Eigenfunctions

The functions $l$ and $l^\sigma$ defined in equations (3.10) and (3.11) are the eigenfunctions of Love waves. For each depth and frequency, different values of eigenfunctions exist corresponding to all roots of $l_{21}(z_0)$ (modes). The motion-stress vector at depth $z_0$ can be defined numerically by normalizing $l(z_0)$ to any arbitrary value. The computation of the eigenfunctions at the next layer interface is done by multiplying the motion-stress vector at depth $z_0$ by $G_1^{-1}$ . The same task is repeated until reaching the top of the half-space. Inside a particular layer, the values of the eigenfunctions are also calculated from the definition of $G_n^{-1}$ (equation (3.18). Examples of eigenfunction variation with depth can be found in Aki and Richards (2002). Among other features, they show that the penetration depth is frequency dependent. For high frequencies, only the most superficial layers are affected by displacements and stresses.