The reflexion of a pulse in a solid sphere

A spherically symmetric source of sound, situated within a solid homogeneous isotropic elastic sphere, emits a short P-pulse of small amplitude. Scalar potentials for the P and SV disturbances are shown to exist and to satisfy wave equations with velocities α and β, respectively. At the free spherical surface the theory of characteristics yields the approximate shape for both P and SV pulses immediately after reflexion (PP and PS), and these approximations with Kirchhoff’s formula give the pulse shapes of PP and PS near the geometrical acoustic arrival time at any point not too near the cusps of the caustics of PP and PS reflexions. This method gives rise to certain integrals analogous to those leading to stationary phase approximations but involving the distribution S', the derivative of the Dirac δ- 'function’, instead of the exponential function. The main result is the calculation of the pulse shape for the field point near the caustics and near the axis. For an incident step pulse figures exhibit the transition in the pulse shape as the field point crosses a caustic or approaches the axis. This calculation involves simple, though laborious, numerical manipulation of complete elliptic integrals. These results are quite general and would apply to any axially symmetric reflexion problem where the pulse length is much shorter than the principal radii of curvature of the reflecting surface. Finally, these formulae are applied to the reflexion in a sphere. The method adopted in this paper has the advantage of directness, and at each stage of the calculation the physical significance of all variables and expressions is easily seen.