Acoustics: Sound Fields, Transducers and Vibration, 2nd Edition

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 ) ,

(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 ) .

(14.137)

We use the following identity for the Laplace operator ...