In this section, we shall apply the boundary integral method to a pulsating cap in a sphere to illustrate its application to an elementary acoustical problem that has already been treated in Section 12.6 using the boundary value method. The geometry of the problem is shown in Fig. 12.16. From Eq. (13.26), we can write the pressure field as a surface integral:

$\begin{array}{c}\tilde{p}\left(r,\theta \right)={\int}_{0}^{2\pi}{\int}_{0}^{\pi}g\left(r,\theta |{r}_{0},{\theta}_{0}\right){|}_{{r}_{0}=R}\frac{\partial}{\partial {r}_{0}}{\tilde{p}\left({r}_{0},{\theta}_{0}\right)|}_{{r}_{0}=R}{R}^{2}\mathrm{sin}{\theta}_{0}d{\theta}_{0}d{\varphi}_{0}\\ -{\int}_{0}^{2\pi}{\int}_{0}^{\pi}{\tilde{p}\left({r}_{0},{\theta}_{0}\right)|}_{{r}_{0}=R}\frac{\partial}{\partial {r}_{0}}\end{array}$

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