Theory

In a vertically heterogeneous, isotropic and elastic medium occupying a half-space, equation (3.9) for Love waves has a solution of the form:

$$u_x =$$ 0

$$u_y = l (k,z,\omega) e^{i(kx-\omega t)}$$ (3.10)

$$u_z =$$ 0

$u_x$ , $u_y$ and $u_z$ are the radial, transversal and vertical components, respectively. $V_L=\frac{\omega}{k}$ (m/s) is the Love velocity at angular frequency $\omega$ (rad/s), $k$ is the wavenumber in the $x$ direction. $l(k,z,\omega)$ is the real amplitude, phase shifts are ignored as only one component is considered. The associated non-null stresses are (from equations (3.6) and (3.8)):

$$\sigma_{xy} = i k \mu(z) l e^{i(kx-\omega t)}$$ (3.11)

$$\sigma_{yz} = \mu(z) \frac{d l}{dz} e^{i(kx-\omega t)}$$

$\mu(z)=\rho(z)V_s(z)^2$ is the shear rigidity. Let call $\left(\mu(z) \frac{d l}{dz}\right)$ by $l^\sigma$ . A motion-stress vector for Love waves ( $\left[l,l^\sigma\right]^T$ ) is defined so that equation of motion (3.9) can be transformed into

$$\frac{d}{dz}\left( \begin{array}{c} l \ l^\sigma \end{array}\rig... ...& 0 \end{array}\right) \left( \begin{array}{c} l \ l^\sigma \end{array}\right)$$ (3.12)

which has the form of equation (3.1). For surface waves, the boundary conditions require that:

$$l\rightarrow 0 \quad \textrm{when} \quad z \rightarrow \infty$$ (3.13)

$$l^\sigma=0 \quad \textrm{at the free surface} \quad z=z_0$$ (3.14)

Because equation (3.12) has the same form as equation (3.1), the solution for the motion-stress vector is given by equation (3.5). The condition on the motion-stress at infinity (equation (3.13)) cannot be introduced directly. It is transformed into a radiation condition that no up-going waves are found in the bottom half-space. For $S_H$ plane waves, the amplitudes of downgoing ( $\grave S_n$ ) and up going ( $\acute S_n$ ) waves traveling across an homogeneous half space are function of the motion-stress vector at the top of the half space ($z=z_n$ ) (Aki and Richards (2002))

$$\left(\begin{array}{c} \grave S_n \ \acute S_n \end{array}\right)= T_n^{-1}\left( \begin{array}{c} l(z_n) \ l^\sigma(z_n) \end{array}\right)$$ (3.15)

where,

$$T_n^{-1} = \frac{1}{2 \nu_n \rho_n} \left( \begin{array}{cc} \nu_n \rho_n e^... ...sn}^2 \\ \nu_n \rho_n e^{-\nu_n z} & e^{-\nu_n z}/ V_{sn}^2 \end{array}\right)$$ (3.16)

$$\nu_n^2 = k^2- \frac{\omega^2}{V_{sn}^2}$$ (3.17)

$V_{sn}$ is the velocity of S waves (m/s). The subscript $n$ denotes parameters defined for layer $n$ as represented in figure 3.1. The motion-stress vector is propagated to $z_0$ by the means of equation (3.5). Equation (3.15) becomes

$$\left(\begin{array}{c} \grave S_n \ \acute S_n \end{array}\right... ...} G(z_1,z_0) \left( \begin{array}{c} l(z_0) \ l^\sigma(z_0) \end{array}\right)$$ (3.18)

where,

$$G(z_n,z_{n-1}) = \left( \begin{array}{cc} cosh [\nu_{n-1} (z_n-z_... ...h [\nu_{n-1} (z_n-z_{n-1})] & cosh [\nu_{n-1} (z_n-z_{n-1})] \end{array}\right)$$ (3.19)

Introducing the boundary conditions (equations (3.13) and (3.14)) into equation (3.18) gives

$$\left(\begin{array}{c} \grave S_n \ 0 \end{array}\right) = T_n^{-1} G_n \ldots{} G_1 \left( \begin{array}{c} l(z_0) \ 0 \end{array}\right)$$

$$= \underbrace{\left(\begin{array}{cc} l_{11} & l_{12} \\ l_{21} & ... ...nd{array}\right)}_{L(z_0)} \left(\begin{array}{c}l(z_0) \ 0 \end{array}\right)$$ (3.20)

which has only non trivial solutions when $l_{21}$ vanishes. The problem of finding the dispersion curves for Love waves is hence reduced to a root search along the slowness or the velocity axis for a given frequency. For a given frequency ( $\frac{\omega}{2\pi}$ ), only a few discrete values are possible for the velocity of the Love surface wave ( $V_L=\left[\frac{\omega}{k(\omega)}\right]_i$ ), corresponding to the dispersion curves of various modes.