ t❡♠ ♦ ♠❡s♠♦ ❢♦r♠❛t♦ q✉❡ ♦ ♦❜t✐❞♦ ♣♦r ●♦r❦♦✈ ♣❛r❛ ✉♠❛ ♣❛rtí❝✉❧❛ ✢✉✐❞❛ ❞❡ ❝♦♠♣r❡ssã♦ ♥ã♦ ❛❜s♦r✈❡❞♦r❛✳

              ◆❛ ❛♥á❧✐s❡ ❞❡ ❛♣r✐s✐♦♥❛♠❡♥t♦ ❞❡ ♣❛rtí❝✉❧❛s✱ é út✐❧ ❞❡✜♥✐r ✉♠ ❢❛t♦r ❞❡ ❝♦♥tr❛st❡ ❛❝úst✐❝♦ ❝♦♠♦

              ρ 1 ( − ρ ❘) ℓ ℓ , ε + f (ε , ε ),

              C(ε s ) ≡ 3 s ✭✺✳✾✮ 2ρ 1 + ρ

              ♦♥❞❡ ❡ss❡ ❢❛t♦r ❝♦♥té♠ ❛s ♣r♦♣r✐❡❞❛❞❡s ❢ís✐❝❛s ❞♦ ♠❡✐♦ ❡ ❞❛ ♣❛rtí❝✉❧❛✳ ❖ ❢❛t♦r ❞❡ ❝♦♥tr❛st❡ ❞❡t❡r♠✐♥❛ s❡ ✉♠❛ ♣❛rtí❝✉❧❛ t❡♠ ✉♠ ♣♦♥t♦ ❞❡ ❡q✉✐❧í❜r✐♦ ❡stá✈❡❧ ❡♠ ✉♠ ♥ó ❞❡ ♣r❡ssã♦ C > 0

              ✭❘✮

              (ε , ε ) ℓ s ♦✉ ❡♠ ✉♥ ❛♥t✐✲♥ó C < 0✳ ❆ ❢✉♥çã♦ f é ❞❛❞❛ ♣❡❧❛s ❊qs✳ ♣❛r❛ ♦ r❡❣✐♠❡ ❞❡ ❜❛✐①❛ ❡ ❛❧t❛ ❢r❡q✉ê♥❝✐❛✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✳

              ✺✳✶✳✸ ❋❡✐①❡ ❞❡ ❇❡ss❡❧ ❛❝úst✐❝♦

              ❋❡✐①❡s ❞❡ ❇❡ss❡❧ ♥ã♦ ❞✐❢r❛t❛♥t❡s✱ ✐st♦ é✱ ❢❡✐①❡s q✉❡ s❡ ♣r♦♣❛❣❛♠ ♣♦r ✉♠❛ ❞✐stâ♥❝✐❛ ✐♥✜♥✐t❛ s❡♠ s♦❢r❡r ❞✐❢r❛çã♦ ♣♦r ✉♠❛ ❛❜❡rt✉r❛ ✐♥✜♥✐t❛✱ ❞❡♥♦t❛♠ ✉♠❛ s♦❧✉çã♦ ❡①❛t❛ ❞❛ ❡q✉❛çã♦ ❤♦♠♦❣ê♥❡❛ ❞❡ ❍❡❧♠❤♦❧t③ ❡①♣r❡ss❛s ❡♠ ❝♦♦r❞❡♥❛❞❛s ❝✐❧í♥❞r✐❝❛s ◆♦ ❡♥t❛♥t♦✱ ❡♠ s✐t✉❛çõ❡s r❡❛✐s✱ ♦ ❡❢❡✐t♦ ❞❡ ❞✐❢r❛çã♦ ♥ã♦ ♣♦❞❡ s❡r ❞❡s♣r❡③❛❞♦✱ ❛♣❡♥❛s ❛♣r♦①✐♠❛çõ❡s ♣❛r❛ ❡ss❡s ❢❡✐①❡s ♣♦❞❡♠ s❡r ♦❜t✐❞♦s ♣♦r ❛❜❡rt✉r❛s ✜♥✐t❛s ♠❛♥t❡♥❞♦ ♦ ♣❡r✜❧ ❞♦ ❢❡✐①❡ ❛♦ ❧♦♥❣♦ ❞❡ ✉♠❛ ❣r❛♥❞❡ ❞✐stâ♥❝✐❛ ❛①✐❛❧✳ ❊ss❡s ❢❡✐①❡s ❣❡r❛❧♠❡♥t❡ sã♦ ❝❤❛♠❛❞♦s ❞❡ ❢❡✐①❡s ❞❡ ❞✐❢r❛çã♦ ❧✐♠✐t❛❞❛✳

              ❈♦♥s✐❞❡r❡ ✉♠ ❢❡✐①❡ ❞❡ ❇❡ss❡❧ ❞❡ n✲és✐♠❛ ♦r❞❡♠ s❡ ♣r♦♣❛❣❛♥❞♦ ❛♦ ❧♦♥❣♦ ❞♦ ❡✐①♦ z ❝♦♠♦ ♠♦str❛❞♦ ♥❛ ❋✐❣✳ ❖ ❢❡✐①❡ ❡♥❝♦♥tr❛ ✉♠❛ ♣❛rtí❝✉❧❛ ♣♦s✐❝✐♦♥❛❞❛ s♦❜r❡ s❡✉ ❡✐①♦✳ ❆ ❡s❝♦❧❤❛ ♥❛t✉r❛❧ ❞❡ ✉♠ ❝♦♥❥✉♥t♦ ❞❡ ✈❛r✐á✈❡✐s ♣❛r❛ ❞❡s❝r❡✈❡r ❡ss❡ ❢❡✐①❡ é ♦ s✐st❡♠❛ ❞❡ 2 2

            • y ❝♦♦r❞❡♥❛❞❛s ❝✐❧í♥❞r✐❝❛s (̺, ϕ, z)✱ ♦♥❞❡ ̺ = px é ❛ ❞✐stâ♥❝✐❛ r❛❞✐❛❧ tr❛♥s✈❡rs❛❧ ❡
            • 1 ϕ = tan (y/x)

                é ♦ â♥❣✉❧♦ ❛③✐♠✉t❛❧✳ ❆ ❛♠♣❧✐t✉❞❡ ❞♦ ♣♦t❡♥❝✐❛❧ ❞❡ ✈❡❧♦❝✐❞❛❞❡ ❞♦ ❢❡✐①❡ é

                ❋❡✐①❡ ❞❡ ❇❡ss❡❧ r❡♣r❡s❡♥t❛❞♦ ♣♦r ✉♠❛ s✉♣❡r♣♦s✐çã♦ ❞❡ ♦♥❞❛s ♣❧❛♥❛s ❝♦♠ β s❡♥❞♦ ♦

                ❋✐❣✉r❛ ✺✳✶✿

                â♥❣✉❧♦ ❞❡ ♠❡✐♦ ❝♦♥❡✳ ❖ ❝✐r❝✉❧♦ ✈❡r♠❡❧❤♦ r❡♣r❡s❡♥t❛ ❛ ♣❛rtí❝✉❧❛✳ z

                ❋♦♥t❡✿ ❆✉t♦r✱ ✷✵✶✺✳

                ❞❛❞♦ ♣♦r v

                ✐(kz cos β+nϕ)

                φ (̺, ϕ, z) = J (k̺ sin β), n, ✐♥ ❡ n ✭✺✳✶✵✮ n k ♦♥❞❡ J é ❛ ❢✉♥çã♦ ❞❡ ❇❡ss❡❧ ❞❡ ♦r❞❡♠ n✱ β ∈ [0, π/2] é ♦ â♥❣✉❧♦ ❞❡ ♠❡✐♦ ❝♦♥❡✱ n ∈ Z é ❝♦♥❤❡❝✐❞♦ ❝♦♠♦ ♦ ♥✉♠❡r♦ ❞❡ ♠♦♠❡♥t♦ ❛♥❣✉❧❛r ♦r❜✐t❛❧✳ ❆ ✐♥t❡♥s✐❞❛❞❡ ♠é❞✐❛ ❞❡ ✉♠ ❢❡✐①❡ ❞❡ ❇❡ss❡❧ ❞❡ ♦r❞❡♠ n é ❡♥❝♦♥tr❛❞❛ ❝❛❧❝✉❧❛♥❞♦ ❛ ♣r❡ssã♦✱ ❊q✳ ❡ ❛ ✈❡❧♦❝✐❞❛❞❡ ✐♥❝✐❞❡♥t❡✱ ❊q✳ 2 nJ (k̺ sin β) n 2 I = e + J (k̺ sin β) cos βe , ϕ z

                ✐♥ n ✭✺✳✶✶✮ ϕ z

                ♦♥❞❡ e ❡ e sã♦ ✈❡t♦r❡s ✉♥✐tár✐♦s ❡♠ ❝♦♦r❞❡♥❛❞❛s ❝✐❧í♥❞r✐❝❛✳ ❉❡✈✐❞♦ ❛ s✐♠❡tr✐❛ ❞♦ ♣r♦❜❧❡♠❛✱ s♦♠❡♥t❡ ❛ ❝♦♠♣♦♥❡♥t❡ ❛①✐❛❧ ❞❛ ❢♦rç❛ ❞❡ r❛❞✐❛çã♦

                ❛❝úst✐❝❛ s✉r❣❡ ♥❛ ♣❛rtí❝✉❧❛✳ ❖❜t❡r❡♠♦s ❛ ✐♥t❡♥s✐❞❛❞❡ ❛①✐❛❧ ✐♥❝✐❞❡♥t❡ ❡ ♦ ✢✉①♦ ❞❡ ♠♦♠❡♥t♦ ♥♦ ❝❡♥tr♦ ❞❛ ♣❛rtí❝✉❧❛ ̺ = 0✳ P❛r❛ ✐ss♦✱ s✉❜st✐t✉í♠♦s ❛ ❊q✳ ✱ ❡ ✉t✐❧✐③❛♠♦s ♦ r❡s✉❧t❛❞♦ ❞❡♥tr♦ ❞❛s ❛ss✐♠

                I = I δ cos β, n,z n,0 ✭✺✳✶✷✮

                I 2 v v n, ✐♥ z n = k cos β sin β, ∇ · ρ n, ✐♥ | ✐γ ✭✺✳✶✸✮ nm c

                ♦♥❞❡ δ é ❛ ❢✉♥çã♦ ❞❡❧t❛ ❞❡ ❑r♦♥❡❝❦❡r✱ q✉❡ ✈❛❧❡ ✉♠ s❡ m = n ♦✉ ③❡r♦✱ ❝❛s♦ ❝♦♥trár✐♦✳ n = 2 ± = 0 1 n ❖ ❝♦❡✜❝✐❡♥t❡ γ é ❞❡✜♥✐❞♦ ❝♦♠♦ γ ✱ γ = −1✱ ❡ γ ✱ ♣❛r❛ |n| = 2, 3, . . .

                ◆♦t❡ q✉❡ ❛ ❢♦rç❛ ❞❡ r❛❞✐❛çã♦ ❛❝úst✐❝❛ ❛①✐❛❧ ❞❡✈✐❞♦ ❛ ✉♠ ❢❡✐①❡ ❞❡ ❇❡ss❡❧ ❞❡ ♦r❞❡♠ ③❡r♦ ✭❋❇❖❩✮ é ❣❡r❛❞❛ ♣r✐♥❝✐♣❛❧♠❡♥t❡ ♣❡❧❛ ❛❜s♦rçã♦ ❞❡♥tr♦ ❞❛ ♣❛rtí❝✉❧❛ ✈✐s❝♦❡❧ást✐❝❛✱ ♣♦✐s ❛ ❢♦rç❛ ❣r❛❞✐❡♥t❡ ♥❛ ❊q✳ ♥❛ ❊q✳ ❡♥❝♦♥tr❛♠♦s q✉❡ ❡st❛ ❢♦rç❛ ❡stá r❡❧❛❝✐♦♥❛❞❛ ❝♦♠ ❛ ❢♦rç❛ ❞❡ r❛❞✐❛çã♦ ❛❝úst✐❝❛ ❛①✐❛❧ ❝❛✉s❛❞❛ ♣♦r ✉♠❛ ♦♥❞❛ ♣❧❛♥❛ ♣r♦❣r❡ss✐✈❛✱

                ❋❇❖❩ ❖PP 3 F (ε , ε ) = F (ε , ε ) cos β + ). ℓ s ℓ s ❖(ε ✭✺✳✶✹✮ ❛❜s,z ❛❜s,z

                ❆ss✐♠✱ ❛ ❢♦rç❛ ❞❡ r❛❞✐❛çã♦ ❛❝úst✐❝❛ ❛①✐❛❧ ♣♦❞❡ s❡r ♥❡❣❛t✐✈❛ s❡ ❛ ❝♦♥❞✐çã♦ ❞❛❞❛ ♥❛ ❊q✳ é s❛t✐s❢❡✐t❛✳

                P❛r❛ ♦❜t❡r ❛ ❢♦rç❛ ❞❡ r❛❞✐❛çã♦ ❛❝úst✐❝❛ ❡♠ ✉♠❛ ♣❛rtí❝✉❧❛ ✢✉✐❞❛ ❞❡ ❝♦♠♣r❡ssã♦ ❛ ♣❛rt✐r s s = 0 = 0 ❞❛ ❊q✳ ❖ r❡s✉❧t❛❞♦ ♦❜t✐❞♦ é ♦ ♠❡s♠♦ q✉❡ é ❞❛❞♦ ♥❛ ❊q✳ ✭✹✺✮ ❞❛ ❘❡❢✳ 2 2 2

                8 πa I ρ c 12γ n ρ c

                ✭❋❧✉✐❞❛✮ 3 2 F (ε , ε ) = αεδ ˜ αε ˜ cos β sin β . ℓ s n,0

                ✭✺✳✶✺✮

                ❛❜s,z cos β − 2 2 2

                c 3 ρ c 5ρ c (ρ /ρ + 2) 1 ℓ ℓ 1 1 ❆❣♦r❛✱ ❝♦♥s✐❞❡r❡ ✉♠ ❢❡✐①❡ ❞❡ ❇❡ss❡❧ ❞❡ ♦r❞❡♠ s✉♣❡r✐♦r✱ ❞❡s❝r✐t♦ ♣❡❧❛ ❊q✳ ❝♦♠

                |n| > 0✳ ◆❡ss❡ ❝❛s♦✱ ❛ ✐♥t❡♥s✐❞❛❞❡ ❛①✐❛❧ é ③❡r♦✳ ❆ss✐♠✱ ❛ ❢♦rç❛ ❞❡ r❛❞✐❛çã♦ ❛❝úst✐❝❛ t❛♠❜é♠ t❡♠ ❛ ❝♦♥tr✐❜✉✐çã♦ ❞❛ ❝♦♠♣♦♥❡♥t❡ ❞❡ ❡s♣❛❧❤❛♠❡♥t♦✳ P♦rt❛♥t♦✱ s✉❜st✐t✉í♠♦s ❛s ❊qs✳ ❡ ❡♥❝♦♥tr❛♠♦s q✉❡ s♦♠❡♥t❡ ♦ ❢❡✐①❡ ❞❡ ❇❡ss❡❧ ❞❡ ♣r✐♠❡✐r❛ ♦r❞❡♠ ✭❋❇P❖✮✱ ❝♦♠ n = ±1✱ ♣r♦❞✉③ ❢♦rç❛ ❞❡ r❛❞✐❛çã♦ ❛❝úst✐❝❛ ❛①✐❛❧ ❡♠ ♣❛rtí❝✉❧❛s ❞❡ ❘❛②❧❡✐❣❤ ✱

                ❋❇P❖ ❋❇P❖ ❋❇P❖

                F (ε , ε ) = F + F z ❛❜s,z s❝❛,z ℓ s 2

                ✭❘✮

                f 3 2 ✭■✮ 1 5 = 6F ε cos β sin β g (ε , ε ) + ε . 1 ℓ s ❖ ε ✭✺✳✶✻✮

              • 36

                ✭■✮

                (ε , ε ) ❆ ❢✉♥çã♦ g 1 s é ❞❛❞❛ ♣❡❧❛s ❊q✳ ♣❛r❛ ❛ ❛♣r♦①✐♠❛çã♦ ❞❡ ❜❛✐①❛ ❡ ❛❧t❛ ❢r❡q✉ê♥❝✐❛✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✳

                ❋♦rç❛ ❞❡ r❛❞✐❛çã♦ ❛❝úst✐❝❛ ❛①✐❛❧ ♥❡❣❛t✐✈❛ t❛♠❜é♠ ♣♦❞❡ ♦❝♦rr❡r s♦❜r❡ ❛ ♣❛rtí❝✉❧❛ ♥♦ 2

                ✭■✮ ✭❘✮

                (ε ℓ , ε )+f ε/36 < 0 r❡❣✐♠❡ ❞❡ ❜❛✐①❛ ❢r❡q✉ê♥❝✐❛ s❡ g 1 s 1 ✳ ◗✉❛♥❞♦ ❛ ❞❡♥s✐❞❛❞❡ ❞❛ ♣❛rtí❝✉❧❛ 1 t❡♠ ✉♠ ✈❛❧♦r ❛♣r♦①✐♠❛❞♦ ❞❛ ❞❡♥s✐❞❛❞❡ ❞♦ ✢✉✐❞♦ ✐❞❡❛❧✱ ✐st♦ é ρ ≈ ρ ✱ ❛ ❝♦♥❞✐çã♦ ♣❛r❛ ❛ ❢♦rç❛ ❞❡ r❛❞✐❛çã♦ ❛❝úst✐❝❛ s❡r ♥❡❣❛t✐✈❛ é ❛ ♠❡s♠❛ ♣❛r❛ ✉♠❛ ♦♥❞❛ ♣❧❛♥❛ ♣r♦❣r❡ss✐✈❛ ❞❛❞❛ ♥❛ ❊q✳ ✳ ◆♦t❡ t❛♠❜é♠ q✉❡ ❛ ❢♦rç❛ ❞❡ r❛❞✐❛çã♦ ❛❝úst✐❝❛ ❡♠ ✉♠❛ ♣❛rtí❝✉❧❛ ❡❧ást✐❝❛ ℓ s = τ = 0 ♣♦❞❡ s❡r ♦❜t✐❞❛ ❛tr❛✈és ❞❛ ❊q✳ ❡ ♣♦rt❛♥t♦

                ✭■✮

                g (0, 0) = 0 1 ✳ ❚❡♥❞♦ ❡♠ ✈✐st❛ q✉❡ t❛♥t♦ ♦ ❢❡✐①❡ ❞❡ ❇❡ss❡❧ ❞❡ ♦r❞❡♠ ③❡r♦ ❝♦♠♦ ♦ ❢❡✐①❡ ❞❡ ❇❡ss❡❧

                ❞❡ ♣r✐♠❡✐r❛ ♦r❞❡♠ ♣♦❞❡♠ ♣r♦❞✉③✐r ❢♦rç❛ ❞❡ r❛❞✐❛çã♦ ❛❝úst✐❝❛ ♥❡❣❛t✐✈❛✱ ❛♥❛❧✐s❛♠♦s ❛ ❡st❛❜✐❧✐❞❛❞❡ ❞❛ ❢♦rç❛ ❞❡ r❛❞✐❛çã♦ ❛❝úst✐❝❛ tr❛♥s✈❡rs❛❧✳ ❋❛③❡♥❞♦ ❡ss❛ ❛♥á❧✐s❡✱ q✉❡r❡♠♦s ✈❡r✐✜❝❛r s❡ ♦s ❢❡✐①❡s ❞❡ ❇❡ss❡❧ ❞❡ ♦r❞❡♠ ③❡r♦ ❡ ❞❡ ♣r✐♠❡✐r❛ ♦r❞❡♠ ♣♦❞❡♠ s❡ ❝♦♠♣♦rt❛r ❝♦♠♦ ❢❡✐①❡ tr❛t♦r ✸❉ ♣❛r❛ ♣❡q✉❡♥❛s ♣❛rtí❝✉❧❛s ✈✐s❝♦❡❧ást✐❝❛s✳ P❛r❛ ❛♥❛❧✐s❛r ❛ ❡st❛❜✐❧✐❞❛❞❡ ❞❛ ♣❛rtí❝✉❧❛ ♥❛ ✈✐③✐♥❤❛♥ç❛ ❞♦ ❡✐①♦ ❞♦ ❢❡✐①❡ ✐r❡♠♦s ❝❛❧❝✉❧❛r ❛ ❡♥❡r❣✐❛ ♣♦t❡♥❝✐❛❧✳ P❛r❛ ✐ss♦✱ ❡♠ ♣r✐♠❡✐r♦ ❧✉❣❛r✱ ❝❛❧❝✉❧❛♠♦s ❛ ♣r❡ssã♦ ❡ ❛ ✈❡❧♦❝✐❞❛❞❡ ❞♦ ✢✉✐❞♦ ✐♥❝✐❞❡♥t❡ ❛tr❛✈és ❞❛s ❊qs✳ ♣❛r❛ ✉♠ ❢❡✐①❡ ❞❡ ❇❡ss❡❧ ❞❡ ♦r❞❡♠ n✳ ❊♠ s❡❣✉✐❞❛✱ ♦❜t❡♠♦s ❛ ❡♥❡r❣✐❛ ♣♦t❡♥❝✐❛❧ ❞❛ ❢♦rç❛ ❞❡ r❛❞✐❛çã♦ ❣r❛❞✐❡♥t❡ ❝♦♠♦ ❞❛❞♦ ♥❛ ❊q✳ ❆ss✐♠✱

                ❡♥❝♦♥tr❛♠♦s q✉❡ 2 2 n

                ✭❘✮ ✭❘✮ 2 2 U (̺) = aF f (ε , ε + cos β f J (k̺ sin β) n ℓ s ) − 2 2 1 n

                3 k ̺

                ✭❘✮ 2 2 J (k̺ sin β) sin β , − f 1 n

                ✭✺✳✶✼✮ ♦♥❞❡ ♦ sí♠❜♦❧♦ J ❞❡♥♦t❛ ❛ ❞❡r✐✈❛❞❛ ❝♦♠ r❡❧❛çã♦ ❛♦ ❛r❣✉♠❡♥t♦ ❞❛ ❢✉♥çã♦✳ ❆ ❡♥❡r❣✐❛ ♣♦t❡♥❝✐❛❧ ❞❡✈✐❞♦ ❛♦ ❢❡✐①❡ ❞❡ ❇❡ss❡❧ ❞❡ ♦r❞❡♠ ③❡r♦ ✭n = 0✮ ❡ ❞❡ ♦r❞❡♠ ✉♠ ✭n = ± ✶✮ t❡♠ ✉♠ ❡①tr❡♠♦ ✭♠í♥✐♠♦ ♦✉ ♠á①✐♠♦✮ ♥♦ ❡✐①♦ ❞♦ ❢❡✐①❡ ̺ = 0✳ ❆ ❡♥❡r❣✐❛ ♣♦t❡♥❝✐❛❧ ♠í♥✐♠❛ ❝♦rr❡s♣♦♥❞❡ ❛ ❢♦rç❛ ❞❡ r❛❞✐❛çã♦ ❛❝úst✐❝❛ tr❛♥s✈❡rs❛❧ ❝♦♥✈❡r❣❡♥t❡✱ ❞❡st❛ ❢♦r♠❛✱ s❡ ❛ ❢♦rç❛ ❞❡ r❛❞✐❛çã♦ ❛❝úst✐❝❛ ❛①✐❛❧ é ♥❡❣❛t✐✈❛✱ ♦ ❢❡✐①❡ ❞❡ ❇❡ss❡❧ ❛t✉❛ ❝♦♠♦ ✉♠ ❢❡✐①❡ tr❛t♦r ✸❉ s♦❜r❡ ❛ ♣❛rtí❝✉❧❛✳ P♦rt❛♥t♦✱ ❢❛③❡♥❞♦ ❛ ❞❡r✐✈❛❞❛ ❞❡ s❡❣✉♥❞❛ ♦r❞❡♠ ❞❛ ❊q✳ ❝♦♠ r❡❧❛çã♦ ❛ ̺

                ✱ ❡♥❝♦♥tr❛♠♦s ❛ ❝♦♥❞✐çã♦ ♣❛r❛ ♦❜t❡r ✉♠ ❢❡✐①❡ tr❛t♦r ♣❛r❛ ✉♠ ❢❡✐①❡ ❞❡ ❇❡ss❡❧ ❞❡ ♦r❞❡♠ ③❡r♦ ❡ ❞❡ ♦r❞❡♠ ✉♠✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✱

                ✭❘✮ ✭❘✮

                [3 + 9 cos(2β)]f (ε , ε ) > 0, 1 − 8f ℓ s ✭✺✳✶✽✮

                ✭❘✮ ✭❘✮ 2f (ε ℓ , ε cos(2β) > 0. s 1 ✭✺✳✶✾✮

                ) − 3f ✭❘✮ ✭❘✮ (ε ℓ , ε ) < (3/8)f

                ➱ ✐♥t❡r❡ss❛♥t❡ ♥♦t❛r q✉❡ ♣❛r❛ β = 45 ❡st❛s ❝♦♥❞✐çõ❡s t♦r♥❛♠✲s❡ f s 1

                ✭❘✮

                (ε , ε ) > 0 ❡ f s ✳

                ❯t✐❧✐③❛♥❞♦ ♦ r❡s✉❧t❛❞♦ ❞❛ ✐♥t❡♥s✐❞❛❞❡ ♠é❞✐❛✱ ❡♥❝♦♥tr❛♠♦s q✉❡ ❛ ❢♦rç❛ ❞❡ r❛❞✐❛çã♦ ❛❝úst✐❝❛ tr❛♥s✈❡rs❛❧ é ❞❛❞❛ ♣♦r F (̺) = F + F n,⊥ ❣r❛❞ ❛❜s 2 J (k̺ sin β)

                ✭■✮ n ̺ U n (̺)e ̺ nεf (ε ℓ , ε ) ϕ .

                e

                s ✭✺✳✷✵✮

                = −∂ − 4F 3k̺

                ◆♦t❡ q✉❡ ❛ ❢♦rç❛ ❞❡ r❛❞✐❛çã♦ ❛❝úst✐❝❛ ❞❡✈✐❞♦ ❛♦ ❢❡✐①❡ ❞❡ ❇❡ss❡❧ ❞❡ ♦r❞❡♠ ③❡r♦ ♥ã♦ t❡♠ ❝♦♠♣♦♥❡♥t❡ ♥❛ ❞✐r❡çã♦ ❛③✐♠✉t❛❧✳

                ❆❣♦r❛✱ ✈♦❧t❛♠♦s ♥♦ss❛ ❛t❡♥çã♦ ♣❛r❛ ♦ t♦rq✉❡ ❞❡ r❛❞✐❛çã♦ ❛❝úst✐❝♦ ❛①✐❛❧ ❣❡r❛❞♦ ♥❛ ♣❛rtí❝✉❧❛ ✈✐s❝♦❡❧ást✐❝❛ ♣♦r ✉♠ ❢❡✐①❡ ❞❡ ❇❡ss❡❧✳ ❆♥t❡s ❞❡ ♣r♦ss❡❣✉✐r ❝♦♠ ❡st❛ ❛♥á❧✐s❡✱ t❡♠♦s q✉❡ ♦❜t❡r ❛ ❝♦♠♣♦♥❡♥t❡ tr❛♥s✈❡rs❛❧ ❞❛ ✈❡❧♦❝✐❞❛❞❡ ❞♦ ✢✉✐❞♦ ✐♥❝✐❞❡♥t❡ ♥♦ ❡✐①♦ ❞♦ ❢❡✐①❡✳ ❯t✐❧✐③❛♥❞♦ ❛ ❊q✳ ❝♦♥❝❧✉í♠♦s q✉❡ s♦♠❡♥t❡ ♦ ❢❡✐①❡ ❞❡ ❇❡ss❡❧ ❞❡ ♣r✐♠❡✐r❛ ♦r❞❡♠ ♣♦ss✉✐ ✈❡❧♦❝✐❞❛❞❡ ❞♦ ✢✉✐❞♦ tr❛♥s✈❡rs❛❧ ❡♠ ̺ = 0✱ v

                

              ❋❇P❖ ✐(kz cos β+nϕ)

                v (0, ϕ, z) = nδ sin β , n,±1 ❡ ✭✺✳✷✶✮

                ✐♥,̺

                2 ✐v

                

              ❋❇P❖ ✐(kz cos β+nϕ)

              v (0, ϕ, z) = δ sin β . n,±1 ✐♥,ϕ ❡ ✭✺✳✷✷✮

                2 ❆ ❝♦♠♣♦♥❡♥t❡ ❛①✐❛❧ ❞♦ ♣r♦❞✉t♦ ✈❡t♦r✐❛❧ ❡♥tr❡ ❛ ✈❡❧♦❝✐❞❛❞❡ ❞♦ ✢✉✐❞♦ ✐♥❝✐❞❡♥t❡ ❡ s❡✉ ❝♦♠♣❧❡①♦ ❝♦♥❥✉❣❛❞♦ é ❞❛❞❛ ♣♦r ❡♠ ❝♦♦r❞❡♥❛❞❛s ❝✐❧í♥❞r✐❝❛s ♣♦r ∗ ∗ (v ) z = 2 v ].

