This problem is essentially the same as the last one when d
=
R. However, this time we shall introduce the property of orthogonality to obtain a simple solution. By reciprocity, the resulting expression can be used to obtain the pressure at a point on the sphere due to a source at some point in space. This is a useful model for the diffraction effects of the human head on sound arriving at one ear, assuming a hard sphere model of the head. Unlike the pulsating sphere, only an infinitesimally small part of the surface is oscillating, so that the velocity distribution is described by

$\tilde{u}\left(R,\theta \right)=\{\begin{array}{cc}{\tilde{u}}_{0},& 0\le \theta \le \delta \\ 0,& \delta \le \theta \le \pi \end{array}\text{,}$

(12.39) ...

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