결 론 - 접수평판의내부분열에의한 3 차원음향방사

일반적인 음원에 의해 발생되는 파의 종류를 그림(5.1)에 나타내었으며, 각각의 환경 에 대해 파가 존재하는 조건이 다양하므로 실제 환경에 대해 존재하는 현상을 이해하 기 위해 좀더 많은 연구를 필요로 한다. 특히 구형의 층상화에 대한 음향 에너지 전달 의 경우 이론적인 공식에서 의하면 유체/ 탄성 영역은 크게 소실되어 지는 것으로 알 려져 있지만 실제 환경에 적용하기 위해서는 구형의 층상화에 대해 해석적 해를 구해 야 한다.

본 논문에서 개발된 모델은 기본적으로 세 개의 층으로 구성된 접수 평판의 환경에 대해 적용된 것이지만 좀 더 현실적인 문제에 있어 수중 음향 반향체의 코팅에 의한 반사 손실을 측정하기 위한 모델이나 앞서 언급한 유체 소음 판(plate) 또는 표면 처리 된 판(coated plat)에서 음향 에너지의 방사 형태에 관한 모델링을 위해 확장할 수 있 다.

A ppe ndix

A 1. D ip S lip

▷ Potentials

= M₀

4 w² e^iwt

- 0. 5 s in 2 ( 2 ²+ k²)J₀( kr ) - 2 cos 2 k s in J₁( kr ) - 0. 5 s in 2 k²cos 2 J₂( kr )

e^- ^{| z - z}^s^|k dk

= M₀

4 w² e^iwt

- 1. 5 s in 2 J0( kr ) - cos 2 2k²- k²₂

k s in J₁( kr ) - 0. 5 s in 2 cos 2 J₂( kr )

e^- ^{| z - z}^s^|k dk

= M₀

4 w² e^iwt

+ cos 2 k²

k cos J₁( kr ) - 0. 5 s in 2 k²s in 2 J₂( kr )

e^- ^{| z - z}^s^|k dk

▷ Dis place me nts and s tres s e s

F or zeroth order,

w^m = e^iwt 4 w² M₀

[

0. 5 s in 2 ( 2 ²+ k²) e ^- ^{| z - z}^s^|

]

- 1. 5 s in 2 k²e^- ^{| z - z}^s^| J₀( kr ) k dk

u^m+ v^m= e^iwt 4 w² M₀

0. 5 s in 2 ( 2 ²+ k²) k

e^- ^{| z - z}^s^| - 1. 5 s in 2 k e^- ^{| z - z}^s^|

J₁( kr ) k d k

u^m- v^m= e^iwt 4 w² M₀

- 0. 5 s in 2 ( 2 ²+ k²) k

e^- ^{| z - z}^s^| 1. 5 s in 2 k e^- ^{| z - z}^s^|

J_{- 1}( kr ) k dk

z z= e^iwt 4 w² M₀

- 0. 5 s in 2 ( 2 ²+ k²)( 2 ²- k²) 1

e^- ^{| z - z}^s^| + 3 s in 2 k² e^- ^{| z - z}^s^|

J₀( kr ) k dk

r z+ ^m_z= e^iwt 4 w² M₀

[

+ 1. 5 s in 2 k (^- ^{s in 2} ^{k ( 2} ²²^{+ k}+ k²²^{) e}) e^-^- ^{| z - z}^{| z - z}^s^|^s^|

]

J₁( kr )k dk

r z- ^m_z= e^iwt 4 w² M₀

[

+ s in 2 k ( 2 ²+ k²) e^- ^{| z - z}^s^|

]

- 1. 5 s in 2 k ( ²+ k²) e^- ^{| z - z}^s^| J_{- 1}( kr )k dk

F or s in order,

w^m= e^iwt 4 w² M₀

2 cos 2 k e^- ^{| z - z}^s^| - cos 2 ( 2 k²- k²_m) k

e^- ^{| z - z}^s^|

J₁( kr ) k dk

u^m+ v^m= e^iwt 4 w² M₀

[

- 2 cos 2^{2 cos 2} ^k²k^e²e^-^- ^{| z - z}^{| z - z}^s^|^s^|

]

J₂( kr ) k dk u^m- v^m= e^iwt

4 w² M₀

[

^{- 2 cos 2}+ 2 cos 2 ^k²²^ee^-^- ^{| z - z}^{| z - z}^s^s^|^|

]

J₀( kr ) k dk

z z= e^iwt 4 w² M₀

[

+ 2 cos 2^{- 2 cos 2} k ( k^{k ( k}²²+⁺ ²²) e^)e^-^- ^{| z - z}^{| z - z}^s^s^|^|

