Let the solution to Eq. (14.132) for a shallow spherical shell of radius a be in the form of the eigenfunction expansion

$\tilde{\eta}\left(w,\varphi \right)=\sum _{m=0}^{\infty}\sum _{n=1}^{\infty}{\tilde{A}}_{mn}{\eta}_{mn}\left(w,\varphi \right),$

(14.136)

where the eigenfunctions are given by

${\eta}_{mn}\left(w,\varphi \right)=\mathrm{cos}\left(m\varphi \right)\left({J}_{m}\left({\alpha}_{mn}w/a\right)-{B}_{mn}{I}_{m}\left({\alpha}_{mn}w/a\right)+{C}_{mn}\right).$

(14.137)

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

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