Surface waves in simple models – dispersive waves

Introduction Surface waves are waves that propagate along a boundary and whose amplitudes go to zero as the distance from the boundary goes to infinity. There are two basic types of surface waves, Love and Rayleigh waves, named after the scientists who studied them first. Love's work was directed to the explanation of waves observed in horizontal seismographs, while Rayleigh predicted the existence of the waves with his name. The main difference between the two types of waves is that the motion is of SH type for Love waves, and of P – SV types for Rayleigh waves. A related type of wave, known as Stoneley waves, consists of P – SV inhomogeneous waves that propagate along the boundary between two half-spaces. In this chapter we will consider these three types of waves. As we shall see, the presence of a layer introduces the phenomenon of dispersion , which is characterized by the existence of two velocities, known as the phase and the group velocity, with the property that they are functions of frequency. In §7.6 a detailed analysis of dispersion is presented. The problem of multilayered media will not be considered here, but the groundwork for its analysis has been introduced in §6.9.2. To solve problems involving surface waves it is necessary to go through the steps described in §6.1, namely, write the equation for the displacement at any point in the medium and then apply appropriate boundary conditions.