                ✐♥ ✐ ■♠[v ✐♥,̺ ✭✺✳✷✸✮

                × v ✐♥ ✐♥,ϕ

                ❆ss✐♠✱ ✉s❛♥❞♦ ❊qs✳ ❡♥❝♦♥tr❛♠♦s q✉❡ ♦ t♦rq✉❡ ❞❡ r❛❞✐❛çã♦ ❛❝úst✐❝♦ é ♦r✐❣✐♥❛❞♦ s♦♠❡♥t❡ ♣❡❧♦ ❢❡✐①❡ ❞❡ ❇❡ss❡❧ ❞❡ ♣r✐♠❡✐r❛ ♦r❞❡♠✱ 2 ✭■✮ 2 4

                ❋❖❇❇

                N (ε , ε ) = 6aF nδ ε g (ε , ε ) sin β + ), ℓ n,±1 ℓ

                s 1 s ❖(ε ✭✺✳✷✹✮ ❛❜s,z ✭■✮

                (ε , ε ) ℓ s ♦♥❞❡ ❛ ❢✉♥çã♦ g 1 é ❞❛❞❛ ♣❛r❛ ♦s r❡❣✐♠❡s ❞❡ ❜❛✐①❛ ❡ ❛❧t❛ ❢r❡q✉ê♥❝✐❛ r❡s♣❡❝t✐✈❛♠❡♥t❡✱ ♣❡❧❛s ❊qs✳

                ✭■✮

                (ε , ε ) ℓ s ❆ ❞✐r❡çã♦ ❞♦ t♦rq✉❡ ❞❡ r❛❞✐❛çã♦ ❛❝úst✐❝♦ ❛①✐❛❧ ❞❡♣❡♥❞❡ ❞♦ s✐♥❛❧ ❞❡ ng 1 ✳ ❙❡

                ✭■✮

                g (ε , ε ) > 0 1 ℓ s ✱ ♦ t♦rq✉❡ ❞❡ r❛❞✐❛çã♦ ❛❝úst✐❝♦ ❛①✐❛❧ s❡❣✉❡ ❛ ❞✐r❡çã♦ ❞♦ ♠♦♠❡♥t♦ ❛♥❣✉❧❛r

                ✭■✮

                (ε ℓ , ε ) < 0 ❞♦ ❢❡✐①❡ ❞❡ ❇❡ss❡❧ ❞❡ ♣r✐♠❡✐r❛ ♦r❞❡♠✳ P♦r ♦✉tr♦ ❧❛❞♦✱ q✉❛♥❞♦ g s ✱ ♦ t♦rq✉❡ ❞❡ 1 r❛❞✐❛çã♦ ❛❝úst✐❝♦ ❛①✐❛❧ t❡♠ ❞✐r❡çã♦ ♦♣♦st❛ ❛♦ ♠♦♠❡♥t♦ ❛♥❣✉❧❛r ❞♦ ❢❡✐①❡✳ ◆ós ♥♦s r❡❢❡r✐♠♦s

                ❛ ❡ss❛ s✐t✉❛çã♦ ❝♦♠♦ t♦rq✉❡ ❞❡ r❛❞✐❛çã♦ ❛①✐❛❧ ♥❡❣❛t✐✈♦✳ ❉❡ ❛❝♦r❞♦ ❝♦♠ ❛s ❊qs✳ ❡ é s❛t✐s❢❡✐t❛✳

                

              ✺✳✷ ❘❡s✉❧t❛❞♦s ❡ ❞✐s❝✉ssõ❡s ♣❛r❛ ♣❛rtí❝✉❧❛s ❞❡ ♣♦❧✐❡t✐✲

              ❧❡♥♦

                ❱❛♠♦s ❛✈❛❧✐❛r ❛ ❢♦rç❛ ❡ ♦ t♦rq✉❡ ❞❡ r❛❞✐❛çã♦ ❛❝úst✐❝♦ ♥✉♠❡r✐❝❛♠❡♥t❡ ❝♦♥s✐❞❡r❛♥❞♦ ✉♠❛ ♣❛rtí❝✉❧❛ ❡s❢ér✐❝❛ ❞❡ ♣♦❧✐❡t✐❧❡♥♦ ❞❡ ❜❛✐①❛ ❞❡♥s✐❞❛❞❡ ✭P❊❇❉✮ ❡ ❞❡ ♣♦❧✐❡t✐❧❡♥♦ ❞❡ ❛❧t❛ ❞❡♥s✐❞❛❞❡ ✭P❊❆❉✮✳ P♦❧✐❡t✐❧❡♥♦ é ♦ ♣❧ást✐❝♦ ♠❛✐s ♣r♦❞✉③✐❞♦ ♥♦ ♠✉♥❞♦✳ ■ss♦ ♦ t♦r♥❛ ❛❜✉♥✲ ❞❛♥t❡♠❡♥t❡ ❞✐s♣♦♥í✈❡❧ ♣❛r❛ ❡①♣❡r✐♠❡♥t♦s ❞❡st✐♥❛❞♦s ❛ ✈❡r✐✜❝❛çã♦ ❞❛ t❡♦r✐❛ ❞❡ ❢♦rç❛ ❡ t♦rq✉❡ ❞❡ r❛❞✐❛çã♦ ❛❝úst✐❝♦ ❞❡s❡♥✈♦❧✈✐❞♦ ♥❡st❡ tr❛❜❛❧❤♦✳ P❛r❛ ♣r♦ss❡❣✉✐r ❝♦♠ ❛ ❛✈❛❧✐❛çã♦ ♥✉♠ér✐❝❛✱ ♣r❡❝✐s❛♠♦s ❞❡ t♦❞♦s ♦s ♣❛râ♠❡tr♦s ❛❝úst✐❝♦s q✉❡ ❞❡s❝r❡✈❡♠ ♦ ♣♦❧✐❡t✐❧❡♥♦✳ ❊♠ ❣❡r❛❧✱ ❛ ❞❡♥s✐❞❛❞❡ ❡ ❛ ✈❡❧♦❝✐❞❛❞❡ ❞♦ s♦♠ ❧♦♥❣✐t✉❞✐♥❛❧ ❡ ❞❡ ❝✐s❛❧❤❛♠❡♥t♦ ♣❛r❛ ❡ss❡ ♠❛t❡r✐❛❧ ❛ t❡♠♣❡r❛t✉r❛ ❛♠❜✐❡♥t❡ sã♦ ❞❛❞♦s ❡♠ t❛❜❡❧❛s ♣❛❞rõ❡s✳ ◆♦ ❡♥t❛♥t♦✱ ❛ ♦r❞❡♠ ❢r❛❝✐♦♥ár✐❛ ✈✐s❝♦❡❧ást✐❝❛ ν ❡ ♦s t❡♠♣♦s ❞❡ r❡❧❛①❛çã♦ ❧♦♥❣✐t✉❞✐♥❛❧ τ ❡ ❞❡ ❝✐s❛❧❤❛♠❡♥t♦ τ s ✱ ♥ã♦ sã♦ ❢❛✲ ❝✐❧♠❡♥t❡ ❡♥❝♦♥tr❛❞♦s ♥❛ ❧✐t❡r❛t✉r❛✳ P♦rt❛♥t♦✱ ✈❛♠♦s ❡st✐♠❛r ❡st❡s ♣❛râ♠❡tr♦s ❛ ♣❛rt✐r ❞❡ ❞❛❞♦s ❡①♣❡r✐♠❡♥t❛✐s ❞❡ ❛❜s♦rçã♦ ❧♦♥❣✐t✉❞✐♥❛❧ ❡ ❞❡ ❝✐s❛❧❤❛♠❡♥t♦ ❡♠ ❢✉♥çã♦ ❞❛ ❢r❡q✉ê♥❝✐❛ ♦❜t✐❞❛ ♣♦r ❡s♣❡❝tr♦s❝♦♣✐❛ ❞❡ ✉❧tr❛ss♦♠ ❛♣r❡s❡♥t❛❞♦s ♥❛ ❘❡❢✳

                ✺✳✷✳✶ ❊st✐♠❛t✐✈❛ ❞♦s ♣❛râ♠❡tr♦s ❞♦ ♣♦❧✐❡t✐❧❡♥♦ ℓ s

                P❛r❛ ♦❜t❡r ❛ ♦r❞❡♠ ❢r❛❝✐♦♥ár✐❛ ✈✐s❝♦❡❧ást✐❝❛ ν ❡ ♦s t❡♠♣♦s ❞❡ r❡❧❛①❛çã♦ τ ✱ τ ✱ r❡❛❧✐③❛✲ ♠♦s ✉♠ ❛❥✉st❡ ❞♦s ❞❛❞♦s ❡①♣❡r✐♠❡♥t❛✐s ❞❡ ❛❜s♦rçã♦ ✈❡rs✉s ❢r❡q✉ê♥❝✐❛ ♣❛r❛ ♦ ♣♦❧✐❡t✐❧❡♥♦ ❞❛❞♦ ♥❛s ✜❣✉r❛s ✹ ❡ ✺ ❞❛ ❘❡❢✳ ❆ ❢✉♥çã♦ q✉❡ ✉t✐❧✐③❛♠♦s ♣❛r❛ ❢❛③❡r ♦ ❛❥✉st❡ é ♦❜t✐❞❛ ❛tr❛✈és ❞❛ ♣❛rt❡ ✐♠❛❣✐♥ár✐❛ ❞❛ r❡❧❛çã♦ ❞❡ ❞✐s♣❡rsã♦ ❞❛❞❛ ♥❛ ❊q✳

                ω

                1 ν −

                ✐πν/2

                α sin arg[1 + (ωτ ) ] j j (ω) = − ❡ c j ν

                2 2 2ν 2 1/4 [1 + (ωτ ) cos(πν/2)] + (ωτ ) sin (πν/2) , j j ✭✺✳✷✺✮

                ×

                ❚❛❜❡❧❛ ✺✳✶✿ P❛râ♠❡tr♦s ❡ ♠❡❞✐❞❛s ❞❛ q✉❛❧✐❞❛❞❡ ❞♦ ❛❥✉st❡ ♣❛r❛ ❛ ❢✉♥çã♦ ❞❡ ❛❜s♦rçã♦ ♥❛ ❊q✳ ❡ ❞❛❞♦s ❡①♣❡r✐♠❡♥t❛✐s ❞❡ ❛❜s♦rçã♦ ❧♦♥❣✐t✉❞✐♥❛❧ ❡ ❞❡ ❝✐s❛❧❤❛♠❡♥t♦ ❡①tr❛í❞♦s ❞❛s ❋✐❣✳ ✹ ❡ ✺ ❞❛ ❘❡❢✳ ✳

                P❊❇❉ P❊❆❉ ▼♦❞♦ ❧♦♥❣✐t✉❞✐♥❛❧ 2 − 15 ❚❡♠♣♦ ❞❡ r❡❧❛①❛çã♦ τ ❬s❪

                1.27 × 10 2.27 × 10 ❖r❞❡♠ ❢r❛❝✐♦♥ár✐❛ ✈✐s❝♦❡❧ást✐❝❛ ν ✵✳✸✼ ✵✳✶✺ 4 − 11

                ❊rr♦ ♣❛❞rã♦ 1.79 × 10 1.09 × 10 2 0.9998 0.9998

                ❈♦❡✜❝✐❡♥t❡ ❞❡ ❞❡t❡r♠✐♥❛çã♦ ✭R ✮ ▼♦❞♦ ❞❡ ❝✐s❛❧❤❛♠❡♥t♦ 4 − 7

                ❚❡♠♣♦ ❞❡ r❡❧❛①❛çã♦ τ s ❬s❪ 3.34 × 10 2.39 × 10

                ❖r❞❡♠ ❢r❛❝✐♦♥ár✐❛ ✈✐s❝♦❡❧ást✐❝❛ ν ✵✳✸✼ ✵✳✶✹ 6 − 8 ❊rr♦ ♣❛❞rã♦ 8.56 × 10 3.56 × 10 2

                0.9994 0.9997 ❈♦❡✜❝✐❡♥t❡ ❞❡ ❞❡t❡r♠✐♥❛çã♦ ✭R ✮

                ❋♦♥t❡✿ ❆✉t♦r✱ ✷✵✶✺✳

                ♦♥❞❡ ❵❛r❣✬ é ❛ ❢✉♥çã♦ ❛r❣✉♠❡♥t♦✱ q✉❡ ❞á ♦ â♥❣✉❧♦ ❡♥tr❡ ❛ ❧✐♥❤❛ ❞❡ ♣♦s✐çã♦ ❞❡ ✉♠ ♥ú♠❡r♦ ❝♦♠♣❧❡①♦ ❡ ♦ ❡✐①♦ r❡❛❧✳ ❆ ♣❛rt✐r ❞❛ ❡q✉❛çã♦ é ♣♦ssí✈❡❧ ❞❡t❡r♠✐♥❛r ♦s ♣❛râ♠❡tr♦s ν j

                ❡ ♦ t❡♠♣♦ ❞❡ r❡❧❛①❛çã♦ τ ❝♦♠ j = {ℓ, s} ❛tr❛✈és ❞♦ ❛❥✉st❡ ❞♦s ❞❛❞♦s ❡①♣❡r✐♠❡♥t❛✐s ♦❜t✐❞♦s ♥❛s ✜❣✉r❛s ✹ ❡ ✺ ❞❛ ❘❡❢✳ ❈♦♥str✉í♠♦s ✉♠❛ t❛❜❡❧❛ ❝♦♠ ♦s ❞❛❞♦s ❞❡ ❛❜s♦r✲ çã♦ ❧♦♥❣✐t✉❞✐♥❛✐s ❡ ❞❡ ❝✐s❛❧❤❛♠❡♥t♦ ❞♦ ♣♦❧✐❡t✐❧❡♥♦ ❡♠ ❢✉♥çã♦ ❞❛ ❢r❡q✉ê♥❝✐❛✳ ❖s ❞❛❞♦s sã♦ ❝♦❧❤✐❞♦s ♠❛♥✉❛❧♠❡♥t❡ ❛ ♣❛rt✐r ❞❛s ❋✐❣s✳ ✹ ❡ ✺ ❞❛ ❘❡❢✳ ✉t✐❧✐③❛♥❞♦ ❲❡❜P❧♦t❉✐❣✐t✐③❡r✱ ✉♠ s♦❢t✇❛r❡ ♦♥❧✐♥❡ ❣r❛t✉✐t♦ ❖s ♣❛râ♠❡tr♦s ❞❡ ❛❥✉st❡ sã♦ ♦❜t✐❞♦s ✉t✐❧✐③❛♥❞♦ ♦ s♦❢t✲ ✇❛r❡ ▼❛t❤❡♠❛t✐❝❛ ❛tr❛✈és ❞♦ ❛❧❣♦r✐t♠♦ ❞❡ ▲❡✈❡♥❜❡r❣✲▼❛rq✉❛r❞t ❛♣r❡s❡♥t❛♠♦s ♦s ♣❛râ♠❡tr♦s ❡st✐♠❛❞♦s ❝♦♠ ♦ ❡rr♦ ♣❛❞rã♦ ❝♦rr❡s♣♦♥❞❡♥t❡ ❡ ♦ ❝♦❡✜❝✐❡♥t❡ 2

                ❞❡ ❞❡t❡r♠✐♥❛çã♦ ✭R ✮✳ ❊st❡s ♣❛râ♠❡tr♦s ♣♦❞❡♠ s❡r ❡♥t❡♥❞✐❞♦s ❞❛ s❡❣✉✐♥t❡ ❢♦r♠❛✳ ❖ ❡rr♦ ♣❛❞rã♦ q✉❡ s❡ ❛♣r♦①✐♠❛ ❞❡ ③❡r♦ ❞á ✉♠❛ ✐♥❞✐❝❛çã♦ ❞❡ ❜♦♠ ❛❥✉st❡ ❡ ♦ ❝♦❡✜❝✐❡♥t❡ ❞❡ ❞❡✲ 2 t❡r♠✐♥❛çã♦ R ❢♦r♥❡❝❡ ✉♠❛ ♠❡❞✐❞❛ q✉❡ ❞❡t❡r♠✐♥❛ ❝♦♠♦ ♦ ♠♦❞❡❧♦ ❡①♣❧✐❝❛ ❛ ✈❛r✐❛❜✐❧✐❞❛❞❡ 2 2

                = 1 ❞♦s ❞❛❞♦s✳ ❊ss❡ ❝♦❡✜❝✐❡♥t❡ ✈❛r✐❛ ❞❡ 0 ≤ R ≤ 1✱ ❝♦♠ R ✐♥❞✐❝❛ q✉❡ ♦ ♠♦❞❡❧♦ ❡①♣❧✐❝❛ t♦❞❛ ❛ ✈❛r✐❛çã♦ ❞❡ ❞❛❞♦s ❡♠ t♦r♥♦ ❞❡ s✉❛ ♠é❞✐❛✳ ❆❧é♠ ❞✐ss♦✱ ♣♦❞❡♠♦s ♦❜s❡r✈❛r ♥❛s ❋✐❣✉r❛s ♣❛r❛ ✉♠❛ ♣❛rtí❝✉❧❛ ❞❡ P❊❆❉ ❡ ❞❡ P❊❇❉✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✱ ❡①❝❡❧❡♥t❡ ❝♦♥❝♦r❞â♥❝✐❛ ❡♥tr❡ ♦s ❞❛❞♦s ❡①♣❡r✐♠❡♥t❛✐s ❞❡ ❛❜s♦rçã♦ ❡ ❛ ❢✉♥çã♦ ❞❡ ❛❥✉st❡ ❞❛❞♦ ♣❡❧❛ ❊q✳

                ❆s ♣r♦♣r✐❡❞❛❞❡s ❢ís✐❝❛s ❡ ♦s ♣❛râ♠❡tr♦s ❞❡ ❛❥✉st❡s ❞♦ ♣♦❧✐❡t✐❧❡♥♦ ♦❜t✐❞♦s ❛tr❛✈és ❞❛ ❘❡❢✳ ➱ ✐♠♣♦rt❛♥t❡ ♥♦t❛r q✉❡ ♦ ❡①♣♦❡♥t❡ ❞❛ ❧❡✐ ❞❡ ♣♦tê♥❝✐❛ ❞❡ ❛❜s♦rçã♦ ♣❛r❛ ✭P❊❇❉✮ ❡ ✭P❊❆❉✮ é r❡s♣❡❝t✐✈❛♠❡♥t❡✱ y = 1 − ν/2 = 0.815 ❡ y = 1 + ν = 1.15

                ✳ ❊ss❡s ✈❛❧♦r❡s ❡stã♦ ❡♠ ❡①❝❡❧❡♥t❡ ❝♦♥❝♦r❞â♥❝✐❛ ❝♦♠ ♦ q✉❡ ❢♦✐ ♦❜t✐❞♦ ♥❛ ❘❡❢✳ ✳

                ❆❥✉st❡ ❞❛ ❢✉♥çã♦ ❞❡ ❛❜s♦rçã♦ ❞❛ ❊q✳ ❞♦s ❞❛❞♦s ❡①♣❡r✐♠❡♥t❛✐s ❞❡ ❛❜s♦rçã♦

                ❋✐❣✉r❛ ✺✳✷✿

                

              ❧♦♥❣✐t✉❞✐♥❛❧ ✭❛✮ ❡ ❞❡ ❝✐s❛❧❤❛♠❡♥t♦ ✭❜✮ ❡①tr❛í❞♦s ❞❛s ❋✐❣✳ ✹ ❡ ✺ ❞❛ ❘❡❢✳ ♣❛r❛ ✉♠❛ ♣❛rtí❝✉❧❛ ❞❡

              P♦❧✐❡t✐❧❡♥♦ ❞❡ ❛❧t❛ ❞❡♥s✐❞❛❞❡ ✭P❊❆❉✮✳ ▲❡❣❡♥❞❛✿ ❧✐♥❤❛ só❧✐❞❛ ❛③✉❧ r❡♣r❡s❡♥t❛ ♦ ❛❥✉st❡ ❞❛ ❢✉♥çã♦

              ❞❡ ❛❜s♦rçã♦ ❞❛❞❛ ♥❛ ❊q✳

                (a) .

                (b) .

                ❋♦♥t❡✿ ❆✉t♦r✱ ✷✵✶✺✳

                ❆❥✉st❡ ❞❛ ❢✉♥çã♦ ❞❡ ❛❜s♦rçã♦ ❞❛ ❊q✳ ❞♦s ❞❛❞♦s ❡①♣❡r✐♠❡♥t❛✐s ❞❡ ❛❜s♦rçã♦

                ❋✐❣✉r❛ ✺✳✸✿

                

              ❧♦♥❣✐t✉❞✐♥❛❧ ✭❛✮ ❡ ❞❡ ❝✐s❛❧❤❛♠❡♥t♦ ✭❜✮ ❡①tr❛í❞♦s ❞❛s ❋✐❣✳ ✹ ❡ ✺ ❞❛ ❘❡❢✳ ♣❛r❛ ✉♠❛ ♣❛rtí❝✉❧❛

              ❞❡ P♦❧✐❡t✐❧❡♥♦ ❞❡ ❜❛✐①❛ ❞❡♥s✐❞❛❞❡ ✭P❊❇❉✮✳ ▲❡❣❡♥❞❛✿ ❧✐♥❤❛ só❧✐❞❛ ❛③✉❧ r❡♣r❡s❡♥t❛ ♦ ❛❥✉st❡ ❞❛

              ❢✉♥çã♦ ❞❡ ❛❜s♦rçã♦ ❞❛❞❛ ♥❛ ❊q✳

                (a) .

                (b) .

                ❋♦♥t❡✿ ❆✉t♦r✱ ✷✵✶✺✳

                ❚❛❜❡❧❛ ✺✳✷✿ Pr♦♣r✐❡❞❛❞❡s ❢ís✐❝❛s ❡ ♦s ♣❛râ♠❡tr♦s ❞❡ ❛❥✉st❡s ❞❛ ♣❛rtí❝✉❧❛ ❞❡ ♣♦❧✐❡t✐❧❡♥♦ ❞❡ ❜❛✐①❛ ❞❡♥s✐❞❛❞❡ ✭P❊❇❉✮ ❡ ❞♦ ♣♦❧✐❡t✐❧❡♥♦ ❞❡ ❛❧t❛ ❞❡♥s✐❞❛❞❡ ✭P❊❆❉✮✳ ❖s ❞❛❞♦s ❢♦r❛♠ ❡①tr❛í❞♦s à t❡♠♣❡r❛t✉r❛ ❛♠❜✐❡♥t❡ ✭20 ❈✮ ♥❛ ❘❡❢✳

                P♦❧✐❡t✐❧❡♥♦ 1 [ ] 3 P❊❇❉ P❊❆❉ ❉❡♥s✐❞❛❞❡ ρ ❦❣✴♠ ✽✾✻ ✾✸✵ ❱❡❧♦❝✐❞❛❞❡ ❞♦ s♦♠ ❧♦♥❣✐t✉❞✐♥❛❧ c ❬♠✴s❪ ✷✺✻✻ ✷✸✽✵ ❱❡❧♦❝✐❞❛❞❡ ❞♦ s♦♠ ❝✐s❛❧❤❛♠❡♥t♦ c s ❬♠✴s❪ ✶✷✼✸ ✾✽✼ − − 2 15

                ❚❡♠♣♦ ❞❡ r❡❧❛①❛çã♦ ❧♦♥❣✐t✉❞✐♥❛❧ τ ❬s❪ 1.27 × 10 2.27 × 10 − − 4 7 ❚❡♠♣♦ ❞❡ r❡❧❛①❛çã♦ ❞❡ ❝✐s❛❧❤❛♠❡♥t♦ τ s ❬s❪ 3.34 × 10 2.39 × 10 ❖r❞❡♠ ❢r❛❝✐♦♥ár✐❛ ✈✐s❝♦❡❧ást✐❝❛ ν ✵✳✸✼ ✵✳✶✺

                ❋♦♥t❡✿ ❆✉t♦r✱ ✷✵✶✺✳ ✺✳✷✳✷ ❆♥á❧✐s❡ ❞♦s ♣❛râ♠❡tr♦s ❞❡ ❡s❝❛❧❛

                ❈♦♥s✐❞❡r❡ ✉♠❛ ♣❛rtí❝✉❧❛ ✈✐s❝♦❡❧ást✐❝❛ s✉s♣❡♥s❛ ♥❛ á❣✉❛ ❡♠ t❡♠♣❡r❛t✉r❛ ❛♠❜✐❡♥t❡✳ 3 = 1000 kg/m

                ❆ á❣✉❛ é ❝❛r❛❝t❡r✐③❛❞❛ ♣♦r s✉❛ ❞❡♥s✐❞❛❞❡ ρ ❡ ♣❡❧❛ ✈❡❧♦❝✐❞❛❞❡ ❞❡ s♦♠ 2 = 1500 m/s

                = 30 kW/m ❛❞✐❛❜át✐❝❛ c ✳ ❖s ❢❡✐①❡s ✐♥❝✐❞❡♥t❡s tê♠ ♣✐❝♦ ❞❡ ✐♥t❡♥s✐❞❛❞❡ I ✳ ◆♦ss❛ ❛♥á❧✐s❡ ❛❜r❛♥❣❡ ✉♠❛ ❢❛✐①❛ ❞❡ ❢r❡q✉ê♥❝✐❛ ❞❡ 10 Hz ❛ 10 MHz✳ ■st♦ ❝♦rr❡s♣♦♥❞❡ ❛ ✉♠❛ ❢❛✐①❛ ❞❡ ❢r❡q✉ê♥❝✐❛ r❛③♦á✈❡❧ ♣❛r❛ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦s ❞❡ ❡①♣❡r✐♠❡♥t♦s ❞❡ ♠❛♥✐♣✉❧❛çã♦ ❞❡ ♣❛rtí❝✉❧❛s ❡ ❞✐s♣♦s✐t✐✈♦s ❞❡ ♣✐♥ç❛s ❛❝úst✐❝❛s✳

                ❆ ❛♥á❧✐s❡ ❞❛ ♣❛rtí❝✉❧❛ ❞❡ ♣♦❧✐❡t✐❧❡♥♦ ❞❡ ❛❧t❛ ❞❡♥s✐❞❛❞❡ ✭P❊❆❉✮ ❝♦♠ r❛✐♦ a = 1 mm ❛❜r❛♥❣❡♥❞♦ ✉♠❛ ❢❛✐①❛ ❞❡ ❢r❡q✉ê♥❝✐❛ ❞❡ 10 Hz ❛ 100 kHz✱ s❡rá ❝♦♥s✐❞❡r❛❞♦ ❛♣r♦①✐♠❛çã♦ ❞❡ ❜❛✐①❛ ❢r❡q✉ê♥❝✐❛✳ ◆❡st❡ ❝❛s♦✱ ♦s ♣❛râ♠❡tr♦s ❞❡ ❡s❝❛❧❛ ❡♥❝♦♥tr❛♠✲s❡ ♥♦s ✐♥t❡r✈❛❧♦s

                0.012 < ε < 0.047, 0.18 < ε < 0.75,

                s 5

                ✭✺✳✷✻✮ < ε < 0.42. 4.1 × 10

                ❆ ❛♥á❧✐s❡ ❞❡ ❛❧t❛ ❢r❡q✉ê♥❝✐❛ é ❢❡✐t❛ ❝♦♥s✐❞❡r❛♥❞♦ ✉♠❛ ♣❛rtí❝✉❧❛ ❞❡ P❊❇❉ ❞❡ r❛✐♦ a = 10 µm ❡♠ ✉♠❛ ❢❛✐①❛ ❞❡ ❢r❡q✉ê♥❝✐❛ ❞❡ 2 ❛ 10 MHz✳ ◆❡ss❡ ❝❛s♦✱ ♦s ♣❛râ♠❡tr♦s ❞❡

                ❡s❝❛❧❛ s❛t✐s❢❛③❡♠ ❛♦s ✐♥t❡r✈❛❧♦s 1 0.0065 < ε < 0.012, 1

                0.025 < ε < 0.046, ✭✺✳✷✼✮

                

              s

              0.084 < ε < 0.42.