]

J₁( kr ) k dk

r z+ ^m_z= e^iwt 4 w² M₀

- 4 cos 2 k²e^- ^{| z - z}^s^| + cos 2 k²

( 3 ³+ k²) e^- ^{| z - z}^s^|

J₂( kr )k dk

r z- ^m_z= e^iwt 4 w² M₀

4 cos 2 k²e^- ^{| z - z}^s^| - cos 2 ( 2 ³+ k⁴

+ k²) e^- ^{| z - z}^s^|

J₀( kr )k dk

F or cos 2 order,

w^m = e^iwt 4 w² M₀

[

+ 0. 5 s in 2 k²e^- ^{| z - z}^s^|

]

- 0. 5 s in 2 k²e^- ^{| z - z}^s^| J₂( kr ) k d k

u^m+ v^m= e^iwt 4 w² M₀

0. 5 s in 2 k³

e^- ^{| z - z}^s^| - 0. 5 s in 2 k³

e^- ^{| z - z}^s^|

J₃( kr ) k dk

u^m- v^m= e^iwt 4 w² M₀

- 0. 5 s in 2 k³

e^- ^{| z - z}^s^| - 0. 5 s in 2 ( k²- 2 ²) k

e^- ^{| z - z}^s^|

J₁( kr ) k dk

z z= e^iwt 4 w² M₀

- 0. 5 s in 2 ( k²+ ²) k²

e^- ^{| z - z}^s^| + s in 2 k² e^- ^{| z - z}^s^|

J₂( kr ) k d k

r z+ ^m_z= e^iwt 4 w² M₀

[

^-+ s in 2^{s in 2} ^kk³³^ee^-^- ^{| z - z}^{| z - z}^s^s^|^|

]

J₃( kr )k dk

r z- ^m_z= e^iwt 4 w² M₀

[

- ^{s in 2}s in 2 ^kk³^e²^-e^-^{| z - z}^{| z - z}^s^| ^s^|

]

J₁( kr )k dk

A .2 S trike- S lip

▷ Potentials

= - M₀

4 w² e^iwt

[

s in s in 2 k²J2( kr )

]

- 2 cos cos k J₁( kr ) e^- ^{| z - z}^s^|k dk

r= - M0

4 w² e^iwt

[

s in s in 2 k J₁( kr )

]

- cos cos J₀( kr ) e^- ^{| z - z}^s^|k dk

= - M₀ 4 w² e^iwt

[

s in s in 2 k J1( kr )

]

+ cos cos J0( kr ) e^- ^{| z - z}^s^|k dk

z= M0

4 w² e^iwt

0 cos cos k J₁( kr ) e ^{| z - z}^s^|k d k

또는,

= - M₀

4 w² e^iwt

[

s in s in 2 k²J₂( kr )

]

- 2 cos cos k J1( kr ) e^- ^{| z - z}^s^|k dk

= M₀

4 w² e^iwt

cos cos k 2k²- k²₂ k J₁( kr ) - s in cos 2 J₂( kr )

e^- ^{| z - z}^s^|k d k

= M0

4 w² e^iwt

cos s in k²_m k J₁( kr ) + s in cos 2 k²J₂( kr )

e^- ^{| z - z}^s^|k dk

▷ Dis place me nts and s tres s e s

F or cos order,

w^m= e^iwt 4 w² M0

- 2 cos k e^- ^{| z - z}^s^| cos ( 2 k²- k²_m ) k

e^- ^{| z - z}^s^|

J1( kr ) k dk

u^m+ v^m= e^iwt 4 w² M₀

[

^{- 2 cos}+ 2 cos ^kk²²^ee^-^- ^{| z - z}^{| z - z}^s^s^|^|

]

J₂( kr ) k dk u^m- v^m= e^iwt

4 w² M0

[

^{+ 2 cos}- 2 cos ^k²²^ee^-^- ^{| z - z}^{| z - z}^s^s^|^|

]

J0( kr ) k dk

z z= e^iwt 4 w² M₀

2 cos k ( 2k²- k²_m)e^- ^{| z - z}^s^| - 2 cos k ( 2k²- k²_m) e^- ^{| z - z}^s^|

J₁( kr ) k dk

r z+ ^m_z= e^iwt 4 w² M₀

4 cos k²e^- ^{| z - z}^s^| - cos ( k⁴

+ 3 k²) e^- ^{| z - z}^s^|

J₂( kr )k d k

r z- ^m_z= e^iwt 4 w² M₀

- 4 cos k²e^- ^{| z - z}^s^| cos ( 2 ³+ k⁴

+ k²) e ^- ^{| z - z}^s^|

J₀( kr )k dk

F or s in 2 order,

w^m = e^iwt 4 w² M₀

[

-^{s in}s in ^kk²²^ee^-^- ^{| z - z}^{| z - z}^s^|^s^|

]

