Elastic waves in a semi-infinite body

A new method is introduced for studying the propagation of elastic waves in isotropic bodies, based on the Kirchhoff potentials borrowed from electromagnetism. By means of this method we identify and characterize the elastic waves generated in a semi-infinite (half-space) body by the action of an external force localized on, or beneath, the body surface. The method implies coupled integral equations for the wave amplitudes, which we solve for both cases mentioned above. For a force localized on the body surface we identify two transverse waves, corresponding to the two polarizations (normal and parallel to the propagation plane). The longitudinal waves appear as eigenmodes. The waves produced by a force localized beneath the surface are stationary waves along the normal to the surface. We compute the surface displacement in both cases and the force exerted on the surface by a force localized beneath. All these quantities exhibit a characteristic decrease with the distance on the body surface and an oscillatory behaviour. We discuss briefly some possibilities of extending the present method to include the effect of the inhomogeneities on the waves propagation.