Axisymmetric wave propagation on a conical shell

Equations are derived to describe the propagation of axially symmetric waves generated at the apex of a conical shell, taking into account he coupling between longitudinal and transverse waves. A form of approximate solution is proposed, based upon the Green's function for wave propagation on a fiat plate but containing several variational parameters, the number of which can be reduced to one if additional approximations are made. It is shown that the region of the cone near the vertex moves amost like a rigid body, but the motion becomes wavelike at a distance from the vertex which decreases rapidly with increasing frequency, and at large distances the propagation is essentially the same as on a fiat plate. The one-parameter approximation is investigated numerically for two realistic situations, and appropriate solutions are presented and discussed.