The Love Wave in a Heterogeneous Medium

Abstract

The transversal wave propagated along the surface of a semi-infinitely elastic body, the elasticity of which increases linearly with the depth, is discussed. The effect of the variation of density is also examined. The mathematical treatment is reduced to find a function φ(z) which satisfies
Putting kz=x, kd=h and where b₀ means the value of √μ/ρ at the surface, we may assume the following solution: The path of integration K is shown in Fig. 1.
The dispersion curve is determined by h and α which are connected with each other by a relation: It is easily verified that α must be larger than unity, which is to be consistent with (1).
When h is comparatively small, it is convenient to transform (1) into the following form: and τ exceeds unity so slightly as to make e^-hN negligibly small. Assuming various values of α for an assigned value of h and operating the integration numerically, we may obtain the value of α wanted.
The white circles on the dispersion curve in Fig. 3 show the values obtained by this method. whereas the black ones are those obtained by the following method.
If h is comparatively large and so α becomes also large, we may obtain the integration (1) in special cases when α is an integer, and the corresponding value of h may be calculated. Therefore we may ees the tendency of the dispersion curve. In the limit of L (wave length) →0, dV/dL tends to ∞.
Finally, it is verified that, assuming the distribution of the density in the form ρ=r+sz (r and s being constants), the conditions of existence of surface wave are r>O and. Under these conditions, the property of dispersion is characterised by the equations and the numerical relation between α and h coincides with that in the case of constant density. (As for the notation, refer to the equations (18)_??_(22).)