J₂( kr ) k d k u^m+ v^m= e^iwt

4 w² M₀

s in k³

e^- ^{| z - z}^s^| - s in k³

e^- ^{| z - z}^s^|

J₃( kr ) k d k

u^m- v^m= e^iwt 4 w² M₀

- s in k³

e^- ^{| z - z}^s^| - s in ( k²- 2 ²) k

e^- ^{| z - z}^s^|

J₁( kr ) k d k

z z= e^iwt 4 w² M₀

- s in ( 2k²- k²m)k²

e^- ^{| z - z}^s^| + 2 s in k² e^- ^{| z - z}^s^|

J₂( kr ) k d k

r z+ ^m_z= e^iwt 4 w² M₀

[

^{- 2 s in}+ 2 s in ^kk³³^ee^-^- ^{| z - z}^{| z - z}^s^s^|^|

]

J₃( kr )k dk

r z- ^m_z= e^iwt 4 w² M₀

[

- 2 s in^{2 s in} ^kk³^e²^-e^-^{| z - z}^{| z - z}^s^| ^s^|

]

J₁( kr )k d k

A .3 T ens ile Crac k

M₁ = M₀, ₁= 90。

M₂ = M₀, ₂= + 90。 , an d M₃ = + 2

M₀, ₃ =

ex cept that M1 = - M0 w hen m = 2, cos 2 order.

▷ Potentials

k = 1 '_k( r , , z : _k, M_k)

k = 1 '_{r , k}( r , , z : _k, M_k)

k = 1 ' _{, k}( r , , z : _k, M_k)

k = 1 'z , k( r , , z : k, Mk)

w here the potentials w ith prim e (' ), w hich is a s ingle couple w ithout m om ent, are

'_k= - M_ke^iwt 4 w² ⁰

[

+ 0. 25 cos 2^{- 0. 25( k}²k^{- 2}( k²+ 2²⁾ ²)

]

^J⁰^{( kr )}

- s in 2 _ks in k J₁( kr ) - 0. 25 ( 1 - cos 2 k) cos 2 k²J2( kr )

e^- ^{| z - z}^s^|k dk

'_{r , k}= - M_ke^iwt 4 w² ⁰

- 0. 5 s in 2 _ks in J₀( kr )

[

0. 25( 1 - cos 2 _k)

- 0. 25 ( 1 - cos 2 _k) cos 2 k

]

^J⁰^{( kr )}

e^- ^{| z - z}^s^|k dk

' _{, k}= - M_ke^iwt

4 w² ⁰

[

- 0. 5 s in 2 _kcos J₀( kr )

]

0. 25( 1 - cos 2 _k) cos 2 k J₀( kr ) e^- ^{| z - z}^s^|k dk

'_{z , k}= - M_ke^iwt

4 w² ⁰

[

- 0. 5 ( 1 + cos 2 k) J0( kr )

]

- 0. 5 s in 2 _ks in k J₀( kr ) e^- ^{| z - z}^s^|k dk

▷ Dis place me nts and s tres s e s

w^m( r , z ) =

k = 1w^m_k ( r , z ) u^m( r , z ) + v^m( r , z ) =

k = 1u^mk ( r , z ) + v^mk ( r , z )

‥

F or zeros order,

w^m_k = e^iwt 4 w² M_k

[ 0. 25( k²- 2 ²) - 0. 25 cos 2 _k( k²+ 2 ²) ] e^- ^{| z - z}^s^| ( 0. 25 + 0. 7 5 cos 2 _k) k²e ^{| z - z}^s^|

J₀( kr ) kd k

u^m_k+ v^m_k = e^iwt 4 w² M_k

0. 25( k²- 2 ²) k

- 0. 25 cos 2 _k( k²+ 2 ²) k e^- ^{| z - z}^s^| ( 0. 25 + 0. 7 5 cos 2 _k) k e ^{| z - z}^s^|

J₁( kr ) kdk

u^m_k- v^m_k = e^iwt 4 w² M_k

- 0. 25( k²- 2 ²) k

- 0. 25 cos 2 k( k²+ 2 ²) k e^- ^{| z - z}^s^| - ( 0. 25 + 0. 75 cos 2 k) k e ^{| z - z}^s^|

