Calculation of wave fields using elastodynamic reciprocity

We consider axisymmetric time-harmonic wave motions generated by point-load excitation. Such problems are conventionally solved by the use of integral transform techniques. A typical example is the wave motion generated by a point force in an elastic layer. In this article, it is shown that, in a much simpler manner, the unknown modal coefficients for a superposition of wave modes can be conveniently obtained by the use of the Betti-Rayleigh reciprocity theorem. In this integral relation, which connects two elastodynamic states, one of the states is a wave mode of the actual wave field, whereas the other is an appropriately selected auxiliary solution. Simple expressions for the unknown coefficients quickly follow. The approach is also applied to the surface wave field generated by a time-harmonic normal point load on a half-space, where the amplitude constant of the generated surface wave motion is obtained.