                ◆❛ ❛♥á❧✐s❡ ❛ s❡❣✉✐r✱ ♦ r❛✐♦ ❞❛ ♣❛rtí❝✉❧❛ a ❡ ♦ ❝♦♠♣r✐♠❡♥t♦ ❞❡ ♦♥❞❛ ✐♥❝✐❞❡♥t❡ λ sã♦ , δ

                ♠✉✐t♦ ♠❛✐♦r❡s ❞♦ q✉❡ ♦s ❧✐♠✐t❡s ❞❡ ❢r♦♥t❡✐r❛ tér♠✐❝❛ ❡ ✈✐s❝♦s❛✿ a, λ ≫ δ t ✈ ✳ ❆❧é♠ ❞✐ss♦✱ ✈❛♠♦s ❝♦♠♣❛r❛r ❛ ❛♣r♦①✐♠❛çã♦ ❞❛ ❢♦rç❛ ❞❡ r❛❞✐❛çã♦ ❛❝úst✐❝❛ ❛①✐❛❧ ❞❡✈✐❞♦ ❛ ♦♥❞❛ z ♣❧❛♥❛ ♣r♦❣r❡ss✐✈❛ ❡ ❞❡ ✉♠ ❢❡✐①❡ ❞❡ ❇❡ss❡❧ ❞❡ ♣r✐♠❡✐r❛ ♦r❞❡♠ ✭❞❡♥♦t❛❞♦s ♣♦r F ✮ ❞❛❞♦s✱

                r❛❞,z

                r❡s♣❡❝t✐✈❛♠❡♥t❡✱ ♥❛s ❊qs✳ ❝♦♠ ❛ ❢♦rç❛ ❞❡ r❛❞✐❛çã♦ ❛❝úst✐❝❛ ❛①✐❛❧ F ♥❛ ❛♣r♦①✐♠❛çã♦ ♠♦♥♦♣♦❧♦✲❞✐♣♦❧♦ ❞❛❞❛ ♥❛ ❊q✳ ❖ ❡rr♦ ❛❜s♦❧✉t♦ ❡♥tr❡ ❛s ❢ór♠✉❧❛s ❞❡

                ❢♦rç❛ ❞❡ r❛❞✐❛çã♦ ❛❝úst✐❝❛ é ❞❡✜♥✐❞♦ ❝♦♠♦ z |F r❛❞,z − F |

                , ∆ ≡

                ✭✺✳✷✽✮ F

                = 6.28 µN ♦♥❞❡ ❛ ♠❛❣♥✐t✉❞❡ ❞❛ ❢♦rç❛ ❞❡ r❛❞✐❛çã♦ ❛❝úst✐❝❛ é F ✱ 6.28 nN ♣❛r❛ ❛s ♣❛rtí❝✉❧❛s P❊❆❉ ❡ P❊❇❉✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✳ ◗✉❛♥❞♦ ❝♦♥s✐❞❡r❛♠♦s ♦ t♦rq✉❡ ❞❡ r❛❞✐❛çã♦ ❛❝úst✐❝♦✱ ♦ ❡rr♦ ❛❜s♦❧✉t♦ é ❞❡✜♥✐❞♦ ❞❛ s❡❣✉✐♥t❡ ❢♦r♠❛✿

                ❋❇P❖

                N

                r❛❞,z

                − N

                ❛❜s,z .

                ∆ ≡ ✭✺✳✷✾✮ aF

                ✺✳✷✳✸ ❖♥❞❛s ♣r♦❣r❡ss✐✈❛s

                ◆❛s ❋✐❣✉r❛s ✐❧✉str❛♠♦s ❛ ❢♦rç❛ ❞❡ r❛❞✐❛çã♦ ❛❝úst✐❝❛ ❛①✐❛❧ ✈❡rs✉s ❢r❡q✉ê♥❝✐❛ ❣❡r❛❞❛ ♥❛s ♣❛rtí❝✉❧❛s ❞❡ P❊❇❉ ❡ P❊❆❉✳ ❆ ❢♦rç❛ ❞❡ r❛❞✐❛çã♦ ♣♦r ✉♠❛ ♦♥❞❛ ♣❧❛♥❛ ♣r♦✲ ❣r❡ss✐✈❛ ✭❖PP✮ é ♠♦str❛❞❛ ♥❛ ❋✐❣✉r❛ ❛♣r❡s❡♥t❛ ♦ r❡s✉❧t❛❞♦ ♣❛r❛ ✉♠ ❢❡✐①❡ ❞❡ ❇❡ss❡❧ ❞❡ ♣r✐♠❡✐r❛ ♦r❞❡♠ ✭❋❇P❖✮ ❝♦♠ n = 1 ❡ â♥❣✉❧♦ ❞❡ ♠❡✐♦ ❝♦♥❡ β = 45

                ✳ ❖❜s❡r✈❡ q✉❡ ❛ ❢♦rç❛ ❞❡ r❛❞✐❛çã♦ ❛①✐❛❧ s♦❜r❡ ❛ ♣❛rtí❝✉❧❛ ❞❡ P❊❆❉ é ♥❡❣❛t✐✈❛ ♣❛r❛

                ❛♠❜♦s ♦s ❢❡✐①❡s✱ ❖PP ❡ ❋❇P❖✱ ❝♦♠♦ ♣♦❞❡ s❡r ✈✐st♦ ♥❛s ❋✐❣✉r❛s ■st♦ ♣♦❞❡r✐❛ t❡r s✐❞♦ ❛♥t❡❝✐♣❛❞♦✱ ❞❡ ❛❝♦r❞♦ ❝♦♠ ❛ ❚❛❜❡❧❛ ♦s t❡♠♣♦s ❞❡ r❡❧❛①❛çã♦ ❞♦ P❊❆❉ s❛t✐s❢❛③❡♠ ❛ ❝♦♥❞✐çã♦ τ ℓ s ✳ ❉❡st❛ ❢♦r♠❛✱ ❛ ❝♦♥❞✐çã♦ ❞❡ ❢♦rç❛ ❞❡ r❛❞✐❛çã♦ ♥❡❣❛t✐✈❛

                ≪ τ ❊q✳ é s❛t✐s❢❡✐t❛✳ ❆❧é♠ ❞✐ss♦✱ ❛ ❞❡♥s✐❞❛❞❡ ❞♦ P❊❆❉ é ♣ró①✐♠❛ ❞❛ ❞❡♥s✐❞❛❞❡ ❞❛ á❣✉❛✱ ✐st♦ ❛ss❡❣✉r❛ q✉❡ ❛ ❊q✳ t❛♠❜é♠ é ❝♦♥❞✐çã♦ ❞❡ ❢♦rç❛ ❞❡ r❛❞✐❛çã♦ ♥❡❣❛t✐✈❛ ♣❛r❛ ♦ ❢❡✐①❡ ❞❡ ❇❡ss❡❧ ❞❡ ♣r✐♠❡✐r❛ ♦r❞❡♠✳ z

                ❆ ❢♦rç❛ ❞❡ r❛❞✐❛çã♦ ❛❝úst✐❝❛ F ❞❛❞❛ ♥❛ ❊q✳ ♣❛r❛ ♦♥❞❛ ♣❧❛♥❛ ♣r♦❣r❡ss✐✈❛ ❡ ♥❛ ❊q✳ ♣❛r❛ ♦ ❢❡✐①❡ ❞❡ ❇❡ss❡❧ ❞❡ ♣r✐♠❡✐r❛ ♦r❞❡♠✱ t❡♠ ❜♦❛ ❝♦♥❝♦r❞â♥❝✐❛ ❝♦♠ ❛ ❢ór♠✉❧❛ ❞❛ ❛♣r♦①✐♠❛çã♦ ♠♦♥♦♣♦❧♦✲❞✐♣♦❧♦ ❞❛ ❊q✳ ♣❛r❛ ❢r❡q✉ê♥❝✐❛s ❛té 72 kHz✱ ♦✉ ε = 0.30✱ ❝♦♠♦ ♣♦❞❡ s❡r ✈✐st♦ ♥❛s ❋✐❣✉r❛s r❡s♣❡❝t✐✈❛♠❡♥t❡ ✳ ◆❡st❛ ❢❛✐①❛✱ ♦ ❡rr♦ 3

                ❛❜s♦❧✉t♦ ♣❛r❛ ♦♥❞❛ ♣❧❛♥❛ ♣r♦❣r❡ss✐✈❛ é ❞❡ ∆ ∼ ε ✱ ❡♥q✉❛♥t♦ q✉❡ ♣❛r❛ ♦ ❢❡✐①❡ ❞❡ ❇❡ss❡❧ 5 ❞❡ ♣r✐♠❡✐r❛ ♦r❞❡♠ é ❞❡ ∆ ∼ ε ✳ ■st♦ ❡stá ❞❡ ❛❝♦r❞♦ ❝♦♠ ♦ ❡rr♦ ♣r❡✈✐st♦ ♥❛s ❊qs✳ ❡ r❡s♣❡❝t✐✈❛♠❡♥t❡✳ ❊♥tr❡t❛♥t♦✱ ❞❡✈❡♠♦s t❡r ❡♠ ♠❡♥t❡ q✉❡ ♦ ❧✐♠✐t❡ ❞❡ ❡s♣❛❧❤❛♠❡♥t♦ ❘❛②❧❡✐❣❤ é ✈á❧✐❞♦ ♣❛r❛ ε < 0.3✱ ♦✉ ❝♦♠♦ ♠❡♥❝✐♦♥❛❞♦ ❛♥t❡r✐♦r♠❡♥t❡ ♣❛r❛ ❢r❡q✉ê♥❝✐❛s ♠❡♥♦r❡s q✉❡ 72 kHz✳

                ❆ ❢♦rç❛ ❞❡ r❛❞✐❛çã♦ ❛❝úst✐❝❛ ❛①✐❛❧ s♦❜r❡ ❛ ♣❛rtí❝✉❧❛ ❞❡ P❊❇❉ é ♣♦s✐t✐✈❛ ❡♠ ❛♠❜♦s ♦s ❝❛s♦s✱ ❝♦♠♦ ♣♦❞❡ s❡r ✈✐st♦ ♥❛s ❋✐❣✉r❛s ❖ ❡rr♦ ❛❜s♦❧✉t♦ ♣❛r❛ ❛ ♦♥❞❛ 2

                ♣❧❛♥❛ ♣r♦❣r❡ss✐✈❛ é ❞❛ ♦r❞❡♠ ∆ ∼ ε ♥♦ ✐♥t❡r✈❛❧♦ ❞❡ 2 à 7.2 MHz✱ ♦✉ 0.08 < ε < 0.3✳ 5 ◆❡ss❡ ✐♥t❡r✈❛❧♦✱ ♦ ❡rr♦ ❛❜s♦❧✉t♦ ♣❛r❛ ♦ ❢❡✐①❡ ❞❡ ❇❡ss❡❧ ❞❡ ♣r✐♠❡✐r❛ ♦r❞❡♠ é ❞❡ ∆ ∼ ε ✳ ◆♦✈❛♠❡♥t❡✱ ✐st♦ ❝♦♥❝♦r❞❛ ❝♦♠ ♦ ❡rr♦ ♠❡♥❝✐♦♥❛❞♦ ♥❛s ❊qs✳

                ✺✳✷✳✹ ❋❡✐①❡ ❞❡ ❇❡ss❡❧ tr❛t♦r

                ❯♠❛ ✈❡③ q✉❡ ♦ ❢❡✐①❡ ❞❡ ❇❡ss❡❧ ❞❡ ♣r✐♠❡✐r❛ ♦r❞❡♠ ♣r♦❞✉③ ❢♦rç❛ ❞❡ r❛❞✐❛çã♦ ❛①✐❛❧ ♥❡❣❛t✐✈❛ s♦❜r❡ ✉♠❛ ♣❛rtí❝✉❧❛ ❞❡ P❊❆❉✱ ✈❛♠♦s ❛♥❛❧✐s❛r ❛ ❢♦rç❛ ❞❡ r❛❞✐❛çã♦ ❛❝úst✐❝❛

                ❋♦rç❛ ❞❡ r❛❞✐❛çã♦ ❛❝úst✐❝❛ ✈❡rs✉s ❢r❡q✉ê♥❝✐❛ ❡①❡r❝✐❞❛ s♦❜r❡ ✉♠❛ ♣❛rtí❝✉❧❛ ❞❡ ♣♦❧✐❡✲

                ❋✐❣✉r❛ ✺✳✹✿

                

              t✐❧❡♥♦ ❞❡ ❛❧t❛ ❞❡♥s✐❞❛❞❡ ✭P❊❆❉✮ ❡ ❞❡ ✉♠❛ ♣❛rtí❝✉❧❛ ❞❡ ♣♦❧✐❡t✐❧❡♥♦ ❞❡ ❜❛✐①❛ ❞❡♥s✐❞❛❞❡ ✭P❊❇❉✮✳

              ❆s ❝♦♥✜❣✉r❛çõ❡s sã♦✿ ✉♠❛ ♦♥❞❛ ♣❧❛♥❛ ♣r♦❣r❡ss✐✈❛ ❛❣✐♥❞♦ s♦❜r❡ ✉♠❛ ♣❛rtí❝✉❧❛ ✭❛✮ ❞❡ P❊❆❉ ❡ ✭❜✮

              ❞❡ P❊❇❉✳ ❆s ♣❛rtí❝✉❧❛s ❞❡ P❊❆❉ ❡ P❊❇❉ sã♦ s✉s♣❡♥s❛ ❡♠ á❣✉❛ ❡ t❡♠ r❛✐♦ ❞❡ a = 1.0, 0.01 mm✱

              r❡s♣❡❝t✐✈❛♠❡♥t❡✳ ❖s ♣❛râ♠❡tr♦s ❢ís✐❝♦s ❞❛s ♣❛rtí❝✉❧❛s sã♦ ❞❛❞♦s ♥❛ ❚❛❜❡❧❛ ❖s ❢❡✐①❡s ✐♥❝✐✲

              2 = 30 kW/m

                

              ❞❡♥t❡s tê♠ ♣✐❝♦ ❞❡ ✐♥t❡♥s✐❞❛❞❡ ❞❡ I ✳ ▲❡❣❡♥❞❛✿ ❧✐♥❤❛ só❧✐❞❛ ❛③✉❧ r❡♣r❡s❡♥t❛ ❛ ❢♦rç❛

              ❞❡ r❛❞✐❛çã♦ ❛✈❛❧✐❛❞❛ ♥❛ ❛♣r♦①✐♠❛çã♦ ♠♦♥♦♣♦❧♦✲❞✐♣♦❧♦ ❞❛❞❛ ♥❛ ❊q✳ ❧✐♥❤❛ ✈❡r♠❡❧❤❛ tr❛❝❡✲

              ❥❛❞❛ r❡♣r❡s❡♥t❛ ❛ ❢♦rç❛ ❞❡ r❛❞✐❛çã♦ ♥♦s r❡❣✐♠❡s ❞❡ ❜❛✐①❛ ❡ ❛❧t❛ ❢r❡q✉ê♥❝✐❛ ❛tr❛✈és ❞❛ ❊qs✳

              ♣❛r❛ ♦♥❞❛ ♣❧❛♥❛ ♣r♦❣r❡ss✐✈❛✳ ❖s ✐♥s❡rt❡s ♠♦str❛♠ ♦ ❡rr♦ r❡❧❛t✐✈♦ ❡♥tr❡ ❛ ❢ór♠✉❧❛s ❞❡ ❢♦rç❛ ❞❡

              r❛❞✐❛çã♦✳ ❆ ❧✐♥❤❛ ✈❡rt✐❝❛❧ ♣♦♥t✐❧❤❛❞❛ ❞❡❧✐♠✐t❛ ❛ r❡❣✐ã♦ ❞❛ ❛♣r♦①✐♠❛çã♦ ❘❛②❧❡✐❣❤✱ ε < 0.3✳ (a)

                PEAD ão aç di

                1.59 ra

                1.27 uto,

                0.95 a de

                0.63

                Erro absol Frequência [MHz] (b)

                PEBD

                0.63

                0.47 uto,

                0.31 ão [nN]

                0.15 aç di

                Erro absol ra a de Forç

                Frequência [MHz] ❋♦♥t❡✿ ❆✉t♦r✱ ✷✵✶✺✳

                tr❛♥s✈❡rs❛❧ q✉❡ é ✐❧✉str❛❞❛ ♥❛ ❋✐❣✳ ❉❡✜♥✐♠♦s ❛ ❢r❡q✉ê♥❝✐❛ ❞❡ 50 kHz✳ ❖ ❝❛♠♣♦ ✈❡t♦r✐❛❧ ❞❛ ❢♦rç❛ ❞❡ r❛❞✐❛çã♦ tr❛♥s✈❡rs❛❧ é ✐❧✉str❛❞♦ ♣❡❧❛s s❡t❛s✱ ❡ ❛ ❡♥❡r❣✐❛ ♣♦t❡♥❝✐❛❧ ❞❛

                ❋✐❣✉r❛ ✺✳✺✿

                ❋♦rç❛ ❞❡ r❛❞✐❛çã♦ ❛❝úst✐❝❛ ✈❡rs✉s ❢r❡q✉ê♥❝✐❛ ❡①❡r❝✐❞❛ s♦❜r❡ ✉♠❛ ♣❛rtí❝✉❧❛ ❞❡ ♣♦❧✐❡✲

              t✐❧❡♥♦ ❞❡ ❛❧t❛ ❞❡♥s✐❞❛❞❡ ✭P❊❆❉✮ ❡ ❞❡ ✉♠❛ ♣❛rtí❝✉❧❛ ❞❡ ♣♦❧✐❡t✐❧❡♥♦ ❞❡ ❜❛✐①❛ ❞❡♥s✐❞❛❞❡ ✭P❊❇❉✮✳

              ❆s ❝♦♥✜❣✉r❛çõ❡s sã♦✿ ❢❡✐①❡ ❞❡ ❇❡ss❡❧ ❞❡ ♣r✐♠❡✐r❛ ♦r❞❡♠ ❝♦♠ n = 1 ❡ β = 45 ❛❣✐♥❞♦ s♦❜r❡ ✉♠❛

              ♣❛rtí❝✉❧❛ ❞❡ ✭❛✮ P❊❆❉ ❡ ✭❜✮ ❞❡ P❊❇❉✳ ❆s ♣❛rtí❝✉❧❛s ❞❡ P❊❆❉ ❡ P❊❇❉ sã♦ s✉s♣❡♥s❛ ❡♠ á❣✉❛

              ❡ t❡♠ r❛✐♦ ❞❡ a = 1.0, 0.01 mm✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✳ ❖s ♣❛râ♠❡tr♦s ❢ís✐❝♦s ❞❛s ♣❛rtí❝✉❧❛s sã♦ ❞❛❞♦s

              ♥❛ ❚❛❜❡❧❛ ❖s ❢❡✐①❡s ✐♥❝✐❞❡♥t❡s tê♠ ♣✐❝♦ ❞❡ ✐♥t❡♥s✐❞❛❞❡ ❞❡ I

                = 30 kW/m 2 ✳ ▲❡❣❡♥❞❛✿ ❧✐♥❤❛

              só❧✐❞❛ ❛③✉❧ r❡♣r❡s❡♥t❛ ❛ ❢♦rç❛ ❞❡ r❛❞✐❛çã♦ ❛✈❛❧✐❛❞❛ ♥❛ ❛♣r♦①✐♠❛çã♦ ♠♦♥♦♣♦❧♦✲❞✐♣♦❧♦ ❞❛❞❛ ♥❛

              ❊q✳ ❧✐♥❤❛ ✈❡r♠❡❧❤❛ tr❛❝❡❥❛❞❛ r❡♣r❡s❡♥t❛ ❛ ❢♦rç❛ ❞❡ r❛❞✐❛çã♦ ♥♦s r❡❣✐♠❡s ❞❡ ❜❛✐①❛ ❡ ❛❧t❛

              ❢r❡q✉ê♥❝✐❛ ❛tr❛✈és ❞❛ ❊q✳ ♣❛r❛ ❢❡✐①❡ ❞❡ ❇❡ss❡❧ ❞❡ ♣r✐♠❡✐r❛ ♦r❞❡♠✳ ❖s ✐♥s❡rt❡s ♠♦str❛♠ ♦

              ❡rr♦ r❡❧❛t✐✈♦ ❡♥tr❡ ❛ ❢ór♠✉❧❛s ❞❡ ❢♦rç❛ ❞❡ r❛❞✐❛çã♦✳ ❆ ❧✐♥❤❛ ✈❡rt✐❝❛❧ ♣♦♥t✐❧❤❛❞❛ ❞❡❧✐♠✐t❛ ❛ r❡❣✐ã♦

              ❞❛ ❛♣r♦①✐♠❛çã♦ ❘❛②❧❡✐❣❤✱ ε < 0.3✳ (a)

                (b) PEAD PEBD

                0.15

                0.31

                0.47

                0.63

                0.15

                0.31

                0.47

                uto, Erro absol uto,

                Frequência [MHz] Frequência [MHz]

                Forç a de ra di aç ão [pN] Forç a de ra di aç ão [nN]

                ❋♦♥t❡✿ ❆✉t♦r✱ ✷✵✶✺✳

                ❢♦rç❛ ❞❡ r❛❞✐❛çã♦ U 1 ❞❛❞❛ ♥❛ ❊q✳ ♣♦❞❡ s❡r ✈✐st❛ ❛tr❛✈és ❞♦ ❣rá✜❝♦ ❞❡ ❝♦♥t♦r♥♦✳ ◆♦t❡

                ♦ ❢❡✐①❡ ❞❡ ❇❡ss❡❧ ❞❡ ♣r✐♠❡✐r❛ ♦r❞❡♠ ♣r♦❞✉③ ✉♠❛ ❢♦rç❛ ❞❡ r❛❞✐❛çã♦ ❛❝úst✐❝❛ tr❛♥s✈❡rs❛❧ ❝♦♥✈❡r❣❡♥t❡ ♥❛ ❞✐r❡çã♦ ❞♦ s❡✉ ❡✐①♦ ♥❛ r❡❣✐ã♦ ❞❡ −2 ≤ ❦①✱ ❦② ≤ 2✱ ♦♥❞❡ ❛ ♣♦s✐çã♦ ❝❡♥tr❛❧ ❞♦ ❢❡✐①❡ é ✉♠❛ r❡❣✐ã♦ ❞❡ ❡q✉✐❧í❜r✐♦ ❡stá✈❡❧ ♣❛r❛ ❛ ❡s❢❡r❛✳ ❉❡st❛ ❢♦r♠❛✱ ❡❧❡ s❡ ❝♦♠♣♦rt❛ ❝♦♠♦ ❢❡✐①❡ tr❛t♦r ✭❢❡✐①❡ ❝❛♣❛③ ❞❡ ❛tr❛✐r ♦❜❥❡t♦s✮ ✸❉ ❝♦♠♣❧❡t♦ ❛❣✐♥❞♦ s♦❜r❡ ❛ ♣❛rtí❝✉❧❛ ❞❡ P❊❆❉✳ ❉❡ ❛❝♦r❞♦ ❝♦♠ ❛ ❊q✳ ❛ ♠❛❣♥✐t✉❞❡ ❞❛ ❢♦rç❛ ❞❡ r❛❞✐❛çã♦ ❛❝úst✐❝❛ tr❛♥s✈❡rs❛❧ 1 /a

                é ❛♣r♦①✐♠❛❞❛♠❡♥t❡ ❛ ♠❛❣♥✐t✉❞❡ ❞❛ ❡♥❡r❣✐❛ ♣♦t❡♥❝✐❛❧ ❞✐✈✐❞✐❞♦ ♣❡❧♦ r❛✐♦ ❞❛ ♣❛rtí❝✉❧❛ U ✳ ❆ss✐♠✱ ✉♠❛ ❡st✐♠❛t✐✈❛ ❞♦ ♣✐❝♦ ❞❡ ♠❛❣♥✐t✉❞❡ ❞❛ ❢♦rç❛ ❞❡ r❛❞✐❛çã♦ ❛❝úst✐❝❛ ❞á 5 µN✳ ■ss♦ é ❝❡r❝❛ ❞❡ ♠✐❧ ✈❡③❡s ♠❛✐♦r q✉❡ ❛ ❢♦rç❛ ❞❡ r❛❞✐❛çã♦ ❛❝úst✐❝❛ ❛①✐❛❧ ♠♦str❛❞♦ ♥❛ ❋✐❣✳ ♣❛r❛ 50 kHz✳

                ❈❛♠♣♦ ✈❡t♦r✐❛❧ ✭s❡t❛s✮ ❞❛ ❢♦rç❛ ❞❡ r❛❞✐❛çã♦ tr❛♥s✈❡rs❛❧ ♥♦ ♣❧❛♥♦ xy✱ ❡①❡r❝✐❞♦ s♦❜r❡

                ❋✐❣✉r❛ ✺✳✻✿

                

              ❛ ♣❛rtí❝✉❧❛ ❞❡ P❊❆❉ ❝♦♠ r❛✐♦ a = 1 mm ♥❛ á❣✉❛✳ ❖ ❣rá✜❝♦ ❞❡ ❝♦♥t♦r♥♦ ❝♦rr❡s♣♦♥❞❡ ❛ ❡♥❡r❣✐❛

              ♣♦t❡♥❝✐❛❧ ❞❛ ❢♦rç❛ ❞❡ r❛❞✐❛çã♦ U 1 ❞❛❞❛ ♥❛ ❊q✳ ◆♦ ❡♥t❛♥t♦✱ ❛q✉✐ ✜①❛♠♦s ❛

              ❢r❡q✉ê♥❝✐❛ ♣❛r❛ 50 kHz✳ ❖ ❝❛♠♣♦ ✈❡t♦r✐❛❧ ❞❛ ❢♦rç❛ ❞❡ r❛❞✐❛çã♦ é ❝❛❧❝✉❧❛❞♦ ❛tr❛✈és ❞❛ ❊q✳

                [nJ]