J_{- 1}( kr ) kdk

z z , k= e^iwt 4 w² M_k

- k²+ ²

[

- 0. 25 cos 2^{0. 25( k}²^{- 2}_k( k²+ 2²⁾ ²)

]

^e^- ^{| z - z}^s^|

- ( 0. 5 + 1. 5 cos 2 _k) k²e ^{| z - z}^s^|

J₀( kr ) kdk

r z , k+ ^m_{, k} = e^iwt 4 w² M_k

[

+ 0. 5 cos 2^{- 0. 5( k}²_k^{- 2}( k²+ 2²^{) k}²) k

]

^e^- ^{| z - z}^s^|

- ( 0. 25 + 0. 75 cos 2 k) k ( s²+ ²) e ^{| z - z}^s^|

J₁( kr ) kdk

r z , k- ^m_{, k} = e^iwt 4 w² M_k

[

- 0. 5 cos 2^{0. 5( k}²^{- 2}_k( k²+ 2²^{) k} ²) k

]

^e^- ^{| z - z}^s^|

( 0. 25 + 0. 7 5 cos 2 k) k ( s²+ ²) e ^{| z - z}^s^|

J_{- 1}( kr ) kdk

F or s in order,

w^m_k = e^iwt 4 w² M_k

k s in 2 _ke^- ^{| z - z}^s^| - 0. 5 k

( ²+ k²) e^- ^{| z - z}^s^|

J₁( kr ) kdk

u^mk+ v^mk = e^iwt 4 w² Mk

+ k²s in 2 _ke^- ^{| z - z}^s^| - k²s in 2 _ke^- ^{| z - z}^s^|

J2( kr ) kdk

u^m_k- v^m_k = e^iwt 4 w² M_k

- 2 s in 2 _k k²e^- ^{| z - z}^s^| + 2 s in 2 _k ²e^- ^{| z - z}^s^|

J₀( kr ) kdk

z z , k= e^iwt 4 w² Mk

- 2 s in 2 _k k ( k²+ ²) e^- ^{| z - z}^s^| + 2 s in 2 _k k ( k²+ ²) e^- ^{| z - z}^s^|

J1( kr ) kd k

r z , k+ ^m_{, k} = e^iwt 4 w² M_k

- 2 s in 2 _k k²e^- ^{| z - z}^s^| + 0. 5 s in 2 _k k²

( 2 ²+ k²) e^- ^{| z - z}^s^|

J₂( kr ) kdk

r z , k- ^m_{, k} = e^iwt 4 w² M_k

2 s in 2 _k k²e^- ^{| z - z}^s^| - s in 2 _k( ³+ 0. 5k⁴

+ 0. 5 k²) e^- ^{| z - z}^s^|

J₀( kr ) kdk

F or cos 2 order

w^m_k = e^iwt 4 w² M_k

+ 0. 25 k²( 1 - cos 2 k) e^- ^{| z - z}^s^| - 0. 25 k²( 1 - cos 2 _k)e^- ^{| z - z}^s^|

J₂( kr ) kdk

u^m_k+ v^m_k = e^iwt 4 w² M_k

+ 0. 25k³

( 1 - cos 2 _k) e^- ^{| z - z}^s^| - 0. 25k³

( 1 - cos 2 _k) e^- ^{| z - z}^s^|

J₃( kr ) kdk

u^m_k- v^m_k = e^iwt 4 w² M_k

- 0. 25k³

( 1 - cos 2 k) e^- ^{| z - z}^s^| - 0. 25 ( k²- 2 ²)k

( 1 - cos 2 _k) e ^- ^{| z - z}^s^|

J₁( kr ) kd k

z z , k= e^iwt 4 w² M_k

- 0. 25( k²+ ²)k²

( 1 - cos 2 _k) e^- ^{| z - z}^s^| + 0. 5k² ( 1 - cos 2 _k) e^- ^{| z - z}^s^|

J₂( kr ) kdk

r z , k+ ^m_{, k} = e^iwt 4 w² M_k

- 0. 5 k³( 1 - cos 2 _k) e^- ^{| z - z}^s^| + 0. 5 k³( 1 - cos 2 _k) e^- ^{| z - z}^s^|

J₃( kr ) kd k

r z , k- ^m_{, k} = e^iwt 4 w² M_k

[

- 0. 5 k^{+ 0. 5 k}³^e²e^-^- ^{| z - z}^{| z - z}^s^|^s^|

]

J₁( kr ) kdk

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문서에서 접수평판의내부분열에의한 3 차원음향방사 (페이지 55-68)