Pressure and Shear Waves Generated by an Explosion in an Elastic Sphere *

Summary We seek to determine the elastic wave motion at a point in a homogeneous isotropic sphere resulting from an explosive sound at another arbitrary point in the sphere. The surface of the sphere is supposed to be traction‐free. A bilateral Laplace transformation in time is applied to the governing differential equations and boundary conditions. This results in a solution expressed as Helmholtz potentials consisting of two slowly convergent series of spherical Bessel functions and Legendre polynomials. For wave numbers up to certain critical value, real poles of the potentials exist giving rise to excited oscillations. However, for wave numbers greater than this critical value, integral representations of the series are obtained through the transformation of Watson. Then using the asymptotic representations of Debye for large wave number and order, the integrals are evaluated by Laplace's method to give P and S wave which are interpreted as ‘distributions’. The stress wave motion solution in the mechanical system is shown to be analogous to the wave motion in the geometric optics system.