                4

                5

                2

                4

                3

                ky

                2

                2

                1

                4

                4

                2

                2

                4 kx

                ❋♦♥t❡✿ ❆✉t♦r✱ ✷✵✶✺✳ ✺✳✷✳✺ ❖♥❞❛ ♣❧❛♥❛ ❡st❛❝✐♦♥ár✐❛

                ◆❛ ❋✐❣✉r❛ ❛♣r❡s❡♥t❛♠♦s ❛ ❢♦rç❛ ❞❡ r❛❞✐❛çã♦ ❣r❛❞✐❡♥t❡ ❞❡✈✐❞♦ ❛ ✉♠❛ ♦♥❞❛ ♣❧❛♥❛ ❡st❛❝✐♦♥ár✐❛ ❞✐str✐❜✉í❞❛ ❛♦ ❧♦♥❣♦ ❞♦ ❡✐①♦ z✳ ❖s r❡s✉❧t❛❞♦s ❞♦ ♠♦❞❡❧♦ ❞❡ ❑❡❧✈✐♥✲❱♦✐❣t ❢r❛❝✐♦♥ár✐♦ ✭♠❡✐♦ ❞✐s♣❡rs✐✈♦✮ é ❝♦♠♣❛r❛❞♦ ❝♦♠ ✉♠ só❧✐❞♦ ❡❧ást✐❝♦ s❡♠ ❛❜s♦rçã♦ ✭♠❡✐♦ ♥ã♦

                ❞✐s♣❡rs✐✈♦✮✳ ❆ ♣❛rt✐r ❞❛ ❊q✳ s❛❜❡♠♦s q✉❡ ❛ ❛❜s♦rçã♦ ♥ã♦ ❛❢❡t❛ ❛ ❢♦rç❛ ❞❡ r❛❞✐❛çã♦ ❣r❛❞✐❡♥t❡✳ ❆♣❡s❛r ❞✐ss♦✱ ❛ ❞✐s♣❡rsã♦ ♥♦ ✐♥t❡r✐♦r ❞❛ ♣❛rtí❝✉❧❛ ♣♦❞❡ ❛❧t❡r❛r ❝♦♥s✐❞❡r❛✈❡❧✲ ♠❡♥t❡ ❛ ❛♠♣❧✐t✉❞❡ ❞♦ ♣✐❝♦ ❞❛ ❢♦rç❛ ❞❡ r❛❞✐❛çã♦ ❣r❛❞✐❡♥t❡✳ ◆❛ ❋✐❣✳ ✭❛✮✱ ❡♥❝♦♥tr❛♠♦s q✉❡ ♦ ♠♦❞❡❧♦ ✈✐s❝♦❡❧ást✐❝♦ ♣r❡✈ê ✉♠ ♣✐❝♦ ❞❛ ❢♦rç❛ ❞❡ r❛❞✐❛çã♦ ❛❝úst✐❝❛ ♥❛ ♣❛rtí❝✉❧❛ ❞❡ P❊❆❉ 26 % ♠❡♥♦r q✉❡ ❞♦ ♠♦❞❡❧♦ só❧✐❞♦ ❡❧ást✐❝♦✳ ❊ss❛ ❞✐❢❡r❡♥ç❛ ♣❛r❛ ❛ ♣❛rtí❝✉❧❛ ❞❡ P❊❇❉ ❝♦♠♦ ♠♦str❛❞♦ ♥❛ ❋✐❣✳ ✭❜✮ é ❞❡ 180 %✳

                ◆♦t❡ q✉❡ ♦ ❢❛t♦r ❞❡ ❝♦♥tr❛st❡ ❞❛❞♦ ♥❛ ❊q✳ é ♠❛✐♦r q✉❡ ③❡r♦ ✭C > 0✮✱ ❞❡t❡r♠✐♥❛♥❞♦ q✉❡ ❛ ♣❛rtí❝✉❧❛ ✭P❊❆❉ ❡ P❊❇❉✮ t❡♠ ✉♠ ♣♦♥t♦ ❞❡ ❡q✉✐❧í❜r✐♦ ❡stá✈❡❧ ❡♠ ✉♠ ♥ó ❞❡ ♣r❡ssã♦✱ ❋✐❣✳ ✭❛✮ ❡ ✭❜✮✳

                ❋♦rç❛ ❞❡ r❛❞✐❛çã♦ ❛❝úst✐❝❛ ♣r♦❞✉③✐❞❛ ♣♦r ✉♠❛ ♦♥❞❛ ♣❧❛♥❛ ❡st❛❝✐♦♥ár✐❛ ❛♦ ❧♦♥❣♦

                ❋✐❣✉r❛ ✺✳✼✿

                

              ❞♦ ❡✐①♦ z ❡♠ ✭❛✮ ✉♠❛ ♣❛rtí❝✉❧❛ ❞❡ P❊❆❉ ♣❛r❛ 50 kHz ❡ ✭❜✮ ✉♠❛ ♣❛rtí❝✉❧❛ ❞❡ P❊❇❉ ♣❛r❛

                

              2 MHz ✱ ✐♠❡rs♦ ❡♠ á❣✉❛✳ ❖s ♣❛râ♠❡tr♦s ❢ís✐❝♦s ❞❛ ♣❛rtí❝✉❧❛ sã♦ ❞❛❞♦s ♥❛ ❚❛❜❡❧❛ ❆ ♦♥❞❛

              ♣❧❛♥❛ ❡st❛❝✐♦♥ár✐❛ t❡♠ ♣✐❝♦ ❞❡ ♣r❡ssã♦ ❞❡ 0.3 MPa✳ ❆ ❢♦rç❛ ❞❡ r❛❞✐❛çã♦ ❢♦✐ ❝♦♠♣✉t❛❞❛ ✉s❛♥❞♦ ❛

              ✭❘✮ ✭❊✮

                = f ❊q✳ P❛r❛ ♦ ♠♦❞❡❧♦ ❡❧ást✐❝♦ ❡♠ ♣❡r❞❛s✱ ❛ss✉♠✐♠♦s q✉❡ f ✳ Pressão (a) ão N] 2 Viscoelástica Elástica 0.18 ] a de ra Ponto de Forç di -2 aprisionamento -0.18 0.00 o [MPa ssã Pre Elástica Pressão -

              • 3 2 -1

                (b)

              • 1 2 3 di ão ra [ nN] 0.0 0.2 Viscoelástica 0.00

                  0.19 ] a de Forç -0.2 aprisionamento Ponto de -0.19 Pre o [MPa ssã -

                • 3 2 -1

                  kz

                • 1 2 3 ❋♦♥t❡✿ ❆✉t♦r✱ ✷✵✶✺✳

                    ✺✳✷✳✻ ❚♦rq✉❡ ❞❡ r❛❞✐❛çã♦ ♣♦r ✉♠ ❢❡✐①❡ ❞❡ ❇❡ss❡❧

                    ◆❛ ❋✐❣✉r❛ ✐❧✉str❛♠♦s ♦ t♦rq✉❡ ❞❡ r❛❞✐❛çã♦ ❛❝úst✐❝♦ ❛①✐❛❧ ❣❡r❛❞♦ ♣♦r ✉♠ ❢❡✐①❡ ❞❡ ❇❡ss❡❧ ❞❡ ♣r✐♠❡✐r❛ ♦r❞❡♠ ❝♦♠ (n = 1) s♦❜r❡ ✭❛✮ ✉♠❛ ♣❛rtí❝✉❧❛ ❞❡ P❊❆❉ ❡ ✭❜✮ ✉♠❛ ♣❛rtí❝✉❧❛ ❞❡ P❊❇❉✳ ❆s ♣❛rtí❝✉❧❛s sã♦ s✉s♣❡♥s❛s s♦❜r❡ ♦ ❡✐①♦ ❞♦ ❢❡✐①❡ ✭̺ = 0✮✳ ❖ ❢❡✐①❡ ❞❡ ❇❡ss❡❧ ❞❡ ♣r✐♠❡✐r❛ ♦r❞❡♠ t❡♠ ♦s ♠❡s♠♦ ♣❛râ♠❡tr♦s ❞❡s❝r✐t♦s ♥❛ ❋✐❣✉r❛ ❖ t♦r✲ q✉❡ ❞❡ r❛❞✐❛çã♦ ❛❝úst✐❝♦ ❛①✐❛❧ ❡①❡r❝✐❞♦ s♦❜r❡ ✉♠❛ ♣❛rtí❝✉❧❛ ❞❡ P❊❆❉ ✭r❡❣✐♠❡ ❞❡ ❜❛✐①❛ ❢r❡q✉ê♥❝✐❛✮ é ♥❡❣❛t✐✈♦ ❡ ❛✉♠❡♥t❛ ♠♦♥♦t♦♥✐❝❛♠❡♥t❡ ❡♠ ♠❛❣♥✐t✉❞❡ ❝♦♠ ❛ ❢r❡q✉ê♥❝✐❛ ❝♦♠♦ ♣♦❞❡ s❡r ✈✐st♦ ♥❛ ❋✐❣✉r❛ ✭❛✮✳ ❆té 72 kHz ✭♦✉ ε < 0.3✮ ❡♥❝♦♥tr❛♠♦s q✉❡ ❛ ❛♣r♦①✐♠❛çã♦ 4

                    ❞❡ ❜❛✐①❛ ❢r❡q✉ê♥❝✐❛ é ✈á❧✐❞❛ ❝♦♠ ❡rr♦ ❛❜s♦❧✉t♦ ❞❡ ∆ ∼ ε ❞❡ ❛❝♦r❞♦ ❝♦♠ ❛ ❊q✳ P❛r❛ ♣❛rtí❝✉❧❛s ❞❡ P❊❇❉✱ ❋✐❣✉r❛ ✭❜✮✱ ♦ t♦rq✉❡ ❞❡ r❛❞✐❛çã♦ ❛❝úst✐❝♦ ❛①✐❛❧ é ♣♦s✐t✐✈♦ ❡ ❛✉♠❡♥t❛ ♠♦♥♦t♦♥✐❝❛♠❡♥t❡ ❝♦♠ ❛ ❢r❡q✉ê♥❝✐❛✳ ◆♦ ✐♥t❡r✈❛❧♦ ❞❡ 2 ❛ 7.2 MHz✱ ♦ ❡rr♦ ❛❜s♦❧✉t♦ 4

                    é ∆ ∼ ε ✱ q✉❡ ❡stá ❞❡ ❛❝♦r❞♦ ❝♦♠ ❛ ❊q✳

                    

                  ✺✳✷✳✼ ❈♦♠♣❛r❛çã♦ ❝♦♠ ♣❛rtí❝✉❧❛s s❡♠ ❛❜s♦rçã♦ ❞❡ ❝✐s❛❧❤❛♠❡♥t♦

                    ◆❡st❛ s❡çã♦✱ ❢❛r❡♠♦s ✉♠❛ ❝♦♠♣❛r❛çã♦ ❞❛ ❢♦rç❛ ❡ ❞♦ t♦rq✉❡ ❞❡ r❛❞✐❛çã♦ ❛❝úst✐❝♦ s♦❜r❡ ✉♠❛ ♣❛rtí❝✉❧❛ ❞❡ P❊❆❉ ❡ ✉♠❛ ♣❛rtí❝✉❧❛ ❞❡ P❊❇❉ ❝♦♥s✐❞❡r❛♥❞♦✲❛ ❝♦♠♦ ✉♠ ♠❛t❡r✐❛❧ só❧✐❞♦✱ ✢✉✐❞♦ ❝♦♠ ❛❜s♦rçã♦ ❧♦♥❣✐t✉❞✐♥❛❧ ❡ ✈✐s❝♦❡❧ást✐❝♦✳ ❆ ❢♦rç❛ ❞❡ r❛❞✐❛çã♦ é ❣❡r❛❞❛ ♣♦r ✉♠❛ ♦♥❞❛ ♣❧❛♥❛ ♣r♦❣r❡ss✐✈❛✱ ✉♠ ❢❡✐①❡ ❞❡ ❇❡ss❡❧ ❞❡ ♦r❞❡♠ ③❡r♦ ❡ ✉♠ ❢❡✐①❡ ❞❡ ❇❡ss❡❧ ❞❡ ♣r✐♠❡✐r❛ ♦r❞❡♠✳ ❖ â♥❣✉❧♦ ❞❡ ♠❡✐♦ ❝♦♥❡ é ❞❡ β = 45 ✳ ❆s ♦♥❞❛s s❡ ♣r♦♣❛❣❛♠ ❛♦ ❧♦♥❣♦ ❞♦ ❡✐①♦ z✳

                    ➱ ✐♠♣♦rt❛♥t❡ ♥♦t❛r q✉❡ ♥♦ r❡❣✐♠❡ ❞❡ ❛❧t❛ ❢r❡q✉ê♥❝✐❛ ♥ã♦ é ♣♦ssí✈❡❧ ❢❛③❡r ❛ ❝♦♠♣❛r❛çã♦ ❝♦♠ ✉♠❛ ♣❛rtí❝✉❧❛ ✢✉✐❞❛ ❝♦♠♣r❡ssí✈❡❧✱ ♣♦✐s ♦ t❡♠♣♦ ❞❡ r❡❧❛①❛çã♦ ❡♠ ✉♠ ✢✉✐❞♦ é s❡♠♣r❡ ♣❡q✉❡♥♦✳ ❉❡st❛ ❢♦r♠❛✱ ♥♦ r❡❣✐♠❡ ❞❡ ❛❧t❛ ❢r❡q✉ê♥❝✐❛ ❛ ♣❛rtí❝✉❧❛ ♥ã♦ ♣♦❞❡ s❡ ❝♦♠♣♦rt❛r ❝♦♠♦ ✉♠ ✢✉✐❞♦ ❝♦♠♣r❡ssí✈❡❧✳ ❆ ❛♠♣❧✐t✉❞❡ ❞❛ ❢♦rç❛ ❞❡ r❛❞✐❛çã♦ s♦❜r❡ ✉♠❛ ♣❛rtí❝✉❧❛ ❞❡ P❊❆❉ ✭r❡❣✐♠❡ ❞❡ ❜❛✐①❛ ❢r❡q✉ê♥❝✐❛✮ é ♠♦str❛❞♦ ♥❛ ✜❣✉r❛ ❆ ❢♦rç❛ ❞❡ r❛❞✐❛çã♦ ❞❡s❝r✐t❛ ♣❡❧♦ ♠♦❞❡❧♦ ✈✐s❝♦❡❧ást✐❝♦ ♠♦str❛ ✉♠ ❝♦♠♣♦rt❛♠❡♥t♦ ❞✐❢❡r❡♥t❡ ❝♦♠ r❡❧❛çã♦ ❛♦s ♦✉tr♦s ♠❛t❡r✐❛✐s ✭❡❧ást✐❝❛ ❡ ✢✉✐❞♦ ❝♦♠♣r❡ssí✈❡❧✮✳ P❛r❛ ♦ ♠♦❞❡❧♦ ✈✐s❝♦❡❧ást✐❝♦ ❛ ❢♦rç❛ é ♥❡❣❛t✐✈❛ ♣❛r❛ t♦❞♦s ♦s ❢❡✐①❡s ❛❝úst✐❝♦s ✭❖PP✱ ❋❇❖❩✱ ❋❇P❖✮✱ ✐st♦ é✱ ❛ ❢♦rç❛ ❛t✉❛ ♥❛ ❞✐r❡çã♦ ♦♣♦st❛ ❞♦ ❢❡✐①❡ ✐♥❝✐❞❡♥t❡✳ ➱ ✐♠♣♦rt❛♥t❡ r❡ss❛❧t❛r q✉❡ ❢♦rç❛ ❞❡ r❛❞✐❛çã♦ ❛❝úst✐❝❛ ♥❡❣❛t✐✈❛ ❞❡✈✐❞♦ ❛ ✉♠❛ ♦♥❞❛ ♣❧❛♥❛ ❢♦✐ ♣r❡✈✐st♦ ❛♥t❡r✐♦r♠❡♥t❡ ❡♠ ✉♠❛ ♣❛rtí❝✉❧❛ ❞❡ ♦✉r♦ r❡✈❡st✐❞❛ ❝♦♠ ✉♠❛ ❝❛♠❛❞❛ ❞❡ ✉♠ ♣♦❧í♠❡r♦ ✭✈✐s❝♦❡❧ást✐❝♦✮ ❆té ♦ ♠♦♠❡♥t♦ ❋♦rç❛ ❞❡ r❛❞✐❛çã♦ ♥❡❣❛t✐✈❛ ❞❡✈✐❞♦ ❛♦ ✉♠ ❢❡✐①❡ ❞❡ ❇❡ss❡❧ só ❢♦✐ ♣r❡✈✐st♦ ♣❛r❛ ❡s❢❡r❛s ✢✉✐❞❛s ♥ã♦ ❛❜s♦r✈❡❞♦r❛s ♥♦ r❡❣✐♠❡ ❞❡ ❡s♣❛❧❤❛♠❡♥t♦ ▼✐❡✱ ka ∼ 1

                    ◆❛ ❋✐❣✳ ✭❜✮✱ ❝♦♠♣❛r❛♠♦s ❛ ❢♦rç❛ ❞❡ r❛❞✐❛çã♦ ❡①❡r❝✐❞❛ s♦❜r❡ ✉♠❛ ♣❛rtí❝✉❧❛ ✈✐s❝♦❡✲ ❧ást✐❝❛ ❡ ✉♠❛ só❧✐❞❛ ❡❧ást✐❝❛✳ ◆♦t❡ q✉❡ ✐♥❞❡♣❡♥❞❡♥t❡ ❞♦ ♠❛t❡r✐❛❧✱ ❡❧ást✐❝♦ ♦✉ ✈✐s❝♦❡❧ást✐❝♦✱ ❛ ❢♦rç❛ ❞❡ r❛❞✐❛çã♦ ❛❝úst✐❝❛ é s❡♠♣r❡ ♣♦s✐t✐✈❛ ♥❡ss❡ r❡❣✐♠❡✳ ❆❧é♠ ❞✐ss♦✱ ❛ ❢♦rç❛ ❞❡ r❛✲ ❞✐❛çã♦ é ❜❛st❛♥t❡ ❞✐❢❡r❡♥t❡ ❞❡♣❡♥❞❡♥❞♦ ❞❛ ❞❡s❝r✐çã♦ ❞♦ ♠❛t❡r✐❛❧ ❞❛ ♣❛rtí❝✉❧❛ ❞❡ P❊❇❉ ✭❡❧ást✐❝♦ ♦✉ ✈✐s❝♦❡❧ást✐❝♦✮✳ ❖ ♠♦❞❡❧♦ ❝♦♥s✐❞❡r❛♥❞♦ ❛ ♣❛rtí❝✉❧❛ ❞❡ P❊❇❉ ❝♦♠♦ ❡❧ást✐❝❛ = τ = 0 ♣♦❞❡ s❡r ♦❜t✐❞♦ ❞❛ ❊q✳ ❊ss❡ ♠♦❞❡❧♦ ♣r❡✈ê q✉❡ ❛ ❛♠♣❧✐t✉❞❡ ❞❛ ❢♦rç❛ ❞❡ r❛❞✐❛çã♦ é ❝❡r❝❛ ❞❡ 30 ✈❡③❡s ♠❡♥♦r q✉❡ ♦ ♠♦❞❡❧♦ ✈✐s❝♦✲ ❡❧ást✐❝♦ ♣❛r❛ ✉♠❛ ♦♥❞❛ ♣❧❛♥❛ ♣r♦❣r❡ss✐✈❛ ❡ ✉♠ ❢❡✐①❡ ❞❡ ❇❡ss❡❧ ❞❡ ♦r❞❡♠ ③❡r♦✳ ❊♥tr❡t❛♥t♦✱ ♣❛r❛ ♦ ❢❡✐①❡ ❞❡ ❇❡ss❡❧ ❞❡ ♣r✐♠❡✐r❛ ♦r❞❡♠✱ ❛ ❞✐❢❡r❡♥ç❛ ❡♥tr❡ ❡st❡s ♠♦❞❡❧♦s é ❞❡ ❝❡r❝❛ ❞❡

                    ❚♦rq✉❡ ❞❡ r❛❞✐❛çã♦ ❛❝úst✐❝♦ ✈❡rs✉s ❢r❡q✉ê♥❝✐❛ ❡①❡r❝✐❞♦ ♣♦r ✉♠ ❢❡✐①❡ ❞❡ ❇❡ss❡❧

                    ❋✐❣✉r❛ ✺✳✽✿

                    

                  ❞❡ ♣r✐♠❡✐r❛ ♦r❞❡♠ s♦❜r❡ ✭❛✮ ✉♠❛ ♣❛rtí❝✉❧❛ ❞❡ P❊❆❉ ❡ ✭❜✮ ✉♠❛ ♣❛rtí❝✉❧❛ ❞❡ P❊❇❉ ❞❡ r❛✐♦

                  a = 1.0, 0.01 mm ✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✱ ✐♠❡rs♦ ❡♠ á❣✉❛✳ ❖s ♣❛râ♠❡tr♦s ❢ís✐❝♦s ❞❛s ♣❛rtí❝✉❧❛s sã♦

                  2 = 30 kW/m

                  ❞❛❞♦s ♥❛ ❚❛❜❡❧❛ ❖ ❢❡✐①❡ ✐♥❝✐❞❡♥t❡ t❡♠ s❡✉ ♣✐❝♦ ❞❡ ✐♥t❡♥s✐❞❛❞❡ ❞❡ I ✳ ▲❡❣❡♥❞❛✿

                  ❧✐♥❤❛ só❧✐❞❛ ❛③✉❧ é ♦ t♦rq✉❡ ❞❡ r❛❞✐❛çã♦ ❛✈❛❧✐❛❞♦ ♥❛ ❛♣r♦①✐♠❛çã♦ ♠♦♥♦♣♦❧♦✲❞✐♣♦❧♦ ❞❛❞♦ ♥❛

                  ❊q✳ ❖s ✐♥s❡rt❡s ♠♦str❛♠ ♦ ❡rr♦ r❡❧❛t✐✈♦ ❡♥tr❡ ❛s ❢ór♠✉❧❛s ❞❡ t♦rq✉❡✳

                  ❆ ❧✐♥❤❛ ✈❡rt✐❝❛❧ ♣♦♥t✐❧❤❛❞❛ ❞❡❧✐♠✐t❛ ❛ r❡❣✐ã♦ ❞❛ ❛♣r♦①✐♠❛çã♦ ❘❛②❧❡✐❣❤ ε < 0.3✳ PEAD

                    (a) ] m ão [nN.m aç

                    4.77 di ra rro

                    3.18 de E

                    1.59 orque

                    Frequência [MHz] PEBD (b)

                    3.98 ]

                    3.18 E rro

                    2.38

                    1.59

                    0.79 ão [pN. m aç di ra de orque T

                    Frequência [MHz] ❋♦♥t❡✿ ❆✉t♦r✱ ✷✵✶✺✳

                    ❞✉❛s ✈❡③❡s✳ ❖ t♦rq✉❡ ❞❡ r❛❞✐❛çã♦ ❛①✐❛❧ ❣❡r❛❞♦ ♣♦r ✉♠ ❢❡✐①❡ ❞❡ ❇❡ss❡❧ ❞❡ ♣r✐♠❡✐r❛ ♦r❞❡♠ s♦❜r❡ ✉♠❛

                    ♣❛rtí❝✉❧❛ ❞❡ P❊❆❉ ✭só❧✐❞♦ ✈✐s❝♦❡❧ást✐❝♦✮ s✉s♣❡♥s❛s ❡♠ á❣✉❛ s♦❜r❡ ♦ ❡✐①♦ ❞♦ ❢❡✐①❡ é ❝♦♠✲

                    ❋♦rç❛ ❞❡ r❛❞✐❛çã♦ ❛❝úst✐❝❛ ♣r♦❞✉③✐❞❛ ♣♦r ✉♠❛ ♦♥❞❛ ♣❧❛♥❛ ♣r♦❣r❡ss✐✈❛✱ ❢❡✐①❡ ❞❡

                    ❋✐❣✉r❛ ✺✳✾✿

                    

                  ❇❡ss❡❧ ❞❡ ♦r❞❡♠ ③❡r♦ ❡ ❢❡✐①❡ ❞❡ ❇❡ss❡❧ ❞❡ ♣r✐♠❡✐r❛ ♦r❞❡♠ ❡①❡r❝✐❞❛ s♦❜r❡ ✉♠❛ ♣❛rtí❝✉❧❛ ❞❡

                  ✭❛✮ P❊❆❉ ❡ ✭❜✮ P❊❇❉ ❞❡ r❛✐♦s a = 1.0, 0.01 mm✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✳ ❖s ❢❡✐①❡s ✐♥❝✐❞❡♥t❡s t❡♠

                  2 = 30 kW/m

                  ❢r❡q✉ê♥❝✐❛ ❞❡ ✭❛✮ 2 MHz ❡ ✭❜✮ 50 kHz✱ ♣✐❝♦ ❞❡ ✐♥t❡♥s✐❞❛❞❡ I ✱ ❡ â♥❣✉❧♦ ❞❡ ♠❡✐♦ ❝♦♥❡

                  β = 45 ✳ ❖s ✈❛❧♦r❡s ❞❛s ❛♠♣❧✐t✉❞❡s ❞❛s ❢♦rç❛s sã♦ ♠♦str❛❞♦s ♣ró①✐♠♦s ♦✉ ❞❡♥tr♦ ❞❡ s✉❛s ❜❛rr❛s

                  ❝♦rr❡s♣♦♥❞❡♥t❡s✳ (a) Fluida absorvente Elástica di ão [nN] 50 9.6 24.76 6.75 17.5 0.017 Viscoelástica 0.04 -0.158 a de ra Forç -100 -50 -111 -78.48 0.8 OPP FBOZ FBPO (b) Elástica ão [pN] di ra 0.6 0.4 0.61 0.43 Viscoelástica Forç a de 0.0 0.2 0.02 0.015 0.0001 0.00017 OPP FBOZ FBPO

                    ❋♦♥t❡✿ ❆✉t♦r✱ ✷✵✶✺✳

                    ♣❛r❛❞♦ ❝♦♠ ♦ t♦rq✉❡ ❛①✐❛❧ ❝♦♥s✐❞❡r❛♥❞♦ ❛ ♣❛rtí❝✉❧❛ ❞❡ P❊❆❉ ❝♦♠♦ ✢✉✐❞❛ ❛❜s♦r✈❡❞♦r❛✳ ❖ ♠♦❞❡❧♦ ✈✐s❝♦❡❧ást✐❝♦ ♣r❡✈ê ✉♠ t♦rq✉❡ ❞❡ r❛❞✐❛çã♦ ♥❡❣❛t✐✈♦ ✭✐st♦ é✱ ♦ t♦rq✉❡ ❞❡ r❛❞✐❛çã♦ ❛❝úst✐❝♦ ❡♥❝♦♥tr❛✲s❡ ♥♦ s❡♥t✐❞♦ ♦♣♦st♦ ❞♦ ✢✉①♦ ❞❡ ♠♦♠❡♥t♦ ❛♥❣✉❧❛r ♠é❞✐♦ ❞♦ ❢❡✐①❡ ✐♥❝✐✲ ❞❡♥t❡✮ ❝♦♠ ❛♠♣❧✐t✉❞❡ −1.10 nN · mm✱ ❡♥q✉❛♥t♦ q✉❡ ♦ ♠♦❞❡❧♦ ❞❡ ✉♠❛ ♣❛rtí❝✉❧❛ ✢✉✐❞❛ ❞❡ ❛❜s♦rçã♦ ❧♦♥❣✐t✉❞✐♥❛❧ ❞á ✉♠ t♦rq✉❡ ❞❡ r❛❞✐❛çã♦ ♣♦s✐t✐✈♦ ❞❡ 0.24 nN · mm✳ ◆♦✈❛♠❡♥t❡ ❛ ❝♦♠♣❛r❛çã♦ ❝♦♠ ✉♠❛ ♣❛rtí❝✉❧❛ ✢✉✐❞❛ ❝♦♠♣r❡ssí✈❡❧ ♣❛r❛ ♦ r❡❣✐♠❡ ❞❡ ❛❧t❛ ❢r❡q✉ê♥❝✐❛ ♥ã♦ é ♣♦ssí✈❡❧ ♣❡❧♦ ♠❡s♠♦ ♠♦t✐✈♦ ❛♣r❡s❡♥t❛❞♦ ❛♥t❡r✐♦r♠❡♥t❡ ♣❛r❛ ❛ ❝♦♠♣❛r❛çã♦ ❞❛ ❢♦rç❛ ❞❡ r❛❞✐❛çã♦✳

