Eigenfunctions

As for the Love case, the functions $r_1$ to $r^\sigma_2$ defined in equation (3.25) are the eigenfunctions for Rayleigh waves. From equation (3.32), it is obvious that

$$\frac{r_1(z_0)}{r_2(z_0)}=-\frac{r_{12}(z_0)}{r_{11}(z_0)}$$ (3.35)

The ratio of eigenfunctions $r_1$ and $r_2$ is hence fixed to a constant that depends upon the values of the elements of matrix $R(z_0)$ , itself, a function of the mode and the frequency for which the Rayleigh velocity has been calculated. The motion-stress vector at depth $z_0$ can be defined numerically, normalizing either $r_1$ or $r_2$ to any arbitrary value. The computation of the eigenfunctions at any arbitrary depth is done in the same way as for the Love case. The elements of $G$ are not given here, but it can calculated by an eigenvalue decomposition of matrix $A$ (equations (3.27) and (3.1)).

The eigenfunctions at the surface are useful for computing the ellipticity of Rayleigh waves (section 3.2). It will be shown how to calculate $\frac{r_{12}(z_0)}{r_{11}(z_0)}$ without the complete knowledge of the elements of matrix $R(z_0)$ .