Velocity Equation of Love Waves Propagated in Multi-layered Media

Abstract

The derivation of the velocity equations of surface waves propagating in multi-layered structure is not hard in its principle. However, the practical computation is extremely complicated when there are many layers. To remove this difficulty, several methods have been suggested, which are all connected with the problems involving P and SV waves. If we put μ=0 into these calculations, the solutions turn out to be those of sound waves in liquid, and by an appropriate change of variables the solution for Love waves is obtained. In this paper, however, a simpler and a more direct method having a clear physical meaning is presented.
We start from the solution in the uppermost layer. Adopting the solution satisfying the condition of the free surface we can obtain the displacement and the stress at the surface of separation between this and the next layer. Two boundary conditions, continuity of displacement and stress, give the solution in the next layer. Repeating the process for each of the layers we can finally obtain the solution in the last medium, namely the semi-infinite medium. This solution must converge at the point of infinite depth, and this condition gives the velocity equation of this case.
By a small modification of expressions we can easily obtain the equation for the problem of a liquid medium as well as the solutions with or without semi-infinite half space.
The problem of sound waves in liquid medium can be at once solved by modifying the above theory using the change of physical quantities given in Table 1.