                    ❖s r❡s✉❧t❛❞♦s ♦❜t✐❞♦s ♥❡ss❡ ❝❛♣ít✉❧♦ ♠♦str❛♠ q✉❡ ❛ ❢♦rç❛ ❞❡ r❛❞✐❛çã♦ ❛❝úst✐❝❛ ❛①✐❛❧ é ♥❡❣❛t✐✈❛ ❡♠ ✉♠❛ ♦♥❞❛ ♣❧❛♥❛ ♣r♦❣r❡ss✐✈❛ ❡ ✉♠ ❢❡✐①❡ ❞❡ ❇❡ss❡❧✱ ❡ss❡ ❢❡♥ô♠❡♥♦ ♦❝♦rr❡ q✉❛♥❞♦ ❛ ❛❜s♦rçã♦ ❞❡ ❝✐s❛❧❤❛♠❡♥t♦ s✉♣❡r❛ ❛ ❛t❡♥✉❛çã♦ ❧♦♥❣✐t✉❞✐♥❛❧ ❝♦♥❢♦r♠❡ ❞❡s❝r✐t♦ ♥❛ ❊q✳ ❆ ❝♦♠♣❛r❛çã♦ ❞❛ ❢♦rç❛ ❞❡ r❛❞✐❛çã♦ s♦❜r❡ ✉♠❛ ♣❛rtí❝✉❧❛ ✈✐s❝♦❡❧ást✐❝❛ ❡ só❧✐❞❛ ❡❧ást✐❝❛ r❡✈❡❧❛♠ ✉♠ ❞❡s✈✐♦ s✐❣♥✐✜❝❛t✐✈♦ ❞♦s ❡st✉❞♦s ❛♥t❡r✐♦r❡s✳ ❆❧é♠ ❞✐ss♦✱ ♠♦str❛♠♦s q✉❡ ✉♠ ❢❡✐①❡ tr❛t♦r ✸❉ ❝♦♠♣❧❡t♦ ❛t✉❛ s♦❜r❡ ✉♠❛ ♣❛rtí❝✉❧❛ P❊❆❉ ❣❡r❛❞♦ ♣❡❧♦ ❢❡✐①❡ ❞❡ ❇❡ss❡❧ ❞❡ ♣r✐♠❡✐r❛ ♦r❞❡♠✳ ◆❛ ❛♥á❧✐s❡ ❞❛ ❢♦rç❛ ❞❡ r❛❞✐❛çã♦ ❛❝úst✐❝❛ ❣❡r❛❞♦ s♦❜r❡ ✉♠❛ ♣❛rtí❝✉❧❛ ✈✐s❝♦❡❧ást✐❝❛ ♣♦r ✉♠❛ ♦♥❞❛ ♣❧❛♥❛ ❡st❛❝✐♦♥ár✐❛✱ ❞❡s✈✐♦s r❡❧❡✈❛♥t❡s s✉r❣✐r❛♠ ❡♠ ❝♦♠♣❛r❛çã♦ ❝♦♠ ♣❛rtí❝✉❧❛ só❧✐❞❛ ❡❧ást✐❝❛✳ ❖❜s❡r✈❛♠♦s ✉♠❛ ❞✐❢❡r❡♥ç❛ s✐❣♥✐✜❝❛t✐✈❛ s♦❜r❡ ❛ ♠❛❣♥✐t✉❞❡ ❞♦ ♣✐❝♦ ❞❛ ❢♦rç❛ ❞❡ r❛❞✐❛çã♦ ❛❝úst✐❝❛ ❝♦♥s✐❞❡r❛♥❞♦ ✉♠❛ ♣❛rtí❝✉❧❛ ❞❡ P❊❇❉✳ ❚♦rq✉❡ ❞❡ r❛❞✐❛çã♦ ❛❝úst✐❝♦ ♥❡❣❛t✐✈♦ ❞❡✈✐❞♦ ❛ ✉♠ ❢❡✐①❡ ❞❡ ❇❡ss❡❧ ❞❡ ♣r✐♠❡✐r❛ ♦r❞❡♠ é ❛♣r❡s❡♥t❛❞♦ ♣❡❧❛ ♣r✐♠❡✐r❛ ✈❡③ ❛ ❝♦♠✉♥✐❞❛❞❡ ❝✐❡♥t✐✜❝❛✳ ❊ss❡s r❡s✉❧t❛❞♦s ❛♣r❡s❡♥t❛❞♦s ♥❡ss❡ ❝❛♣ít✉❧♦ ♣♦❞❡♠ s❡r út❡✐s ♥♦ ❡st✉❞♦ ❞❡ ♠❛♥✐♣✉❧❛çã♦ ❡ s❡♣❛r❛çã♦ ❞❡ ♣❛rtí❝✉❧❛s✳

                    6 ❈♦♥❝❧✉sõ❡s ❡ tr❛❜❛❧❤♦s ❢✉t✉r♦s ✻✳✶ ❈♦♥❝❧✉sõ❡s

                    ❊st❡ tr❛❜❛❧❤♦ t❡✈❡ ❝♦♠♦ ♣r♦♣ós✐t♦ ✐♥✈❡st✐❣❛r ♦s ❢❡♥ô♠❡♥♦s ❞❡ ❢♦rç❛ ❡ t♦rq✉❡ ❞❡ r❛❞✐✲ ❛çã♦ ❛❝úst✐❝♦ s♦❜r❡ ✉♠❛ ♣❛rtí❝✉❧❛ só❧✐❞❛ ✈✐s❝♦❡❧ást✐❝❛✳ ❉❡st❛ ❢♦r♠❛✱ ❢♦✐ ♣♦ssí✈❡❧ ❞❡r✐✈❛r ❡①♣r❡ssõ❡s ❛♥❛❧ít✐❝❛s ♣❛r❛ ❛ ❢♦rç❛ ❡ t♦rq✉❡ ❞❡ r❛❞✐❛çã♦ ❛❝úst✐❝♦ ❡①❡r❝✐❞♦ ♣♦r ✉♠ ❢❡✐①❡ ❝♦♠ ❢r❡♥t❡ ❞❡ ♦♥❞❛ ❛r❜✐trár✐❛ s♦❜r❡ ✉♠❛ ♣❡q✉❡♥❛ ♣❛rtí❝✉❧❛✱ ♥♦ ❝❤❛♠❛❞♦ ❧✐♠✐t❡ ❞❡ ❡s♣❛❧❤❛✲ ♠❡♥t♦ ❘❛②❧❡✐❣❤✱ s✉s♣❡♥s♦ ♥✉♠ ✢✉✐❞♦ ✐❞❡❛❧✳

                    ❆ ❡q✉❛çã♦ ❞❛ ♦♥❞❛ ❧✐♥❡❛r✱ ❝♦♥❤❡❝✐❞❛ ❝♦♠♦ ❡q✉❛çã♦ ❞❡ ❍❡❧♠❤♦❧t③ ❢♦✐ ❞❡r✐✈❛❞❛ ❛tr❛✈és ❞❛s ❡q✉❛çõ❡s ❞❡ ❝♦♥s❡r✈❛çã♦ ♣❛r❛ ✉♠ ✢✉✐❞♦ ✐❞❡❛❧✳ ❆ s♦❧✉çã♦ ❞❡st❛ ❡q✉❛çã♦ ❢♦✐ ❛♣r❡s❡♥t❛❞❛ ❡♠ t❡r♠♦s ❞♦ ♠ét♦❞♦ ❞❡ ♦♥❞❛s ♣❛r❝✐❛✐s ❝♦♠ ♦ ✐♥t✉✐t♦ ❞❡ ♦❜t❡r ♦s ❝❛♠♣♦s ❛❝úst✐❝♦s q✉❡ sã♦ ♥❡❝❡ssár✐♦s ♣❛r❛ r❡s♦❧✈❡r ♦ ♣r♦❜❧❡♠❛ ❞❡ ❡s♣❛❧❤❛♠❡♥t♦ ❧✐♥❡❛r ♣❛r❛ ✉♠❛ ♣❛rtí❝✉❧❛ ✈✐s❝♦❡❧ást✐❝❛ ❡s❢ér✐❝❛✳ ❆❧é♠ ❞✐ss♦✱ ❛ ♣r♦♣❛❣❛çã♦ ❞❡ ♦♥❞❛s ♥♦ ✐♥t❡r✐♦r ❞❛ ♣❛rtí❝✉❧❛ ❢♦✐ ♠♦❞❡❧❛❞♦ ❛tr❛✈és ❞❡ ✉♠❛ ❧❡✐ ❞❡ ❍♦♦❦❡ ❣❡♥❡r❛❧✐③❛❞❛ ✭♠♦❞❡❧♦ ❞❡ ❑❡❧✈✐♥✲❱♦✐❣t ❢r❛❝✐♦♥ár✐♦✮ ❝♦♠ ♦s t❡r♠♦s ❞❡ ♣❡r❞❛s ♣r♦♣♦r❝✐♦♥❛✐s ❛ ❞❡r✐✈❛❞❛ t❡♠♣♦r❛❧ ❢r❛❝✐♦♥ár✐❛✱ ✈❡❥❛ ❛ ❊q✳ ❊ss❡s ❡st✉❞♦s ❞❡ ♣r♦♣❛❣❛çã♦ ❞❡ ♦♥❞❛s ❡♠ ✉♠ ❞❡t❡r♠✐♥❛❞♦ ♠❡✐♦ ✭✢✉✐❞♦ ♦✉ só❧✐❞♦✮ ❢♦✐ ❞❡ s✉♠❛ ✐♠♣♦rtâ♥❝✐❛ ♣❛r❛ ❛ ❝♦♠♣r❡❡♥sã♦ ❞♦ ♣r♦❜❧❡♠❛ ❞❡ ❡s♣❛❧❤❛♠❡♥t♦ ❛❝úst✐❝♦✳ ➱ ✐♠♣♦rt❛♥t❡ r❡ss❛❧t❛r q✉❡ ♦ ❡s♣❛❧❤❛♠❡♥t♦ ❡stá ❞✐r❡t❛♠❡♥t❡ r❡❧❛❝✐♦♥❛❞♦ ❝♦♠ ♦ ❢❡♥ô♠❡♥♦ ❞❡ ❢♦rç❛ ❡ t♦rq✉❡ ❞❡ r❛❞✐❛çã♦ ❛❝úst✐❝♦✳

                    ❊①♣r❡ssõ❡s ❞❡ ❢♦rç❛ ❡ t♦rq✉❡ ❞❡ r❛❞✐❛çã♦ ❛❝úst✐❝♦ ❢♦r❛♠ ❞❡r✐✈❛❞❛s ♥❛ ❛♣r♦①✐♠❛çã♦ ❞❡ ❝❛♠♣♦ ❞✐st❛♥t❡✳ ❊ss❛s ❢ór♠✉❧❛s sã♦ ♠❛✐s s✐♠♣❧❡s ❡ ❞❡♣❡♥❞❡♠ ❞❛s ♣r♦♣r✐❡❞❛❞❡s ❢ís✐❝❛s n ❞♦ ♦❜❥❡t♦✱ ❞❛❞❛s ♣❡❧♦s ❝♦❡✜❝✐❡♥t❡s ❞❡ ❡s♣❛❧❤❛♠❡♥t♦ ❡s❝❛❧❛r❡s s ✱ ❡ ❞❛s ❝❛r❛❝t❡ríst✐❝❛s ❞❛ nm ♦♥❞❛ ✐♥❝✐❞❡♥t❡ ❞❡♥♦t❛❞❛s ♣❡❧♦s ❝♦❡✜❝✐❡♥t❡s ❞❡ ❢♦r♠❛ ❞♦ ❢❡✐①❡ a ✳ ◆♦ r❡❣✐♠❡ ❞❡ ❡s♣❛✲ ❧❤❛♠❡♥t♦ ❘❛②❧❡✐❣❤✱ ❢♦rç❛ ❡ t♦rq✉❡ ❞❡ r❛❞✐❛çã♦ ❛❝úst✐❝♦ ❞❡♣❡♥❞❡ ❛♣❡♥❛s ❞♦s ❝♦❡✜❝✐❡♥t❡s ❞❡ ❡s♣❛❧❤❛♠❡♥t♦ ❞❡ ♠♦♥♦♣♦❧♦ ❡ ❞✐♣♦❧♦✳ ❉❡♣♦✐s ❞❡ r❡❛❧✐③❛r ❡①♣❛♥sõ❡s ❛ss✐♥tót✐❝❛s ❞❡ss❡s ν ν j j ) ) ❝♦❡✜❝✐❡♥t❡s ♥♦s ❧✐♠✐t❡s ❞❡ ❜❛✐①❛ ❢r❡q✉ê♥❝✐❛ (ωτ ≪ 1 ❡ ❛❧t❛ ❢r❡q✉ê♥❝✐❛ (ωτ ≫ 1✱ ❡♥❝♦♥tr❛♠♦s q✉❡ ❛ ❢♦rç❛ ❞❡ r❛❞✐❛çã♦ ❛❝úst✐❝❛ ♣r♦✈♦❝❛❞❛ ♣♦r ✉♠❛ ♦♥❞❛ ♣❧❛♥❛ ♣r♦❣r❡ss✐✈❛ ❡ ❡st❛❝✐♦♥ár✐❛✱ ✉♠ ❢❡✐①❡ ❞❡ ❇❡ss❡❧ ❞❡ ♦r❞❡♠ ③❡r♦ ❡ ❞❡ ♣r✐♠❡✐r❛ ♦r❞❡♠✱ é ❡♠ ❣r❛♥❞❡ ♣❛rt❡ ❞❡✈✐❞♦ ❛♦s ❡❢❡✐t♦s ❞❡ ❛❜s♦rçã♦ ❞❡♥tr♦ ❞❛ ♣❛rtí❝✉❧❛✳ ❖ t♦rq✉❡ ❞❡ r❛❞✐❛çã♦ ❛❝úst✐❝♦ ❣❡r❛❞♦ ♣❡❧♦ ❢❡✐①❡ ❞❡ ❇❡ss❡❧ ❞❡ ♣r✐♠❡✐r❛ ♦r❞❡♠ t❛♠❜é♠ ❢♦✐ ❝❛❧❝✉❧❛❞♦✳

                    ❊♠ ♣❛rt✐❝✉❧❛r✱ ❛♣❧✐❝♦✉✲s❡ ❛ t❡♦r✐❛ ❞❡s❡♥✈♦❧✈✐❞❛ ♥♦ ❈❛♣ít✉❧♦ s♦❜r❡ ✉♠❛ ♣❡q✉❡♥❛ ♣❛r✲ tí❝✉❧❛ ❢❡✐t❛ ❞❡ ✉♠ ♠❛t❡r✐❛❧ ✈✐s❝♦❡❧ást✐❝♦✱ ♣♦❧✐❡t✐❧❡♥♦ ❞❡ ❜❛✐①❛ ❞❡♥s✐❞❛❞❡ ✭P❊❇❉✮ ❡ ♣♦❧✐❡t✐✲ ❧❡♥♦ ❞❡ ❛❧t❛ ❞❡♥s✐❞❛❞❡ ✭P❊❆❉✮✱ s✉s♣❡♥s❛ ❡♠ á❣✉❛ ❞❡✈✐❞♦ ❛ ✉♠❛ ♦♥❞❛ ♣❧❛♥❛ ♣r♦❣r❡ss✐✈❛✱ ♦♥❞❛ ♣❧❛♥❛ ❡st❛❝✐♦♥ár✐❛ ❡ ❢❡✐①❡s ❞❡ ❇❡ss❡❧✳ ❖s r❡s✉❧t❛❞♦s ♦❜t✐❞♦s ♠♦str❛♠ ✉♠ ❞❡s✈✐♦ s✐❣✲ ♥✐✜❝❛t✐✈♦ ❞♦s ❡st✉❞♦s ❛♥t❡r✐♦r❡s ♣❛r❛ ♣❛rtí❝✉❧❛ só❧✐❞❛ ❡❧ást✐❝❛✳ ❆❧é♠ ❞✐ss♦✱ ❞❡s❝♦❜r✐♠♦s q✉❡ ❛ ❢♦rç❛ ❞❡ r❛❞✐❛çã♦ ❛❝úst✐❝❛ ❛①✐❛❧ é ♥❡❣❛t✐✈❛ ❞❡✈✐❞♦ ❛ ✉♠❛ ♦♥❞❛ ♣❧❛♥❛ ♣r♦❣r❡ss✐✈❛ ❡ ✉♠ ❢❡✐①❡ ❞❡ ❇❡ss❡❧✱ ❡ss❡ ❢❡♥ô♠❡♥♦ ♦❝♦rr❡ q✉❛♥❞♦ ❛ ❛❜s♦rçã♦ ❞❡ ❝✐s❛❧❤❛♠❡♥t♦ s✉♣❡r❛ ❛ ❛t❡♥✉❛çã♦ ❧♦♥❣✐t✉❞✐♥❛❧ ❝♦♥❢♦r♠❡ ❞❡s❝r✐t♦ ♥❛ ❊q✳ ❚♦rq✉❡ ❞❡ r❛❞✐❛çã♦ ❛❝úst✐❝♦ ♥❡❣❛✲ t✐✈♦ ❞❡✈✐❞♦ ❛ ✉♠ ❢❡✐①❡ ❞❡ ❇❡ss❡❧ ❞❡ ♣r✐♠❡✐r❛ ♦r❞❡♠ t❛♠❜é♠ ♦❝♦rr❡ q✉❛♥❞♦ ❡st❛ ❝♦♥❞✐çã♦ é s❛t✐s❢❡✐t❛✳ ❆ ❡st❛❜✐❧✐❞❛❞❡ tr❛♥s✈❡rs❛❧ ❞❛ ❢♦rç❛ ❞❡ r❛❞✐❛çã♦ ❛❝úst✐❝❛ ❣❡r❛❞❛ ♣❡❧♦ ❢❡✐①❡ ❞❡ ❇❡ss❡❧ ❞❡ ♣r✐♠❡✐r❛ ♦r❞❡♠ t❛♠❜é♠ ❢♦✐ ✐♥✈❡st✐❣❛❞❛✳ ▼♦str❛♠♦s q✉❡ ✉♠ ❢❡✐①❡ tr❛t♦r ✸❉ ❝♦♠♣❧❡t♦ ❛t✉❛ s♦❜r❡ ✉♠❛ ♣❛rtí❝✉❧❛ P❊❆❉✳ ◆❛ ❛♥á❧✐s❡ ❞❛ ❢♦rç❛ ❞❡ r❛❞✐❛çã♦ ❛❝úst✐❝❛ ❣❡r❛❞♦ s♦❜r❡ ✉♠❛ ♣❛rtí❝✉❧❛ ✈✐s❝♦❡❧ást✐❝❛ ♣♦r ✉♠❛ ♦♥❞❛ ♣❧❛♥❛ ❡st❛❝✐♦♥ár✐❛✱ ❞❡s✈✐♦s r❡❧❡✲ ✈❛♥t❡s s✉r❣✐r❛♠ ❡♠ ❝♦♠♣❛r❛çã♦ ❝♦♠ ♣❛rtí❝✉❧❛ só❧✐❞❛ ❡❧ást✐❝❛✳ ❖❜s❡r✈❛♠♦s ✉♠❛ ❞✐❢❡r❡♥ç❛ s✐❣♥✐✜❝❛t✐✈❛ s♦❜r❡ ❛ ♠❛❣♥✐t✉❞❡ ❞♦ ♣✐❝♦ ❞❛ ❢♦rç❛ ❞❡ r❛❞✐❛çã♦ ❛❝úst✐❝❛ ❝♦♥s✐❞❡r❛♥❞♦ ✉♠❛ ♣❛rtí❝✉❧❛ ❞❡ P❊❇❉✳ ■ss♦ ❛❝♦♥t❡❝❡ ❞❡✈✐❞♦ ❛ ♣❛rtí❝✉❧❛ ✈✐s❝♦❡❧ást✐❝❛ s❡r ✉♠ ♠❡✐♦ ❞✐s♣❡rs✐✈♦✳

                    P♦r ✜♠✱ ❛ ❢♦rç❛ ❡ t♦rq✉❡ ❞❡ r❛❞✐❛çã♦ ❛❝úst✐❝♦ ❡①❡r❝✐❞♦ s♦❜r❡ ✉♠❛ ♣❡q✉❡♥❛ ♣❛rtí❝✉❧❛ ✈✐s❝♦❡❧ást✐❝❛ ❢♦r❛♠ t❡♦r✐❝❛♠❡♥t❡ ✐♥✈❡st✐❣❛❞♦s✳ ❆s♣❡❝t♦s ✐♥✉s✐t❛❞♦s ❞❡ss❡s ❢❡♥ô♠❡♥♦s t❛✐s ❝♦♠♦ ❢♦rç❛s ❡ t♦rq✉❡s ❞❡ r❛❞✐❛çã♦ ❛❝úst✐❝♦ ❛①✐❛✐s ♥❡❣❛t✐✈♦s ❢♦r❛♠ ♣r❡✈✐st♦s ❡ ❞✐s❝✉t✐❞♦s✳ ➱ ✐♠♣♦rt❛♥t❡ s❛❧✐❡♥t❛r q✉❡ ♦s r❡s✉❧t❛❞♦s ❛q✉✐ ❛♣r❡s❡♥t❛❞♦s ♣♦❞❡♠ s❡r r❡❧❡✈❛♥t❡s ♣❛r❛ ♦ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❞❡ ♥♦✈♦s ❞✐s♣♦s✐t✐✈♦s q✉❡ ✉t✐❧✐③❛♠ té❝♥✐❝❛s ❞❡ ♠❛♥✐♣✉❧❛çã♦ ❞❡ ♣❛rtí❝✉❧❛s ❞❡✈✐❞♦ ❛ ❝❛♠♣♦s ❛❝úst✐❝♦s✳

                    ✻✳✷ ❚r❛❜❛❧❤♦s ❢✉t✉r♦s

                    ❆ ♣❛rt✐r ❞❡st❡ tr❛❜❛❧❤♦✱ é ♣♦ssí✈❡❧ ❛♣r❡s❡♥t❛r ✉♠ ♠ét♦❞♦ ❞❡ ❛♣r✐s✐♦♥❛♠❡♥t♦ ❞❡ ♣❛rtí✲ ❝✉❧❛s ✈✐s❝♦❡❧ást✐❝❛s ♣♦r ✉♠ tr❛♥s❞✉t♦r ❢♦❝❛❧✐③❛❞♦✳ ❊ss❡ ♠ét♦❞♦ ♣♦❞❡ s❡r ✈❛❧✐❞❛❞♦ ❛tr❛✈és ❞❡ s✐♠✉❧❛çõ❡s ❝♦♠♣✉t❛❝✐♦♥❛✐s✳ ❆❧é♠ ❞✐ss♦✱ ♣♦❞❡♠♦s ❢❛③❡r ♦ ❡st✉❞♦ ❞❡ ❞✐♥â♠✐❝❛ ❞❡ ♣❛r✲ tí❝✉❧❛s ♣❛r❛ ♠❡❧❤♦r ❛♥❛❧✐s❛r ♦♥❞❡ ♦❝♦rr❡ ♦ ❛♣r✐s✐♦♥❛♠❡♥t♦ ❞❛s ♣❛rtí❝✉❧❛s✳ ❊ss❡ t✐♣♦ ❞❡ ✐♥✈❡st✐❣❛çã♦ ♣♦❞❡ s❡r ❢✉♥❞❛♠❡♥t❛❧ ♣❛r❛ ❞❡s❡♥✈♦❧✈❡r ❡str❛té❣✐❛s ❡✜❝✐❡♥t❡s ❞❡ ♠❛♥✐♣✉❧❛çã♦ ❞❡ ♣❛rtí❝✉❧❛s✳

                    ❈♦♠ r❡❧❛çã♦ ❛ ✐♥t❡r❛çã♦ ❞❡ ♦♥❞❛s ❛❝úst✐❝❛s ❝♦♠ ✉♠ ❝♦♥❥✉♥t♦ ❞❡ ♣❛rtí❝✉❧❛s✱ ❤á ❛ ♥❡❝❡ss✐❞❛❞❡ ❞❡ ✉♠ ❡st✉❞♦ ❞❡t❛❧❤❛❞♦ ❞❡ ❢♦rç❛ ❡ t♦rq✉❡ ❞❡ r❛❞✐❛çã♦ ❝♦♥s✐❞❡r❛♥❞♦ ♣❛rtí❝✉❧❛s ❢❡✐t❛s ❞❡ ✉♠ ♠❛t❡r✐❛❧ ✈✐s❝♦❡❧ást✐❝♦✳ ❊♠ ♣❛rt✐❝✉❧❛r✱ ♦s r❡s✉❧t❛❞♦s ❛ s❡r❡♠ ♦❜t✐❞♦s ♣♦❞❡♠ s❡r ❝r✉❝✐❛✐s ♣❛r❛ s❡♣❛r❛çã♦ ❞❡ ♣❛rtí❝✉❧❛s ✉t✐❧✐③❛♥❞♦ ✉♠ ❢❡✐①❡ ❢♦❝❛❧✐③❛❞♦✱ ✈✐st♦ q✉❡ ♦s r❡s✉❧t❛❞♦s ♦❜t✐❞♦s ♥❡st❡ tr❛❜❛❧❤♦ ♣❛r❛ ✉♠❛ ú♥✐❝❛ ♣❛rtí❝✉❧❛ ♠♦str❛♠ q✉❡ ❡❧❛ ♣♦❞❡ s❡r tr❛♥s❧❛❞❛ ♦✉ ❛tr❛í❞❛ ❞❡♣❡♥❞❡♥❞♦ ❞❛s ❝❛r❛❝t❡ríst✐❝❛s ❢ís✐❝❛s ❞❛ ♣❛rtí❝✉❧❛✳

                    ❘❡❢❡rê♥❝✐❛s

                    ❬✶❪ ❚❖❘❘✱ ●✳ ❘✳ ❚❤❡ ❛❝♦✉st✐❝ r❛❞✐❛t✐♦♥ ❢♦r❝❡✳ ❆♠❡r✐❝❛❧ ❏♦✉r♥❛❧ ♦❢ P❤②s✐❝s✱ ✈✳ ✺✷✱ ♣✳ ✹✵✷✕✹✵✽✱ ✶✾✽✹✳

                    ❬✷❪ ▲❊❊✱ ❈✳ P✳❀ ❲❆◆●✱ ❚✳ ●✳ ❆❝♦✉st✐❝ r❛❞✐❛t✐♦♥ ♣r❡ss✉r❡✳ ❏♦✉r♥❛❧ ♦❢ t❤❡ ❆❝♦✉st✐❝❛❧ ❙♦❝✐❡t② ♦❢ ❆♠❡r✐❝❛✱ ✈✳ ✾✹✱ ♣✳ ✶✵✾✾✕✶✶✵✾✱ ✶✾✾✸✳

                    ❬✸❪ ❲❯✱ ❏✳ ❘✳ ❆❝♦✉st✐❝❛❧ t✇❡❡③❡rs✳ ❏♦✉r♥❛❧ ♦❢ t❤❡ ❆❝♦✉st✐❝❛❧ ❙♦❝✐❡t② ♦❢ ❆♠❡r✐❝❛✱ ✈✳ ✽✾✱ ♣✳ ✷✶✹✵✕✷✶✹✸✱ ✶✾✾✶✳

