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Acoustics: Sound Fields, Transducers and Vibration, 2nd Edition by Tim Mellow, Leo Beranek

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14.17. Green's function for a shallow spherical shell

Let the solution to Eq. (14.132) for a shallow spherical shell of radius a be in the form of the eigenfunction expansion
η ˜ ( w , ϕ ) = m = 0 n = 1 A ˜ m n η m n ( w , ϕ ) ,
image (14.136)
where the eigenfunctions are given by
η m n ( w , ϕ ) = cos ( m ϕ ) ( J m ( α m n w / a ) B m n I m ( α m n w / a ) + C m n ) .
image (14.137)
We use the following identity for the Laplace operator ...

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