On the theory of head waves

Abstract When a combined longitudinal and transverse disturbance, diverging from a localized source, strikes a plane boundary between two solid elastic media, several systems of head waves and second-order boundary waves are generated, each associated with grazing incidence of one or the other of the reflected or refracted waves. Associated with grazing incidence of P 1 P 2 , the refracted P-wave, is the head wave system comprising P 1 P 2 P 1 (the "refracted wave" of seismic prospectors), and P 1 P 2 S 1 (a transverse head wave) in the upper medium, and P 1 P 2 S 2 (a transverse head wave) in the lower medium. There is no boundary wave in the lower medium. These three waves, with the second-order term of P 1 thorn 2 (the first-order term is zero on the boundary) satisfy conditions of continuity of stress and displacement at the boundary. Moreover, the energy of the three head waves is derived completely from the second-order component of P 1 P 2 , which possesses a component of energy flow normal to the boundary. The amplitudes of P 1 P 2 P 1 , P 1 P 2 S 1 and P 1 P 2 S 2 are calculated for certain cases.