                    ❬✹❪ ❚❊❘❯❨❯❑■ ❑❖❩❯❑❆✱ ❚❖❘❯ ❚❯❩■❯❚■✱ ❍✳ ▼✳❀ ❋❯❑❯❉❆✱ ❚✳ ❈♦♥tr♦❧ ♦❢ ❛ st❛♥✲ ❞✐♥❣ ✇❛✈❡ ✜❡❧❞ ✉s✐♥❣ ❛ ❧✐♥❡✲❢♦❝✉s❡❞ tr❛♥s❞✉❝❡r ❢♦r t✇♦✲❞✐♠❡♥s✐♦♥❛❧ ♠❛♥✐♣✉❧❛t✐♦♥ ♦❢ ♣❛rt✐❝❧❡s✳ ❏❛♣❛♥❡s❡ ❏♦✉r♥❛❧ ♦❢ ❆♣♣❧✐❡❞ P❤②s✐❝s✱ ✈✳ ✸✼✱ ♣✳ ✷✾✼✹✕✷✾✼✽✱ ✶✾✾✽✳

                    ❬✺❪ ❚✳ ▲❆❯❘❊▲▲✱ ❋✳ P✳❀ ◆■▲❙❙❖◆✱ ❆✳ ❈❤✐♣ ✐♥t❡❣r❛t❡❞ str❛t❡❣✐❡s ❢♦r ❛❝♦✉st✐❝ s❡♣❛r❛t✐♦♥ ❛♥❞ ♠❛♥✐♣✉❧❛t✐♦♥ ♦❢ ❝❡❧❧s ❛♥❞ ♣❛rt✐❝❧❡s✳ ❈❤❡♠✐❝❛❧ ❙♦❝✐❡t② ❘❡✈✐❡✇s✱ ✈✳ ✸✻✱ ♣✳ ✹✾✷✕✺✵✻✱ ✷✵✵✼✳

                    ❬✻❪ ❏❯◆●❲❖❖ ▲❊❊✱ ❙❍■❆✲❨❊◆ ❚❊❍✱ ❆❇❘❆❍❆▼ ▲❊❊✱ ❍❨❯◆●✱ ❍✳ ❑❀ ▲❊❊✱ ❈❀ ❙❍❯◆●✱ ❑✳ ❑✳ ❙✐♥❣❧❡ ❜❡❛♠ ❛❝♦✉st✐❝ tr❛♣♣✐♥❣✳ ❆♣♣❧✐❡❞ P❤②s✐❝s ▲❡tt❡rs✱ ❲♦♦❞❜✉r②✱ ✈✳ ✾✺✱ ♣✳ ✵✼✸✼✵✶✱ ✷✵✵✾✳

                    ❬✼❪ ❙❍■✱ ❏✳❀ ❆❍▼❊❉✱ ❉✳❀ ▼❆❖✱ ❳✳❀ ▲■◆✱ ❙✳❀ ▲❆❲■❚✱ ❆✳❀ ❍❯❆◆●✱ ❚✳ ❆❝♦✉st✐❝ t✇❡❡③❡rs✿ ♣❛tt❡r♥✐♥❣ ❝❡❧❧s ❛♥❞ ♠✐❝r♦♣❛rt✐❝❧❡s ✉s✐♥❣ st❛♥❞✐♥❣ s✉r❢❛❝❡ ❛❝♦✉st✐❝ ✇❛✈❡s ✭❙❙❆❲✮✳ ▲❛❜ ♦♥ ❛ ❈❤✐♣✱ ✈✳ ✾✱ ♣✳ ✷✽✾✵✕✷✽✾✺✱ ✷✵✵✾✳

                    ❬✽❪ ▲❊◆❙❍❖❋✱ ❆✳❀ ▲❆❯❘❊▲▲✱ ❚✳ ❈♦♥t✐♥✉♦✉s s❡♣❛r❛t✐♦♥ ♦❢ ❝❡❧❧s ❛♥❞ ♣❛rt✐❝❧❡s ✐♥ ♠✐✲ ❝r♦✢✉✐❞✐❝ s②st❡♠s✳ ❈❤❡♠✐❝❛❧ ❙♦❝✐❡t② ❘❡✈✐❡✇s✱ ✈✳ ✸✾✱ ♥✳ ✸✱ ♣✳ ✶✷✵✸✕✶✷✶✼✱ ✷✵✶✵✳

                    ❬✾❪ ❋❘■❊◆❉✱ ❏✳❀ ❨❊❖✱ ▲✳ ❨✳ ▼✐❝r♦s❝❛❧❡ ❛❝♦✉st♦✢✉✐❞✐❝s✿ ▼✐❝r♦✢✉✐❞✐❝s ❞r✐✈❡♥ ✈✐❛ ❛❝♦✉s✲ t✐❝s ❛♥❞ ✉❧tr❛s♦♥✐❝s✳ ❘❡✈✐❡✇ ▼♦❞❡r♥ P❤②s✐❝s✱ ✈✳ ✽✸✱ ♣✳ ✻✹✼✕✼✵✹✱ ✷✵✶✶✳ ❬✶✵❪ ❲❆◆●✱ ❩✳❀ ❩❍❊✱ ❏✳ ❘❡❝❡♥t ❛❞✈❛♥❝❡s ✐♥ ♣❛rt✐❝❧❡ ❛♥❞ ❞r♦♣❧❡t ♠❛♥✐♣✉❧❛t✐♦♥ ❢♦r

                    ❧❛❜✲♦♥✲❛✲❝❤✐♣ ❞❡✈✐❝❡s ❜❛s❡❞ ♦♥ s✉r❢❛❝❡ ❛❝♦✉st✐❝ ✇❛✈❡s✳ ▲❛❜ ❈❤✐♣✱ ✈✳ ✶✶✱ ♥✳ ✼✱ ♣✳ ✶✷✽✵✕✶✷✽✺✱ ✷✵✶✶✳

                    ❬✶✶❪ ▲❊◆❙❍❖❋✱ ❆✳❀ ▼❆●◆❯❙❙❖◆✱ ❈✳❀ ▲❆❯❘❊▲▲✱ ❚✳ ❆❝♦✉st♦✢✉✐❞✐❝s ✽✿ ❆♣♣❧✐❝❛t✐♦♥s ♦❢ ❛❝♦✉st♦♣❤♦r❡s✐s ✐♥ ❝♦♥t✐♥✉♦✉s ✢♦✇ ♠✐❝r♦s②st❡♠s✳ ▲❛❜ ♦♥ ❛ ❈❤✐♣✱ ✈✳ ✶✷✱ ♣✳ ✶✷✶✵✕ ✶✷✷✸✱ ✷✵✶✷✳

                    ❬✶✷❪ ❉■◆●✱ ❳✳❀ ▲■◆✱ ❙✳✲❈✳ ❙✳❀ ❑■❘❆▲❨✱ ❇✳❀ ❨❯❊✱ ❍✳❀ ▲■✱ ❙✳❀ ❈❍■❆◆●✱ ■✳✲❑✳❀ ❙❍■✱ ❏✳❀ ❇❊◆❑❖❱■❈✱ ❙✳ ❏✳❀ ❍❯❆◆●✱ ❚✳ ❏✳ ❖♥✲❝❤✐♣ ♠❛♥✐♣✉❧❛t✐♦♥ ♦❢ s✐♥❣❧❡ ♠✐❝r♦♣❛rt✐✲ ❝❧❡s✱ ❝❡❧❧s✱ ❛♥❞ ♦r❣❛♥✐s♠s ✉s✐♥❣ s✉r❢❛❝❡ ❛❝♦✉st✐❝ ✇❛✈❡s✳ Pr♦❝❡❡❞✐♥❣s ♦❢ t❤❡ ◆❛t✐♦♥❛❧ ❆❝❛❞❡♠② ♦❢ ❙❝✐❡♥❝❡s ♦❢ t❤❡ ❯♥✐t❡❞ ❙t❛t❡s ♦❢ ❆♠❡r✐❝❛✱ ❲❛s❤✐♥❣t♦♥✱ ✈✳ ✶✵✾✱ ♣✳ ✶✶✶✵✺✕ ✶✶✶✵✾✱ ✷✵✶✷✳

                    ❬✶✸❪ ❈❍❖❊✱ ❨✳❀ ❑■▼✱ ❏✳ ❲✳❀ ❙❍❯◆●✱ ❑✳ ❑✳❀ ❑■▼✱ ❊✳ ❙✳ ▼✐❝r♦♣❛rt✐❝❧❡ tr❛♣♣✐♥❣ ✐♥ ❛♥ ✉❧tr❛s♦♥✐❝ ❇❡ss❡❧ ❜❡❛♠✳ ❆♣♣❧✐❡❞ P❤②s✐❝s ▲❡tt❡rs✱ ❲♦♦❞❜✉r②✱ ✈✳ ✾✾✱ ♣✳ ✷✸✸✼✵✹✱ ✷✵✶✶✳

                    ❬✶✹❪ ●▲❨◆◆❊✲❏❖◆❊❙✱ P✳❀ ❇❖▲❚❘❨❑✱ ❘✳ ❏✳❀ ❍■▲▲✱ ▼✳ ❆❝♦✉st♦✢✉✐❞✐❝s ✾✿ ▼♦❞❡❧❧✐♥❣ ❛♥❞ ❛♣♣❧✐❝❛t✐♦♥s ♦❢ ♣❧❛♥❛r r❡s♦♥❛♥t ❞❡✈✐❝❡s ❢♦r ❛❝♦✉st✐❝ ♣❛rt✐❝❧❡ ♠❛♥✐♣✉❧❛t✐♦♥✳ ▲❛❜ ♦♥ ❛ ❈❤✐♣✱ ✈✳ ✶✷✱ ♥✳ ✽✱ ♣✳ ✶✹✶✼✕✶✹✷✻✱ ✷✵✶✷✳

                    ❬✶✺❪ ❈❖❯❘❚◆❊❨✱ ❈✳ ❘✳ P✳❀ ❉❊▼❖❘❊✱ ❈✳ ❊✳ ▼✳❀ ❲❯✱ ❍✳❀ ●❘■◆❊◆❑❖✱ ❆✳❀ ❲■▲❈❖❳✱ P✳ ❉✳❀ ❈❖❈❍❘❆◆✱ ❙✳❀ ❉❘■◆❑❲❆❚❊❘✱ ❇✳ ❲✳ ■♥❞❡♣❡♥❞❡♥t tr❛♣♣✐♥❣ ❛♥❞ ♠❛♥✐♣✉✲ ❧❛t✐♦♥ ♦❢ ♠✐❝r♦♣❛rt✐❝❧❡s ✉s✐♥❣ ❞❡①t❡r♦✉s ❛❝♦✉st✐❝ t✇❡❡③❡rs✳ ❆♣♣❧✐❡❞ P❤②s✐❝s ▲❡tt❡rs✱ ❲♦♦❞❜✉r②✱ ✈✳ ✶✵✹✱ ♣✳ ✶✺✹✶✵✸✱ ✷✵✶✹✳

                    ❬✶✻❪ ❚❘■◆❍✱ ❊✳ ❍✳ ❈♦♠♣❛❝t ❛❝♦✉st✐❝ ❧❡✈✐t❛t✐♦♥ ❞❡✈✐❝❡ ❢♦r st✉❞✐❡s ✐♥ ✢✉✐❞ ❞②♥❛♠✐❝s ❛♥❞ ♠❛t❡r✐❛❧ s❝✐❡♥❝❡ ✐♥ t❤❡ ❧❛❜♦r❛t♦r② ❛♥❞ ♠✐❝r♦❣r❛✈✐t②✳ ❘❡✈✐❡✇ ♦❢ ❙❝✐❡♥t✐✜❝ ■♥str✉✲ ♠❡♥ts✱ ❲♦♦❞❜✉r②✱ ✈✳ ✺✻✱ ♣✳ ✷✵✺✾✕✷✵✻✺✱ ✶✾✽✺✳

                    ❬✶✼❪ ❇❘❆◆❉❚✱ ❊✳ ❍✳ ❆❝♦✉st✐❝ ♣❤②s✐❝s✿ ❙✉s♣❡♥❞❡❞ ❜② s♦✉♥❞✳ ◆❛t✉r❡✱ ▲♦♥❞♦♥✱ ✈✳ ✹✶✸✱ ♣✳ ✹✼✹✕✹✼✺✱ ✷✵✵✶✳

                    ❬✶✽❪ ❋❖❘❊❙❚■✱ ❉✳❀ ◆❆❇❆❱■✱ ▼✳❀ ❑▲■◆●❆❯❋✱ ▼✳❀ P❖❯▲■❑❆❑❖❙✱ ❆✳ ❋✳ ❉✳ ❆❝♦✉s✲ t♦♣❤♦r❡t✐❝ ❝♦♥t❛❝t❧❡ss tr❛♥s♣♦rt ❛♥❞ ❤❛♥❞❧✐♥❣ ♦❢ ♠❛tt❡r ✐♥ ❛✐r✳ Pr♦❝❡❡❞✐♥❣s ♦❢ t❤❡ ◆❛t✐♦♥❛❧ ❆❝❛❞❡♠② ♦❢ ❙❝✐❡♥❝❡s ♦❢ t❤❡ ❯♥✐t❡❞ ❙t❛t❡s ♦❢ ❆♠❡r✐❝❛✱ ❲❛s❤✐♥❣t♦♥✱ ✈✳ ✶✶✵✱ ♣✳ ✶✷✺✹✾✕✶✷✺✺✹✱ ✷✵✶✸✳

                    ❬✶✾❪ ❆◆❉❘❆❉❊✱ ▼✳ ❆✳ ❇✳❀ P❊❘❊❩✱ ◆✳❀ ❆❉❆▼❖❲❙❑■✱ ❏✳ ❈✳ P❛rt✐❝❧❡ ♠❛♥✐♣✉❧❛t✐♦♥ ❜② ❛ ♥♦♥✲r❡s♦♥❛♥t ❛❝♦✉st✐❝ ❧❡✈✐t❛t♦r✳ ❆♣♣❧✐❡❞ P❤②s✐❝s ▲❡tt❡rs✱ ❲♦♦❞❜✉r②✱ ✈✳ ✶✵✻✱ ♣✳ ✵✶✹✶✵✶✱ ✷✵✶✺✳

                    ❬✷✵❪ ❱❖▲❑❊✲❙❊P❯▲❱❊❉❆✱ ❑✳❀ ❙❆◆❚■▲▲❆◆✱ ❆✳ ❖✳❀ ❇❖❯▲▲❖❙❆✱ ❘✳ ❘✳ ❚r❛♥s❢❡r ♦❢ ❛♥❣✉❧❛r ♠♦♠❡♥t✉♠ t♦ ♠❛tt❡r ❢r♦♠ ❛❝♦✉st✐❝❛❧ ✈♦rt✐❝❡s ✐♥ ❢r❡❡ ✜❡❧❞✳ P❤②s✐❝❛❧ ❘❡✈✐❡✇ ▲❡tt❡rs✱ ❲♦♦❞❜✉r②✱ ✈✳ ✶✵✵✱ ♣✳ ✵✷✹✸✵✷✱ ✷✵✵✽✳

                    ❬✷✶❪ ❆◆❍❆❯❙❊❘✱ ❆✳❀ ❲❯◆❊◆❇❯❘●❊❘✱ ❘✳❀ ❇❘❆❙❙❊▲❊❚✱ ❊✳ ❆❝♦✉st✐❝ r♦t❛t✐♦♥❛❧ ♠❛♥✐♣✉❧❛t✐♦♥ ✉s✐♥❣ ♦r❜✐t❛❧ ❛♥❣✉❧❛r ♠♦♠❡♥t✉♠ tr❛♥s❢❡r✳ P❤②s✐❝❛❧ ❘❡✈✐❡✇ ▲❡tt❡rs✱ ❲♦♦❞❜✉r②✱ ✈✳ ✶✵✾✱ ♣✳ ✵✸✹✸✵✶✱ ✷✵✶✷✳

                    ❬✷✷❪ ❙❈❍❲❆❘❩✱ ❚✳❀ ❍❆❍◆✱ P✳❀ P❊❚■❚✲P■❊❘❘❊✱ ●✳❀ ❉❯❆▲✱ ❏✳ ❘♦t❛t✐♦♥ ♦❢ ✜❜❡rs ❛♥❞ ♦t❤❡r ♥♦♥✲s♣❤❡r✐❝❛❧ ♣❛rt✐❝❧❡s ❜② t❤❡ ❛❝♦✉st✐❝ r❛❞✐❛t✐♦♥ t♦rq✉❡✳ ▼✐❝r♦✢✉✐❞✐❝s ❛♥❞ ◆❛♥♦✢✉✐❞✐❝s✱ ✈✳ ✶✽✱ ♣✳ ✻✺✕✼✾✱ ✷✵✶✺✳

                    ❬✷✸❪ ❑■◆●✱ ▲✳ ❱✳ ❖♥ t❤❡ ❛❝♦✉st✐❝ r❛❞✐❛t✐♦♥ ♣r❡ss✉r❡ ♦♥ s♣❤❡r❡s✳ Pr♦❝❡❡❞✐♥❣s ♦❢ t❤❡ ❘♦②❛❧ ❙♦❝✐❡t② ✭▲♦♥❞♦♥✮✱ ✈✳ ✶✹✼✱ ♥✳ ✽✻✶✱ ♣✳ ✷✶✷✕✷✹✵✱ ✶✾✸✹✳

                    ❬✷✹❪ ❊▼❇▲❊❚❖◆✱ ❚✳ ❋✳ ❲✳ ▼❡❛♥ ❢♦r❝❡ ♦♥ ❛ s♣❤❡r❡ ✐♥ ❛ s♣❤❡r✐❝❛❧ s♦✉♥❞ ✜❡❧❞✳ ■✳ ✭❚❤❡✲ ♦r❡t✐❝❛❧✮✳ ❏♦✉r♥❛❧ ♦❢ t❤❡ ❆❝♦✉st✐❝❛❧ ❙♦❝✐❡t② ♦❢ ❆♠❡r✐❝❛✱ ✈✳ ✷✻✱ ♣✳ ✹✵✕✹✺✱ ✶✾✺✹✳

                    ❬✷✺❪ ❨❖❙■❖❑❆✱ ❑✳❀ ❑❆❲❆❙■▼❆✱ ❨✳ ❆❝♦✉st✐❝ r❛❞✐❛t✐♦♥ ♣r❡ss✉r❡ ♦♥ ❛ ❝♦♠♣r❡ss✐❜❧❡ s♣❤❡r❡✳ ❆❝✉st✐❝❛✱ ✈✳ ✺✱ ♣✳ ✶✻✼✕✶✼✸✱ ✶✾✺✺✳ ❬✷✻❪ ❲❊❙❚❊❘❱❊▲❚✱ P✳ ❏✳ ❆❝♦✉st✐❝ r❛❞✐❛t✐♦♥ ♣r❡ss✉r❡✳ ❏♦✉r♥❛❧ ♦❢ t❤❡ ❆❝♦✉st✐❝❛❧ ❙♦❝✐❡t②

                    ♦❢ ❆♠❡r✐❝❛✱ ✈✳ ✷✾✱ ♣✳ ✷✻✕✷✾✱ ✶✾✺✼✳ ❬✷✼❪ ●❖❘❑❖❱✱ ▲✳ P✳ ❖♥ t❤❡ ❋♦r❝❡s ❆❝t✐♥❣ ♦♥ ❛ ❙♠❛❧❧ P❛rt✐❝❧❡ ✐♥ ❛♥ ❆❝♦✉st✐❝❛❧ ❋✐❡❧❞

                    ✐♥ ❛♥ ■❞❡❛❧ ❋❧✉✐❞✳ ❙♦✈✐❡t P❤②s✐❝s ✕ ❉♦❦❧❛❞②✱ ✈✳ ✻✱ ♥✳ ✾✱ ♣✳ ✼✼✸✕✼✼✺✱ ✶✾✻✷✳ ❬✷✽❪ ◆❨❇❖❘●✱ ❲✳ ▲✳ ❘❛❞✐❛t✐♦♥ ♣r❡ss✉r❡ ♦♥ ❛ s♠❛❧❧ r✐❣✐❞ s♣❤❡r❡✳ ❏♦✉r♥❛❧ ♦❢ t❤❡ ❆❝♦✉s✲ t✐❝❛❧ ❙♦❝✐❡t② ♦❢ ❆♠❡r✐❝❛✱ ✈✳ ✹✷✱ ♣✳ ✾✹✼✕✾✺✷✱ ✶✾✻✼✳ ❬✷✾❪ ❍❆❙❊●❆❲❆✱ ❚✳❀ ❨❖❙■❖❑❆✱ ❑✳ ❆❝♦✉st✐❝ r❛❞✐❛t✐♦♥ ❢♦r❝❡ ♦♥ ❛ s♦❧✐❞ ❡❧❛st✐❝ s♣❤❡r❡✳

                    ❏♦✉r♥❛❧ ♦❢ t❤❡ ❆❝♦✉st✐❝❛❧ ❙♦❝✐❡t② ♦❢ ❆♠❡r✐❝❛✱ ✈✳ ✹✻✱ ♣✳ ✶✶✸✾✕✶✶✹✸✱ ✶✾✻✾✳ ❬✸✵❪ ❲❯✱ ❏✳❀ ❉❯✱ ●✳ ❆❝♦✉st✐❝ r❛❞✐❛t✐♦♥ ❢♦r❝❡ ♦♥ ❛ s♠❛❧❧ ❝♦♠♣r❡ss✐❜❧❡ s♣❤❡r❡ ✐♥ ❛ ❢♦❝✉s❡❞

                    ❜❡❛♠✳ ❏♦✉r♥❛❧ ♦❢ t❤❡ ❆❝♦✉st✐❝❛❧ ❙♦❝✐❡t② ♦❢ ❆♠❡r✐❝❛✱ ✈✳ ✽✼✱ ♣✳ ✾✾✼✕✶✵✵✸✱ ✶✾✾✵✳ ❬✸✶❪ ▼❆❘❙❚❖◆✱ P✳ ▲✳ ❆①✐❛❧ r❛❞✐❛t✐♦♥ ❢♦r❝❡ ♦❢ ❛ ❇❡ss❡❧ ❜❡❛♠ ♦♥ ❛ s♣❤❡r❡ ❛♥❞ ❞✐r❡❝t✐♦♥ r❡✈❡rs❛❧ ♦❢ t❤❡ ❢♦r❝❡✳ ❏♦✉r♥❛❧ ♦❢ t❤❡ ❆❝♦✉st✐❝❛❧ ❙♦❝✐❡t② ♦❢ ❆♠❡r✐❝❛✱ ✈✳ ✶✷✵✱ ♣✳

                    ✸✺✶✽✕✸✺✷✹✱ ✷✵✵✻✳ ❬✸✷❪ ▼❆❘❙❚❖◆✱ P✳ ▲✳ ◆❡❣❛t✐✈❡ ❛①✐❛❧ r❛❞✐❛t✐♦♥ ❢♦r❝❡s ♦♥ s♦❧✐❞ s♣❤❡r❡s ❛♥❞ s❤❡❧❧s ✐♥ ❛

                    ❜❡ss❡❧ ❜❡❛♠ ✭▲✮✳ ❏♦✉r♥❛❧ ♦❢ t❤❡ ❆❝♦✉st✐❝❛❧ ❙♦❝✐❡t② ♦❢ ❆♠❡r✐❝❛✱ ✈✳ ✶✷✷✱ ♣✳ ✸✶✻✷✕✸✶✻✺✱ ✷✵✵✼✳

                    ❬✸✸❪ ▼❆❘❙❚❖◆✱ P✳ ▲✳ ❘❛❞✐❛t✐♦♥ ❢♦r❝❡ ♦❢ ❛ ❤❡❧✐❝♦✐❞❛❧ ❇❡ss❡❧ ❜❡❛♠ ♦♥ ❛ s♣❤❡r❡✳ ❏♦✉r♥❛❧ ♦❢ t❤❡ ❆❝♦✉st✐❝❛❧ ❙♦❝✐❡t② ♦❢ ❆♠❡r✐❝❛✱ ✈✳ ✶✷✺✱ ♥✳ ✻✱ ♣✳ ✸✺✸✾✕✸✺✹✼✱ ✷✵✵✾✳

                    ❬✸✹❪ ▼■❚❘■✱ ❋✳ ●✳ ◆❡❣❛t✐✈❡ ❛①✐❛❧ r❛❞✐❛t✐♦♥ ❢♦r❝❡ ♦♥ ❛ ✢✉✐❞ ❛♥❞ ❡❧❛st✐❝ s♣❤❡r❡s ✐❧❧✉♠✐♥❛t❡❞ ❜② ❛ ❤✐❣❤✲♦r❞❡r ❇❡ss❡❧ ❜❡❛♠ ♦❢ ♣r♦❣r❡ss✐✈❡ ✇❛✈❡s✳ ❏♦✉r♥❛❧ ♦❢ P❤②s✐❝s ❆✱ ❇r✐st♦❧✱ ✈✳ ✹✷✱ ♣✳ ✷✹✺✷✵✷✱ ✷✵✵✾✳

                    ❬✸✺❪ ▼■❚❘■✱ ❋✳ ●✳ ❙✐♥❣❧❡ ❇❡ss❡❧ tr❛❝t♦r✲❜❡❛♠ t✇❡❡③❡rs✳ ❲❛✈❡ ▼♦t✐♦♥✱ ✈✳ ✺✶✱ ♥✳ ✻✱ ♣✳ ✾✽✻✕✾✾✸✱ ✷✵✶✹✳

                    ❬✸✻❪ ▼■❚❘■✱ ❋✳ ●✳ ❆❝♦✉st✐❝ r❛❞✐❛t✐♦♥ ❢♦r❝❡ ♦♥ ❛ s♣❤❡r❡ ✐♥ st❛♥❞✐♥❣ ❛♥❞ q✉❛s✐✲st❛♥❞✐♥❣ ③❡r♦✲♦r❞❡r ❇❡ss❡❧ ❜❡❛♠ t✇❡❡③❡rs✳ ❆♥♥❛❧s ♦❢ P❤②s✐❝s✱ ◆❡✇ ❨♦r❦✱ ✈✳ ✸✷✸✱ ♥✳ ✼✱ ♣✳ ✶✻✵✹✕✶✻✷✵✱ ✷✵✵✽✳

                    ❬✸✼❪ ▼■❚❘■✱ ❋✳ ●✳ ▲❛♥❣❡✈✐♥ ❛❝♦✉st✐❝ r❛❞✐❛t✐♦♥ ❢♦r❝❡ ♦❢ ❛ ❤✐❣❤✲♦r❞❡r ❜❡ss❡❧ ❜❡❛♠ ♦♥ ❛ r✐❣✐❞ s♣❤❡r❡✳ ■❊❊❊ ❚r❛♥s❛❝t✐♦♥s ♦♥ ❯❧tr❛s♦♥✐❝s ❋❡rr♦❡❧❡❝tr✐❝s ❛♥❞ ❋r❡q✉❡♥❝② ❈♦♥tr♦❧✱ ✈✳ ✺✻✱ ♣✳ ✶✵✺✾✕✶✵✻✹✱ ✷✵✵✾✳

