Geometrical Aspects of Acoustic Radiation from a Shallow Spherical Cap

Geometrical properties are used to obtain the acoustic field from harmonic and pulse excitation of a shallow spherical cap in a baffle. The axial field from harmonic oscillation is obtained by observing the equivalence with the field from oscillation of a suitably chosen full sphere. This defines a convenient Legendre polynomial expansion of the velocity on the cap and leads to an axial field in the form of a series of products of spherical Bessel and Hankel functions.

Each term corresponds to a Bessel function variation of the potential in the focal plane. The full solution for a uniform pulse excitation is obtained from the geometry of intersecting spheres. The simplicity and finite duration of the pulse solution suggests its use as convolution kernel for general time dependence of the excitation.