                    ❬✸✽❪ ▼■❚❘■✱ ❋✳ ●✳ ❆❝♦✉st✐❝ r❛❞✐❛t✐♦♥ ❢♦r❝❡ ♦❢ ❤✐❣❤✲♦r❞❡r ❜❡ss❡❧ ❜❡❛♠ st❛♥❞✐♥❣ ✇❛✈❡ t✇❡❡③❡rs ♦♥ ❛ r✐❣✐❞ s♣❤❡r❡✳ ❯❧tr❛s♦♥✐❝s✱ ✈✳ ✹✾✱ ♥✳ ✽✱ ♣✳ ✼✾✹✕✼✾✽✱ ✷✵✵✾✳ ❬✸✾❪ ▼■❚❘■✱ ❋✳ ●✳ ❆❝♦✉st✐❝ r❛❞✐❛t✐♦♥ ❢♦r❝❡ ♦♥ ❛♥ ❛✐r ❜✉❜❜❧❡ ❛♥❞ s♦❢t ✢✉✐❞ s♣❤❡r❡s ✐♥

                    ✐❞❡❛❧ ❧✐q✉✐❞s✿ ❊①❛♠♣❧❡ ♦❢ ❛ ❤✐❣❤✲♦r❞❡r ❜❡ss❡❧ ❜❡❛♠ ♦❢ q✉❛s✐✲st❛♥❞✐♥❣ ✇❛✈❡s✳ ❚❤❡ ❊✉r♦♣❡❛♥ P❤②s✐❝❛❧ ❏♦✉r♥❛❧ ❊✿ ❙♦❢t ▼❛tt❡r ❛♥❞ ❇✐♦❧♦❣✐❝❛❧ P❤②s✐❝s✱ ✈✳ ✷✽✱ ♥✳ ✹✱ ♣✳ ✹✻✾✕✹✼✽✱ ✷✵✵✾✳

                    ❬✹✵❪ ▼■❚❘■✱ ❋✳ ●✳ ❆①✐❛❧ ❛♥❞ tr❛♥s✈❡rs❡ ❛❝♦✉st✐❝ r❛❞✐❛t✐♦♥ ❢♦r❝❡s ♦♥ ❛ ✢✉✐❞ s♣❤❡r❡ ♣❧❛❝❡❞ ❛r❜✐tr❛r✐❧② ✐♥ ❇❡ss❡❧ ❜❡❛♠ st❛♥❞✐♥❣ ✇❛✈❡ t✇❡❡③❡rs✳ ❆♥♥❛❧s ♦❢ P❤②s✐❝s✱ ◆❡✇ ❨♦r❦✱ ✈✳ ✸✹✷✱ ♣✳ ✶✺✽✕✶✼✵✱ ✷✵✶✹✳

                    ❬✹✶❪ ❙■▲❱❆✱ ●✳ ❚✳ ❆♥ ❡①♣r❡ss✐♦♥ ❢♦r t❤❡ r❛❞✐❛t✐♦♥ ❢♦r❝❡ ❡①❡rt❡❞ ❜② ❛♥ ❛❝♦✉st✐❝ ❜❡❛♠ ✇✐t❤ ❛r❜✐tr❛r② ✇❛✈❡❢r♦♥t ✭▲✮✳ ❏♦✉r♥❛❧ ♦❢ t❤❡ ❆❝♦✉st✐❝❛❧ ❙♦❝✐❡t② ♦❢ ❆♠❡r✐❝❛✱ ✈✳ ✶✸✵✱ ♥✳ ✻✱ ♣✳ ✸✺✹✶✕✸✺✹✹✱ ✷✵✶✶✳

                    ❬✹✷❪ ❩❍❆◆●✱ ▲✳❀ ▼❆❘❙❚❖◆✱ P✳ ▲✳ ●❡♦♠❡tr✐❝❛❧ ✐♥t❡r♣r❡t❛t✐♦♥ ♦❢ ♥❡❣❛t✐✈❡ r❛❞✐❛t✐♦♥ ❢♦r❝❡s ♦❢ ❛❝♦✉st✐❝❛❧ ❜❡ss❡❧ ❜❡❛♠s ♦♥ s♣❤❡r❡s✳ P❤②s✐❝❛❧ ❘❡✈✐❡✇ ❊✱ ✈✳ ✽✹✱ ♥✳ ✸✱ ♣✳ ✵✸✺✻✵✶✱ ✷✵✶✶✳

                    ❬✹✸❪ ❩❍❆◆●✱ ▲✳❀ ▼❆❘❙❚❖◆✱ P✳ ▲✳ ❆①✐❛❧ r❛❞✐❛t✐♦♥ ❢♦r❝❡ ❡①❡rt❡❞ ❜② ❣❡♥❡r❛❧ ♥♦♥✲ ❞✐✛r❛❝t✐♥❣ ❜❡❛♠s✳ ❏♦✉r♥❛❧ ♦❢ t❤❡ ❆❝♦✉st✐❝❛❧ ❙♦❝✐❡t② ♦❢ ❆♠❡r✐❝❛✱ ✈✳ ✶✸✶✱ ♥✳ ✹✱ ♣✳ ✸✷✾✕✸✸✺✱ ✷✵✶✷✳

                    ❬✹✹❪ ❩❍❆◆●✱ ▲✳❀ ▼❆❘❙❚❖◆✱ P✳ ▲✳ ❖♣t✐❝❛❧ t❤❡♦r❡♠ ❢♦r ❛❝♦✉st✐❝ ♥♦♥✲❞✐✛r❛❝t✐♥❣ ❜❡❛♠s ❛♥❞ ❛♣♣❧✐❝❛t✐♦♥ t♦ r❛❞✐❛t✐♦♥ ❢♦r❝❡ ❛♥❞ t♦rq✉❡✳ ❇✐♦♠❡❞✐❝❛❧ ❖♣t✐❝s ❊①♣r❡ss✱ ✈✳ ✹✱ ♥✳ ✾✱ ♣✳ ✶✻✶✵✕✶✻✶✼✱ ✷✵✶✸✳

                    ❬✹✺❪ ❆❩❆❘P❊❨❱❆◆❉✱ ▼✳ ❆❝♦✉st✐❝ r❛❞✐❛t✐♦♥ ❢♦r❝❡ ♦❢ ❛ ❜❡ss❡❧ ❜❡❛♠ ♦♥ ❛ ♣♦r♦✉s s♣❤❡r❡✳ ❏♦✉r♥❛❧ ♦❢ t❤❡ ❆❝♦✉st✐❝❛❧ ❙♦❝✐❡t② ♦❢ ❆♠❡r✐❝❛✱ ✈✳ ✶✸✶✱ ♥✳ ✻✱ ♣✳ ✹✸✸✼✕✹✸✹✽✱ ✷✵✶✷✳

                    ❬✹✻❪ ❆❩❆❘P❊❨❱❆◆❉✱ ▼✳❀ ❆❩❆❘P❊❨❱❆◆❉✱ ▼✳ ❆♣♣❧✐❝❛t✐♦♥ ♦❢ ❛❝♦✉st✐❝ ❜❡ss❡❧ ❜❡❛♠s ❢♦r ❤❛♥❞❧✐♥❣ ♦❢ ❤♦❧❧♦✇ ♣♦r♦✉s s♣❤❡r❡s✳ ❯❧tr❛s♦✉♥❞ ✐♥ ▼❡❞✐❝✐♥❡ ❛♥❞ ❇✐♦❧♦❣②✱ ✈✳ ✹✵✱ ♥✳ ✷✱ ♣✳ ✹✷✷✕✹✸✸✱ ✷✵✶✹✳

                    ❬✹✼❪ ❇❆❘❊❙❈❍✱ ❉✳❀ ❚❍❖▼❆❙✱ ❏✳✲▲✳❀ ▼❆❘❈❍■❆◆❖✱ ❘✳ ❚❤r❡❡✲❞✐♠❡♥s✐♦♥❛❧ ❛❝♦✉st✐❝ r❛❞✐❛t✐♦♥ ❢♦r❝❡ ♦♥ ❛♥ ❛r❜✐tr❛r✐❧② ❧♦❝❛t❡❞ ❡❧❛st✐❝ s♣❤❡r❡✳ ❏♦✉r♥❛❧ ♦❢ t❤❡ ❆❝♦✉st✐❝❛❧ ❙♦❝✐❡t② ♦❢ ❆♠❡r✐❝❛✱ ✈✳ ✶✸✸✱ ♥✳ ✶✱ ♣✳ ✷✺✕✸✻✱ ✷✵✶✸✳

                    ❬✹✽❪ ❇❆❘❊❙❈❍✱ ❉✳❀ ❚❍❖▼❆❙✱ ❏✳✲▲✳❀ ▼❆❘❈❍■❆◆❖✱ ❘✳ ❙♣❤❡r✐❝❛❧ ✈♦rt❡① ❜❡❛♠s ♦❢ ❤✐❣❤ r❛❞✐❛❧ ❞❡❣r❡❡ ❢♦r ❡♥❤❛♥❝❡❞ s✐♥❣❧❡✲❜❡❛♠ t✇❡❡③❡rs✳ ❏♦✉r♥❛❧ ♦❢ ❆♣♣❧✐❡❞ P❤②s✐❝s✱ ❲♦♦❞❜✉r②✱ ✈✳ ✶✶✸✱ ♣✳ ✶✽✹✾✵✶✱ ✷✵✶✸✳

                    ❬✹✾❪ ❙❆P❖❩❍◆■❑❖❱✱ ❖✳ ❆✳❀ ❇❆■▲❊❨✱ ▼✳ ❘✳ ❘❛❞✐❛t✐♦♥ ❢♦r❝❡ ♦❢ ❛♥ ❛r❜✐tr❛r② ❛❝♦✉st✐❝ ❜❡❛♠ ♦♥ ❛♥ ❡❧❛st✐❝ s♣❤❡r❡ ✐♥ ❛ ✢✉✐❞✳ ❏♦✉r♥❛❧ ♦❢ t❤❡ ❆❝♦✉st✐❝❛❧ ❙♦❝✐❡t② ♦❢ ❆♠❡r✐❝❛✱ ✈✳ ✶✸✸✱ ♥✳ ✷✱ ♣✳ ✻✻✶✕✻✼✻✱ ✷✵✶✸✳

                    ❬✺✵❪ ❙■▲❱❆✱ ●✳ ❚✳ ❆❝♦✉st✐❝ r❛❞✐❛t✐♦♥ ❢♦r❝❡ ❛♥❞ t♦rq✉❡ ♦♥ ❛♥ ❛❜s♦r❜✐♥❣ ❝♦♠♣r❡ss✐❜❧❡ ♣❛rt✐❝❧❡ ✐♥ ❛♥ ✐♥✈✐s❝✐❞ ✢✉✐❞✳ ❏♦✉r♥❛❧ ♦❢ t❤❡ ❆❝♦✉st✐❝❛❧ ❙♦❝✐❡t② ♦❢ ❆♠❡r✐❝❛✱ ✈✳ ✶✸✻✱ ♥✳ ✺✱ ♣✳ ✷✹✵✺✕✷✹✶✸✱ ✷✵✶✹✳

                    ❬✺✶❪ ❲❊■❙❊❘✱ ▼✳ ❆✳ ❍✳❀ ❆P❋❊▲✱ ❘✳ ❊✳❀ ◆❊PP■❘❆❙✱ ❊✳ ❆✳ ■♥t❡r♣❛rt✐❝❧❡ ❢♦r❝❡s ♦♥ r❡❞ ❝❡❧❧s ✐♥ ❛ st❛♥❞✐♥❣ ✇❛✈❡ ✜❡❧❞✳ ❆❝t❛ ❆❝✉st✐❝❛ ❯♥✐t❡❞ ✇✐t❤ ❆❝✉st✐❝❛✱ ✈✳ ✺✻✱ ♣✳ ✶✶✹✕✶✶✾✱ ✶✾✽✹✳

                    ❬✺✷❪ ❩❍❯❑✱ ❆✳ P✳ ❍②❞r♦❞②♥❛♠✐❝ ✐♥t❡r❛❝t✐♦♥ ♦❢ t✇♦ s♣❤❡r✐❝❛❧ ♣❛rt✐❝❧❡s ❞✉❡ t♦ s♦✉♥❞ ✇❛✈❡s ♣r♦♣❛❣❛t✐♥❣ ♣❡r♣❡♥❞✐❝✉❧❛r❧② t♦ t❤❡ ❝❡♥t❡r ❧✐♥❡✳ ❙♦✈✐❡t ❆♣♣❧✐❡❞ ▼❡❝❤❛♥✐❝s✱ ✈✳ ✷✶✱ ♣✳ ✶✶✵✕✶✶✻✱ ✶✾✽✺✳ ✐♥ ❊♥❣❧✐s❤✳

                    ❬✺✸❪ ❩❍❊◆●✱ ❳✳❀ ❆P❋❊▲✱ ❘✳ ❊✳ ❆❝♦✉st✐❝ ✐♥t❡r❛❝t✐♦♥ ❢♦r❝❡s ❜❡t✇❡❡♥ t✇♦ ✢✉✐❞ s♣❤❡r❡s ✐♥ ❛♥ ❛❝♦✉st✐❝ ✜❡❧❞✳ ❏♦✉r♥❛❧ ♦❢ t❤❡ ❆❝♦✉st✐❝❛❧ ❙♦❝✐❡t② ♦❢ ❆♠❡r✐❝❛✱ ✈✳ ✾✼✱ ♣✳ ✷✷✶✽✕✷✷✷✻✱ ✶✾✾✺✳

                    ❬✺✹❪ ❉❖■◆■❑❖❱✱ ❆✳ ❆✳ ❖♥ t❤❡ r❛❞✐❛t✐♦♥ ♣r❡ss✉r❡ ♦♥ s♠❛❧❧ s♣❤❡r❡s✳ ❏♦✉r♥❛❧ ♦❢ t❤❡ ❆❝♦✉st✐❝❛❧ ❙♦❝✐❡t② ♦❢ ❆♠❡r✐❝❛✱ ✈✳ ✶✵✵✱ ♣✳ ✶✷✸✶✕✶✷✸✸✱ ✶✾✾✻✳

                    ❬✺✺❪ ❆❩❆❘P❊❨❱❆◆❉✱ ▼✳❀ ❆▲■❇❆❑❍❙❍■✱ ▼✳ ❆✳❀ ❙❊▲❋✱ ❘✳ ❊✛❡❝ts ♦❢ ♠✉❧t✐✲s❝❛tt❡r✐♥❣ ♦♥ t❤❡ ♣❡r❢♦r♠❛♥❝❡ ♦❢ ❛ s✐♥❣❧❡✲❜❡❛♠ ❛❝♦✉st✐❝ ♠❛♥✐♣✉❧❛t✐♦♥ ❞❡✈✐❝❡✳ ■❊❊❊ ❚r❛♥s❛❝t✐✲ ♦♥s ♦♥ ❯❧tr❛s♦♥✐❝s✱ ❋❡rr♦❡❧❡❝tr✐❝s✱ ❛♥❞ ❋r❡q✉❡♥❝② ❈♦♥tr♦❧✱ ✈✳ ✺✾✱ ♥✳ ✽✱ ♣✳ ✶✼✹✶✕✶✼✹✾✱ ✷✵✶✷✳

                    ❬✺✻❪ ❙■▲❱❆✱ ●✳ ❚✳❀ ❇❘❯❯❙✱ ❍✳ ❆❝♦✉st✐❝ ✐♥t❡r❛❝t✐♦♥ ❢♦r❝❡s ❜❡t✇❡❡♥ s♠❛❧❧ ♣❛rt✐❝❧❡s ✐♥ ❛♥ ✐❞❡❛❧ ✢✉✐❞✳ P❤②s✐❝❛❧ ❘❡✈✐❡✇ ❊✱ ✈✳ ✾✵✱ ♥✳ ✻✱ ♣✳ ✵✻✸✵✵✼✱ ✷✵✶✹✳

                    ❬✺✼❪ ❙❊P❊❍❘■❘❆❍◆❆▼❆✱ ❙✳❀ ▲■▼✱ ❑✳✲▼✳❀ ❈❍❆❯✱ ❋✳ ❙✳ ◆✉♠❡r✐❝❛❧ st✉❞② ♦❢ ✐♥t❡r✲ ♣❛rt✐❝❧❡ r❛❞✐❛t✐♦♥ ❢♦r❝❡ ❛❝t✐♥❣ ♦♥ r✐❣✐❞ s♣❤❡r❡s ✐♥ ❛ st❛♥❞✐♥❣ ✇❛✈❡✳ ❏♦✉r♥❛❧ ♦❢ t❤❡ ❆❝♦✉st✐❝❛❧ ❙♦❝✐❡t② ♦❢ ❆♠❡r✐❝❛✱ ✈✳ ✶✸✼✱ ♣✳ ✷✻✶✹✕✷✻✷✷✱ ✷✵✶✺✳

                    ❬✺✽❪ ▼■❚❘■✱ ❋✳ ●✳ ❆❝♦✉st✐❝ r❛❞✐❛t✐♦♥ ❢♦r❝❡ ❛❝t✐♥❣ ♦♥ ❡❧❛st✐❝ ❛♥❞ ✈✐s❝♦❡❧❛st✐❝ s♣❤❡r✐❝❛❧ s❤❡❧❧s ♣❧❛❝❡❞ ✐♥ ❛ ♣❧❛♥❡ st❛♥❞✐♥❣ ✇❛✈❡ ✜❡❧❞✳ ❯❧tr❛s♦♥✐❝s✱ ✈✳ ✹✸✱ ♥✳ ✽✱ ♣✳ ✻✽✶✕✻✾✶✱ ✷✵✵✺✳

                    ❬✺✾❪ ▼■❚❘■✱ ❋✳ ●✳ ❆❝♦✉st✐❝ r❛❞✐❛t✐♦♥ ❢♦r❝❡ ❛❝t✐♥❣ ♦♥ ❛❜s♦r❜✐♥❣ s♣❤❡r✐❝❛❧ s❤❡❧❧s✳ ❲❛✈❡ ▼♦t✐♦♥✱ ✈✳ ✹✸✱ ♥✳ ✶✱ ♣✳ ✶✷✕✶✾✱ ✷✵✵✺✳

                    ❬✻✵❪ ▼■❚❘■✱ ❋✳ ●✳❀ ❋❊▲▲❆❍✱ ❩✳ ❊✳ ❆✳ ❚❤❡♦r② ♦❢ t❤❡ ❛❝♦✉st✐❝ r❛❞✐❛t✐♦♥ ❢♦r❝❡ ❡①❡rt❡❞ ♦♥ ❛ s♣❤❡r❡ ❜② st❛♥❞✐♥❣ ❛♥❞ q✉❛s✐st❛♥❞✐♥❣ ③❡r♦✲♦r❞❡r ❜❡ss❡❧ ❜❡❛♠ t✇❡❡③❡rs ♦❢ ✈❛r✐❛❜❧❡ ❤❛❧❢✲❝♦♥❡ ❛♥❣❧❡s✳ ■❊❊❊ ❚r❛♥s❛❝t✐♦♥s ♦♥ ❯❧tr❛s♦♥✐❝s✱ ❋❡rr♦❡❧❡❝tr✐❝s✱ ❛♥❞ ❋r❡q✉❡♥❝② ❈♦♥tr♦❧✱ ✈✳ ✺✺✱ ♥✳ ✶✶✱ ♣✳ ✷✹✻✾✕✷✹✼✽✱ ✷✵✵✽✳

                    ❬✻✶❪ ▼■❚❘■✱ ❋✳ ●✳❀ ❋❊▲▲❆❍✱ ❩✳ ❊✳ ❆✳ ❚❤❡ ♠❡❝❤❛♥✐s♠ ♦❢ t❤❡ ❛ttr❛❝t✐♥❣ ❛❝♦✉st✐❝ r❛❞✐❛✲ t✐♦♥ ❢♦r❝❡ ♦♥ ❛ ♣♦❧②♠❡r✲❝♦❛t❡❞ ❣♦❧❞ s♣❤❡r❡ ✐♥ ♣❧❛♥❡ ♣r♦❣r❡ss✐✈❡ ✇❛✈❡s✳ ❚❤❡ ❊✉r♦♣❡❛♥ P❤②s✐❝❛❧ ❏♦✉r♥❛❧ ❊✿ ❙♦❢t ▼❛tt❡r ❛♥❞ ❇✐♦❧♦❣✐❝❛❧ P❤②s✐❝s✱ ✈✳ ✷✻✱ ♥✳ ✹✱ ♣✳ ✸✸✼✕✸✹✸✱ ✷✵✵✽✳

                    ❬✻✷❪ ❘❆❨▲❊■●❍✱ ▲✳ ❖♥ t❤❡ ❡q✉✐❧✐❜r✐✉♠ ♦❢ ❧✐q✉✐❞ ❝♦♥❞✉❝t✐♥❣ ♠❛ss❡s ❝❤❛r❣❡❞ ✇✐t❤ ❡❧❡❝✲ tr✐❝✐t②✳ P❤✐❧♦s♦♣❤✐❝❛❧ ▼❛❣❛③✐♥❡✱ ✈✳ ✶✹✱ ♣✳ ✶✽✻✱ ✶✽✽✷✳ ❬✻✸❪ ▼❆■❉❆◆■❑✱ ●✳ ❚♦rq✉❡s ❞✉❡ t♦ ❛❝♦✉st✐❝❛❧ r❛❞✐❛t✐♦♥ ♣r❡ss✉r❡✳ ❏♦✉r♥❛❧ ♦❢ t❤❡ ❆❝♦✉s✲ t✐❝❛❧ ❙♦❝✐❡t② ♦❢ ❆♠❡r✐❝❛✱ ✈✳ ✸✵✱ ♥✳ ✼✱ ♣✳ ✻✷✵✕✻✷✸✱ ✶✾✺✽✳ ❬✻✹❪ ❙▼■❚❍✱ ❲✳ ❊✳ ❆❝♦✉st✐❝ r❛❞✐❛t✐♦♥ ♣r❡ss✉r❡ ❢♦r❝❡s ❛♥❞ t♦rq✉❡s ❢r♦♠ ❡❧❛st✐❝ s❝❛tt❡r✐♥❣✳

                    ❆✉str❛❧✐❛♥ ❏♦✉r♥❛❧ ♦❢ P❤②s✐❝s✱ ✈✳ ✷✺✱ ♥✳ ✸✱ ♣✳ ✷✼✺✕✷✽✷✱ ✶✾✼✷✳ ❬✻✺❪ ❍❊❋◆❊❘✱ ❇✳ ❚✳❀ ▼❆❘❙❚❖◆✱ P✳ ▲✳ ❆♥ ❛❝♦✉st✐❝❛❧ ❤❡❧✐❝♦✐❞❛❧ ✇❛✈❡ tr❛♥s❞✉❝❡r ✇✐t❤

                    ❛♣♣❧✐❝❛t✐♦♥s ❢♦r t❤❡ ❛❧✐❣♥♠❡♥t ♦❢ ✉❧tr❛s♦♥✐❝ ❛♥❞ ✉♥❞❡r✇❛t❡r s②st❡♠s✳ ❏♦✉r♥❛❧ ♦❢ t❤❡ ❆❝♦✉st✐❝❛❧ ❙♦❝✐❡t② ♦❢ ❆♠❡r✐❝❛✱ ✈✳ ✶✵✻✱ ♣✳ ✸✸✶✸✕✸✸✶✻✱ ✶✾✾✾✳

                    ❬✻✻❪ ❋❆◆✱ ❩✳❀ ▼❊■✱ ❉✳❀ ❨❆◆●✱ ❑✳❀ ❈❍❊◆✱ ❩✳ ❆❝♦✉st✐❝ r❛❞✐❛t✐♦♥ t♦rq✉❡ ♦♥ ❛♥ ✐rr❡❣✉❧❛r❧② s❤❛♣❡❞ s❝❛tt❡r❡r ✐♥ ❛♥ ❛r❜✐tr❛r② s♦✉♥❞ ✜❡❧❞✳ ❏♦✉r♥❛❧ ♦❢ t❤❡ ❆❝♦✉st✐❝❛❧ ❙♦❝✐❡t② ♦❢ ❆♠❡r✐❝❛✱ ✈✳ ✶✷✹✱ ♥✳ ✺✱ ♣✳ ✷✼✷✼✕✷✼✸✷✱ ✷✵✵✽✳

                    ❬✻✼❪ ❩❍❆◆●✱ ▲✳❀ ▼❆❘❙❚❖◆✱ P✳ ▲✳ ❆♥❣✉❧❛r ♠♦♠❡♥t✉♠ ✢✉① ♦❢ ♥♦♥♣❛r❛①✐❛❧ ❛❝♦✉st✐❝ ✈♦rt❡① ❜❡❛♠s ❛♥❞ t♦rq✉❡s ♦♥ ❛①✐s②♠♠❡tr✐❝ ♦❜❥❡❝ts✳ P❤②s✐❝❛❧ ❘❡✈✐❡✇ ❊✱ ✈✳ ✽✹✱ ♥✳ ✻✱ ♣✳ ✵✻✺✻✵✶✱ ✷✵✶✶✳

                    ❬✻✽❪ ❩❍❆◆●✱ ▲✳❀ ▼❆❘❙❚❖◆✱ P✳ ▲✳ ❆❝♦✉st✐❝ r❛❞✐❛t✐♦♥ t♦rq✉❡ ❛♥❞ t❤❡ ❝♦♥s❡r✈❛t✐♦♥ ♦❢ ❛♥❣✉❧❛r ♠♦♠❡♥t✉♠ ✭❧✮✳ ❚❤❡ ❏♦✉r♥❛❧ ♦❢ t❤❡ ❆❝♦✉st✐❝❛❧ ❙♦❝✐❡t② ♦❢ ❆♠❡r✐❝❛✱ ✈✳ ✶✷✾✱ ♥✳ ✹✱ ♣✳ ✶✻✼✾✕✶✻✽✵✱ ✷✵✶✶✳

                    ❬✻✾❪ ▼■❚❘■✱ ❋✳ ●✳❀ ▲❖❇❖✱ ❚✳ P✳❀ ❙■▲❱❆✱ ●✳ ❚✳ ❆①✐❛❧ ❛❝♦✉st✐❝ r❛❞✐❛t✐♦♥ t♦rq✉❡ ♦❢ ❛ ❇❡ss❡❧ ✈♦rt❡① ❜❡❛♠ ♦♥ s♣❤❡r✐❝❛❧ s❤❡❧❧s✳ P❤②s✐❝❛❧ ❘❡✈✐❡✇ ❊✱ ✈✳ ✽✺✱ ♥✳ ✷✱ ♣✳ ✵✷✻✻✵✷✱ ✷✵✶✷✳

                    ❬✼✵❪ ❋✳ ●✳ ▼■❚❘■✱ ❚✳ P✳ ▲✳❀ ❙■▲❱❆✱ ●✳ ❚✳ ❆①✐❛❧ ❛❝♦✉st✐❝ r❛❞✐❛t✐♦♥ t♦rq✉❡ ♦❢ ❛ ❜❡ss❡❧ ✈♦rt❡① ❜❡❛♠ ♦♥ s♣❤❡r✐❝❛❧ s❤❡❧❧s✳ P❤②s✐❝❛❧ ❘❡✈✐❡✇ ❊✱ ✈✳ ✽✺✱ ♣✳ ✵✷✻✻✵✷✱ ✷✵✶✷✳

                    ❬✼✶❪ P✳ ▼✳ ●❆▼▼❊▲✱ ❆✳ ❈✳❀ ❲❆◆●✱ ❚✳ ●✳ ❆ ❤✐❣❤✲♣♦✇❡r❡❞ s✐r❡♥ ❢♦r st❛❜❧❡ ❛❝♦✉st✐❝ ❧❡✈✐t❛t✐♦♥ ♦❢ ❞❡♥s❡ ♠❛t❡r✐❛❧s ✐♥ t❤❡ ❡❛rt❤✬s ❣r❛✈✐t②✳ ❏♦✉r♥❛❧ ♦❢ t❤❡ ❆❝♦✉st✐❝❛❧ ❙♦❝✐❡t② ♦❢ ❆♠❡r✐❝❛✱ ✈✳ ✽✸✱ ♣✳ ✹✾✻✕✺✵✶✱ ✶✾✽✽✳

                    ❬✼✷❪ ❳■❊✱ ❲✳❀ ❲❊■✱ ❇✳ P❛r❛♠❡tr✐❝ st✉❞② ♦❢ s✐♥❣❧❡✲❛①✐s ❛❝♦✉st✐❝ ❧❡✈✐t❛t✐♦♥✳ ❆♣♣❧✐❡❞ P❤②s✐❝s ▲❡tt❡rs✱ ✈✳ ✼✾✱ ♥✳ ✻✱ ♣✳ ✽✽✶✕✽✽✸✱ ✷✵✵✶✳

                    ❬✼✸❪ ❋❖❘❊❙❚■✱ ❉✳❀ P❖❯▲■❑❆❑❖❙✱ ❉✳ ❆❝♦✉st♦♣❤♦r❡t✐❝ ❝♦♥t❛❝t❧❡ss ❡❧❡✈❛t✐♦♥✱ ♦r❜✐t❛❧ tr❛♥s♣♦rt ❛♥❞ s♣✐♥♥✐♥❣ ♦❢ ♠❛tt❡r ✐♥ ❛✐r✳ P❤②s✐❝❛❧ ❘❡✈✐❡✇ ▲❡tt❡rs✱ ❲♦♦❞❜✉r②✱ ✈✳ ✶✶✷✱ ♣✳ ✵✷✹✸✵✶✱ ✷✵✶✹✳

                    ❬✼✹❪ ❏✳ ▼❊◆●✱ ❉✳ ▼❊■✱ ❑✳ ❏✐❛✱ ❩✳ ❋❛♥✱ ❑✳ ❨❛♥❣✳ ❈♦♥t❛❝t❧❡ss ❛♥❞ ♥♦♥✲✐♥✈❛s✐✈❡ ❞❡❧✐✈❡r② ♦❢ ♠✐❝r♦✲♣❛rt✐❝❧❡s ❧②✐♥❣ ♦♥ ❛ ♥♦♥✲❝✉st♦♠✐③❡❞ r✐❣✐❞ s✉r❢❛❝❡ ❜② ✉s✐♥❣ ❛❝♦✉st✐❝ r❛❞✐❛t✐♦♥ ❢♦r❝❡✳ ❯❧tr❛s♦♥✐❝s✱ ✈✳ ✺✹✱ ♣✳ ✶✸✺✵✕✶✸✺✼✱ ✷✵✶✹✳

                    ❬✼✺❪ ▲❊❊✱ ❏✳❀ ▲❊❊✱ ❈✳❀ ❑✐♠✱ ❍✳ ❍✳❀ ❏❛❝♦❜✱ ❆✳❀ ▲❡♠♦r✱ ❘✳❀ ❚❡❤✱ ❙✳❀ ▲❡❡✱ ❆✳❀ ❙❍❯◆●✱ ❑✳ ❑✳ ❚❛r❣❡t❡❞ ❝❡❧❧ ✐♠♠♦❜✐❧✐③❛t✐♦♥ ❜② ✉❧tr❛s♦✉♥❞ ♠✐❝r♦❜❡❛♠✳ ❇✐♦t❡❝❤♥♦❧♦❣② ❛♥❞ ❇✐♦❡♥❣✐♥❡❡r✐♥❣✱ ✈✳ ✶✵✽✱ ♣✳ ✶✻✹✸✕✶✻✺✵✱ ✷✵✶✶✳

                    ❬✼✻❪ ❑✳ ❍✳ ▲❆▼✱ ❍✳ ❙✳ ❍❙❯✱ ❨✳ ▲✐❀ ❈✳ ▲❡❡❀ ❆✳ ▲■◆❀ ◗✳ ❩❍❖❯❀ ❊✳ ❙✳ ❑✳ ❑✳ ❑✳ ❙❍❯◆●✳ ❯❧tr❛❤✐❣❤ ❢r❡q✉❡♥❝② ❧❡♥s❧❡ss ✉❧tr❛s♦♥✐❝ tr❛♥s❞✉❝❡rs ❢♦r ❛❝♦✉st✐❝ t✇❡❡③❡rs ❛♣♣❧✐❝❛t✐♦♥✳ ❇✐♦t❡❝❤♥♦❧♦❣② ❛♥❞ ❇✐♦❡♥❣✐♥❡❡r✐♥❣✱ ✈✳ ✶✶✵✱ ♣✳ ✽✽✶✕✽✽✻✱ ✷✵✶✸✳

                    ❬✼✼❪ ❍✳ ❙✳ ❍❙❯✱ ❋✳ ❩❍❊◆●✱ ❨✳ ▲✐✱ ❈✳ ▲❡❡✱ ◗✳ ❩❤♦✉ ❛♥❞ ❙❍❯◆●✱ ❑✳ ❑✳ ❋♦❝✉s❡❞ ❤✐❣❤ ❢r❡q✉❡♥❝② ♥❡❡❞❧❡ tr❛♥s❞✉❝❡r ❢♦r ✉❧tr❛s♦♥✐❝ ✐♠❛❣✐♥❣ ❛♥❞ tr❛♣♣✐♥❣✳ ❆♣♣❧✐❡❞ P❤②s✐❝s ▲❡tt❡rs✱ ❲♦♦❞❜✉r②✱ ✈✳ ✶✵✶✱ ♣✳ ✵✷✹✶✵✺✱ ✷✵✶✷✳

                    ❬✼✽❪ ❊❱❆◆❉❊❘✱ ▼✳❀ ◆■▲❙❙❖◆✱ ❏✳ ❆❝♦✉st♦✢✉✐❞✐❝s ✷✵✿ ❆♣♣❧✐❝❛t✐♦♥s ✐♥ ❛❝♦✉st✐❝ tr❛♣♣✐♥❣✳ ▲❛❜ ❈❤✐♣✱ ✈✳ ✶✷✱ ♣✳ ✹✻✻✼✕✹✻✼✻✱ ✷✵✶✷✳

                    ❬✼✾❪ ▼✳ ❲■❑▲❯◆❉✱ ❙✳ ◆✳❀ ❍❊❘❚❩✱ ❍✳ ▼✳ ❯❧tr❛s♦♥✐❝ tr❛♣♣✐♥❣ ✐♥ ❝❛♣✐❧❧❛r✐❡s ❢♦r tr❛❝❡✲ ❛♠♦✉♥t ❜✐♦♠❡❞✐❝❛❧ ❛♥❛❧②s✐s✳ ❏♦✉r♥❛❧ ♦❢ ❆♣♣❧✐❡❞ P❤②s✐❝s✱ ✈✳ ✾✵✱ ♥✳ ✶✱ ✷✵✵✶✳

                    ❬✽✵❪ ❲■❑▲❯◆❉✱ ▼✳❀ ❍❊❘❚❩✱ ❍✳ ▼✳ ❯❧tr❛s♦♥✐❝ ❡♥❤❛♥❝❡♠❡♥t ♦❢ ❜❡❛❞✲❜❛s❡❞ ❜✐♦❛✣♥✐t② ❛ss❛②s✳ ▲❛❜ ❈❤✐♣✱ ✈✳ ✻✱ ♣✳ ✶✷✼✾✕✶✷✾✷✱ ✷✵✵✻✳

                    ❬✽✶❪ ❏✳ ❍❯▲❚❙❚❘❖▼✱ ❖✳ ▼❆◆◆❊❇❊❘●✱ ❑✳ ❉❖P❋✱ ▼✳ ❍✳ ❍❊❘❚❩✱ ❍✳ ❇❘■❙▼❆❘❀ ❲■✲ ❑▲❯◆❉✱ ▼✳ Pr♦❧✐❢❡r❛t✐♦♥ ❛♥❞ ✈✐❛❜✐❧✐t② ♦❢ ❛❞❤❡r❡♥t ❝❡❧❧s ♠❛♥✐♣✉❧❛t❡❞ ❜② st❛♥❞✐♥❣✲ ✇❛✈❡ ✉❧tr❛s♦✉♥❞ ✐♥ ❛ ♠✐❝r♦✢✉✐❞✐❝ ❝❤✐♣✳ ❯❧tr❛s♦✉♥❞ ✐♥ ♠❡❞✐❝✐♥❡ ❛♥❞ ❜✐♦❧♦❣②✱ ✈✳ ✸✸✱ ♥✳ ✶✱ ♣✳ ✶✹✺✕✶✺✶✱ ✷✵✵✼✳

                    ❬✽✷❪ ▼✳ ❊❱❆◆❉❊❘✱ ▲✳ ❏❖❍❆◆❙❙❖◆✱ ❚✳ ▲■▲▲■❊❍❖❘◆✱ ❏✳ P■❙❑❯❘✱ ▼✳ ▲■◆❉❱❆▲▲✱ ❙✳ ❏❖❍❆◆❙❙❖◆✱ ▼✳ ❆▲▼◗❱■❙❚✱ ❚✳ ▲❆❯❘❊▲▲❀ ◆■▲❙❙❖◆✱ ❏✳ ◆♦♥✐♥✈❛s✐✈❡ ❛❝♦✉st✐❝ ❝❡❧❧ tr❛♣♣✐♥❣ ✐♥ ❛ ♠✐❝r♦✢✉✐❞✐❝ ♣❡r❢✉s✐♦♥ s②st❡♠ ❢♦r ♦♥❧✐♥❡ ❜✐♦❛ss❛②s✳ ❆♥❛❧②t✐❝❛❧ ❈❤❡♠✐str②✱ ✈✳ ✼✾✱ ♣✳ ✷✾✽✹✕✷✾✾✶✱ ✷✵✵✼✳

                    ❬✽✸❪ ❙✳❇✳◗✳ ❚❘❆◆✱ P✳ ▼✳❀ ❚❍■❇❆❯▲❚✱ P✳ ❋❛st ❛❝♦✉st✐❝ t✇❡❡③❡rs ❢♦r t❤❡ t✇♦❞✐♠❡♥s✐✲ ♦♥❛❧ ♠❛♥✐♣✉❧❛t✐♦♥ ♦❢ ✐♥❞✐✈✐❞✉❛❧ ♣❛rt✐❝❧❡s ✐♥ ♠✐❝r♦✢✉✐❞✐❝ ❝❤❛♥♥❡❧s✳ ❆♣♣❧✐❡❞ P❤②s✐❝s ▲❡tt❡rs✱ ✈✳ ✶✵✶✱ ♣✳ ✶✶✹✶✵✸✱ ✷✵✶✷✳

                    ❬✽✹❪ ❇❖❆▲✱ ❉✳ ❍✳ ▼❊❈❍❆◆■❈❙ ❖❋ ❚❍❊ ❈❊▲▲✳ ✷♥❞✳ ❡❞✳ ❈❛♠❜r✐❞❣❡✱ ❯❑✿ ❈❛♠❜r✐❞❣❡ ❯♥✐✈❡rs✐t② Pr❡ss✱ ✷✵✶✷✳

                    ❬✽✺❪ ❊P❙❚❊■◆✱ P✳ ❙✳❀ ❈❆❘❍❆❘❚✱ ❘✳ ❘✳ ❚❤❡ ❛❜s♦r♣t✐♦♥ ♦❢ s♦✉♥❞ ✐♥ s✉s♣❡♥s✐♦♥ ❛♥❞ ❡♠✉❧s✐♦♥s✿ ■✳ ❲❛t❡r ❢♦❣ ✐♥ ❛✐r✳ ❏♦✉r♥❛❧ ♦❢ t❤❡ ❆❝♦✉st✐❝❛❧ ❙♦❝✐❡t② ♦❢ ❆♠❡r✐❝❛✱ ✈✳ ✷✺✱ ♣✳ ✺✺✸✕✺✻✺✱ ✶✾✺✸✳

                    ❬✽✻❪ ▼❆■◆❆❘❉■✱ ❋✳ ❋❘❆❈❚■❖◆❆▲ ❈❆▲❈❯▲❯❙ ❆◆❉ ❲❆❱❊❙ ■◆ ▲■◆❊❆❘ ❱■❙❈❖✲ ❊▲❆❙❚■❈■❚❨✿ ❆◆ ■◆❚❘❖❉❯❈❚■❖◆ ❚❖ ▼❆❚❍❊▼❆❚■❈❆▲ ▼❖❉❊▲❙✳ ▲♦♥❞♦♥✱ ❯❑✿ ■♠♣❡r✐❛❧ ❈♦❧❧❡❣❡ Pr❡ss✱ ✷✵✶✵✳

                    ❬✽✼❪ ❙❩❆❇❖✱ ❚✳ ▲✳❀ ❲❯✱ ❏✳ ❆ ♠♦❞❡❧ ❢♦r ❧♦♥❣✐t✉❞✐♥❛❧ ❛♥❞ s❤❡❛r ✇❛✈❡ ♣r♦♣❛❣❛t✐♦♥ ✐♥ ✈✐s❝♦❡❧❛st✐❝ ♠❡❞✐❛✳ ❏♦✉r♥❛❧ ♦❢ t❤❡ ❆❝♦✉st✐❝❛❧ ❙♦❝✐❡t② ♦❢ ❆♠❡r✐❝❛✱ ✈✳ ✶✵✼✱ ♥✳ ✺✱ ♣✳ ✷✹✸✼✕✷✹✹✻✱ ✷✵✵✵✳

                    ❬✽✽❪ ❈❍❊◆✱ ❲✳❀ ❍❖▲▼✱ ❙✳ ▼♦❞✐✜❡❞ s③❛❜♦ ✇❛✈❡ ❡q✉❛t✐♦♥ ♠♦❞❡❧s ❢♦r ❧♦ss② ♠❡❞✐❛ ♦❜❡②✐♥❣ ❢r❡q✉❡♥❝② ♣♦✇❡r ❧❛✇✳ ❏♦✉r♥❛❧ ♦❢ t❤❡ ❆❝♦✉st✐❝❛❧ ❙♦❝✐❡t② ♦❢ ❆♠❡r✐❝❛✱ ✈✳ ✶✶✹✱ ♥✳ ✺✱ ♣✳ ✷✺✼✵✕✷✺✼✹✱ ✷✵✵✸✳

                    ❬✽✾❪ ❍❖▲▼✱ ❙✳❀ ◆➘❙❍❖▲▼✱ ❙✳ P✳ ❈♦♠♣❛r✐s♦♥ ♦❢ ❢r❛❝t✐♦♥❛❧ ✇❛✈❡ ❡q✉❛t✐♦♥s ❢♦r ♣♦✇❡r ❧❛✇ ❛tt❡♥✉❛t✐♦♥ ✐♥ ✉❧tr❛s♦✉♥❞ ❛♥❞ ❡❧❛st♦❣r❛♣❤②✳ ❯❧tr❛s♦✉♥❞ ✐♥ ♠❡❞✐❝✐♥❡ ❛♥❞ ❜✐♦❧♦❣②✱ ✈✳ ✹✵✱ ♥✳ ✹✱ ♣✳ ✻✾✺✕✼✵✸✱ ✷✵✶✹✳

                    ❬✾✵❪ ❍❖▲▼✱ ❙✳❀ ◆➘❙❍❖▲▼✱ ❙✳ P✳❀ P❘■❊❯❘✱ ❋✳❀ ❙■◆❑❯❙✱ ❘✳ ❉❡r✐✈✐♥❣ ❢r❛❝t✐♦♥❛❧ ❛❝♦✉s✲ t✐❝ ✇❛✈❡ ❡q✉❛t✐♦♥s ❢r♦♠ ♠❡❝❤❛♥✐❝❛❧ ❛♥❞ t❤❡r♠❛❧ ❝♦♥st✐t✉t✐✈❡ ❡q✉❛t✐♦♥s✳ ❈♦♠♣✉t❡rs ❛♥❞ ▼❛t❤❡♠❛t✐❝s ✇✐t❤ ❆♣♣❧✐❝❛t✐♦♥s✱ ✈✳ ✻✻✱ ♥✳ ✺✱ ♣✳ ✻✷✶✕✻✷✾✱ ✷✵✶✸✳

                    ❬✾✶❪ ▼❆❙❊✱ ●✳ ❊✳ ❚❍❊❖❘❨ ❆◆❉ P❘❖❇▲❊▼❙ ❖❋ ❈❖◆❚■◆❯❯▼ ▼❊❈❍❆◆■❈❙✳ ▼❝●r❛✇✲❍✐❧❧ ❜♦♦❦ ❝♦♠♣❛♥②✱ ✶✾✼✵✳ ✈✳ ✶✳

                    ❬✾✷❪ ❲■▲▲■❆▼❙✱ ❊✳ ●✳ ❋❖❯❘■❊❘ ❆❈❖❯❙❚■❈❙✿ ❙❖❯◆❉ ❘❆❉■❆❚■❖◆ ❆◆❉ ◆❊❆❘✲ ❋■❊▲❉ ❆❈❖❯❙❚■❈❆▲ ❍❖▲❖●❘❆P❍❨✳ ▲♦♥❞♦♥✱ ❯❑✿ ❆❝❛❞❡♠✐❝ ♣r❡ss✱ ✶✾✾✾✳

                    ❬✾✸❪ ❙❊●❊▲✱ ▲✳ ❆✳ ▼❆❚❍❊▼❆❚■❈❙ ❆PP▲■❊❉ ❚❖ ❈❖◆❚■◆❯❯▼ ▼❊❈❍❆◆■❈❙✳ ✷✵✵✼✳

                    ❬✾✹❪ ❆❨❘❊❙✱ ❱✳❀ ●❆❯◆❆❯❘❉✱ ●✳ ❈✳ ❆❝♦✉st✐❝ r❡s♦♥❛♥❝❡ s❝❛tt❡r✐♥❣ ❜② ✈✐s❝♦❡❧❛st✐❝ ♦❜❥❡❝ts✳ ❏♦✉r♥❛❧ ♦❢ t❤❡ ❆❝♦✉st✐❝❛❧ ❙♦❝✐❡t② ♦❢ ❆♠❡r✐❝❛✱ ✈✳ ✽✶✱ ♥✳ ✵✷✱ ♣✳ ✸✵✶✕✸✶✶✱ ✶✾✽✼✳

                    ❬✾✺❪ ●❘❆❋❋✱ ❑✳ ❋✳ ❲❆❱❊ ▼❖❚■❖◆ ■◆ ❊▲❆❙❚■❈ ❙❖▲■❉❙✳ ▼✐♥❡♦❧❛✱ ◆❨ ❯❙❆✿ ❉♦✈❡r✱ ✶✾✼✺✳

                    ❬✾✻❪ ●❘❆❨✱ ❈✳ ●✳❀ ◆■❈❑❊▲✱ ❇✳ ●✳ ❉❡❜②❡ ♣♦t❡♥t✐❛❧ r❡♣r❡s❡♥t❛t✐♦♥ ♦❢ ✈❡❝t♦r ✜❡❧❞s✳ ❆♠❡r✐❝❛♥ ❏♦✉r♥❛❧ ♦❢ P❤②s✐❝s✱ ◆❡✇ ❨♦r❦✱ ✈✳ ✹✻✱ ♥✳ ✼✱ ♣✳ ✼✸✺✕✼✸✻✱ ✶✾✼✽✳

                    ❬✾✼❪ ❏❖❍◆ ❲■▲▲■❆▼ ❙❚❘❯❚❚✱ ❇✳ ❘✳ ❚❍❊ ❚❍❊❖❘❨ ❖❋ ❙❖❯◆❉✳ ❉♦✈❡r P✉❜❧✐❝❛t✐✲ ♦♥s✱ ✶✾✹✺✳ ✈✳ ✶✳

                    ❬✾✽❪ ▼❖❘❙❊✱ P✳ ▼✳❀ ■◆●❆❘❉✱ ❑✳ ❯✳ ❚❍❊❖❘❊❚■❈❆▲ ❆❈❖❯❙❚■❈❙✳ Pr✐♥❝❡t♦♥ ❯♥✐✲ ✈❡rs✐t② Pr❡ss✱ ✶✾✽✼✳

                    ❬✾✾❪ ❋❆❘❆◆✱ ❏✳ ❏✳ ❙♦✉♥❞ s❝❛tt❡r✐♥❣ ❜② s♦❧✐❞ ❝②❧✐♥❞❡rs ❛♥❞ s♣❤❡r❡s✳ ❏♦✉r♥❛❧ ♦❢ t❤❡ ❆❝♦✉st✐❝❛❧ ❙♦❝✐❡t② ♦❢ ❆♠❡r✐❝❛✱ ✈✳ ✷✸✱ ♥✳ ✹✱ ♣✳ ✹✵✺✕✹✶✸✱ ✶✾✺✶✳

                    ❬✶✵✵❪ ❆❘❋❑❊◆✱ ●✳ ❇✳❀ ❲❊❇❊❘✱ ❍✳ ❏✳ ▼❆❚❍❊▼❆❚■❈❆▲ ▼❊❚❍❖❉❙ ❋❖❘ P❍❨❙■✲ ❈■❙❚❙✳ ❙❛♥ ❉✐❡❣♦✱ ❈❆✱ ❯❙❆✿ ❆❝❛❞❡♠✐❝ Pr❡ss✱ ■♥❝✳✱ ✶✾✽✺✳

                    ❬✶✵✶❪ ❲❖▲❋❘❆▼ ❘❊❙❊❆❘❈❍✱ ■✳ ▼❆❚❍❊▼❆❚■❈❆ ✶✵✳ ❱❡rs✐♦♥ ✶✵✳✶✳ ❡❞✳ ❈❤❛♠♣❛✐❣♥✱ ■▲ ❯❙❆✿ ❲♦❧❢r❛♠ ❘❡s❡❛r❝❤✱ ■♥❝✳✱ ✷✵✶✺✳

                    ❬✶✵✷❪ ❲❊❙❚❊❘❱❊▲❚✱ P✳ ❏✳ ❚❤❡♦r② ♦❢ st❡❛❞② ❢♦r❝❡s ❝❛✉s❡❞ ❜② s♦✉♥❞ ✇❛✈❡s✳ ❏♦✉r♥❛❧ ♦❢ t❤❡ ❆❝♦✉st✐❝❛❧ ❙♦❝✐❡t② ♦❢ ❆♠❡r✐❝❛✱ ✈✳ ✷✸✱ ♣✳ ✸✶✷✕✸✶✺✱ ✶✾✺✶✳ ❬✶✵✸❪ ▲Ö❋❙❚❊❉❚✱ ❘✳❀ P❯❚❚❊❘▼❆◆✱ ❙✳ ❚❤❡♦r② ♦❢ ❧♦♥❣ ✇❛✈❡❧❡♥❣t❤ ❛❝♦✉st✐❝ r❛❞✐❛t✐♦♥

                    ♣r❡ss✉r❡✳ ❏♦✉r♥❛❧ ♦❢ t❤❡ ❆❝♦✉st✐❝❛❧ ❙♦❝✐❡t② ♦❢ ❆♠❡r✐❝❛✱ ✈✳ ✾✵✱ ♥✳ ✹✱ ♣✳ ✷✵✷✼✕✷✵✸✸✱ ✶✾✾✶✳

                    ❬✶✵✹❪ ❏❆❈❑❙❖◆✱ ❏✳ ❉✳ ❈▲❆❙❙■❈❆▲ ❊▲❊❈❚❘❖❉❨◆❆▼■❈❙✳ ❏♦❤♥ ❲✐❧❡②✱ ✶✾✾✽✳ ❬✶✵✺❪ ❍❖▲▼✱ ❙✳❀ ❙■◆❑❯❙✱ ❘✳ ❆ ✉♥✐❢②✐♥❣ ❢r❛❝t✐♦♥❛❧ ✇❛✈❡ ❡q✉❛t✐♦♥ ❢♦r ❝♦♠♣r❡ss✐♦♥❛❧ ❛♥❞ s❤❡❛r ✇❛✈❡s✳ ❏♦✉r♥❛❧ ♦❢ t❤❡ ❆❝♦✉st✐❝❛❧ ❙♦❝✐❡t② ♦❢ ❆♠❡r✐❝❛✱ ✈✳ ✶✷✼✱ ♥✳ ✶✱ ♣✳ ✺✹✷✕✺✹✽✱

                    ✷✵✶✵✳ ❬✶✵✻❪ ❑■◆❙▲❊❘✱ ▲✳ ❊✳❀ ❋❘❊❨✱ ❆✳ ❘✳❀ ❈❖PP❊◆❙✱ ❆✳ ❇✳❀ ❙❆◆❉❊❘❙✱ ❏✳ ❱✳ ❋❯◆❉❆✲

                    ▼❊◆❚❆▲❙ ❖❋ ❆❈❖❯❙❚■❈❙✳ ✹t❤✳ ❡❞✳ ◆❡✇ ❨♦r❦✱ ◆❨ ❯❙❆✿ ❏♦❤♥ ❲✐❧❡② ✪ ❙♦♥s✱ ■♥❝✳✱ ✶✾✾✾✳

                    ❬✶✵✼❪ ❏✳ ❉❯❘◆■◆✱ ❏✳❏ ▼■❈❊▲■✱ ❏✳❀ ❊❇❊❘▲❨✱ ❏✳ ❍✳ ❉✐✛r❛❝t✐♦♥✲❢r❡❡ ❜❡❛♠s✳ P❤②s✐❝❛❧ ❘❡✈✐❡✇ ▲❡tt❡rs✱ ❲♦♦❞❜✉r②✱ ✈✳ ✺✽✱ ♥✳ ✶✺✱ ♣✳ ✶✹✾✾ ✕ ✶✺✵✶✱ ✶✾✽✼✳

                    ❬✶✵✽❪ ▼■❚❘■✱ ❋✳❀ ❙■▲❱❆✱ ●✳ ❖✛✲❛①✐❛❧ ❛❝♦✉st✐❝ s❝❛tt❡r✐♥❣ ♦❢ ❛ ❤✐❣❤✲♦r❞❡r ❜❡ss❡❧ ✈♦rt❡① ❜❡❛♠ ❜② ❛ r✐❣✐❞ s♣❤❡r❡✳ ❲❛✈❡ ▼♦t✐♦♥✱ ✈✳ ✹✽✱ ♥✳ ✺✱ ♣✳ ✸✾✷✕✹✵✵✱ ✷✵✶✶✳

                    ❬✶✵✾❪ ❲❯✱ ❏✳ ❉❡t❡r♠✐♥❛t✐♦♥ ♦❢ ✈❡❧♦❝✐t② ❛♥❞ ❛tt❡♥✉❛t✐♦♥ ♦❢ s❤❡❛r ✇❛✈❡s ✉s✐♥❣ ✉❧tr❛s♦♥✐❝ s♣❡❝tr♦s❝♦♣②✳ ❏♦✉r♥❛❧ ♦❢ t❤❡ ❆❝♦✉st✐❝❛❧ ❙♦❝✐❡t② ♦❢ ❆♠❡r✐❝❛✱ ✈✳ ✾✾✱ ♥✳ ✵✺✱ ♣✳ ✷✽✼✶✕ ✷✽✼✺✱ ✶✾✾✻✳

                    ❬✶✶✵❪ ❘❖❍❆❚●■✱ ❆✳ ❲❡❜P❧♦t❉✐❣✐t✐③❡r✿ ❲❡❜ ❜❛s❡❞ t♦♦❧ t♦ ❡①tr❛❝t ❞❛t❛ ❢r♦♠ ♣❧♦ts✱ ✐♠❛✲ ❣❡s✱ ❛♥❞ ♠❛♣s✱ ✷✵✶✺✳

                    ❬✶✶✶❪ ❲❊■❙❙❚❊■◆✱ ❊✳ ❲✳ ▲❡✈❡♥❜❡r❣✲▼❛rq✉❛r❞t ▼❡t❤♦❞✳ ❋r♦♠ ▼❛t❤❲♦r❧❞✕❆ ❲♦❧❢r❛♠ ❲❡❜ ❘❡s♦✉r❝❡✱ ✷✵✶✺✳

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