On the criteria of existence of eigenfrequencies corresponding to Love waves
The geodynamical problem of formation of the so called surface Love waves can be reduced to the following spectral problem: \[ -U''+\left(\frac{(\sqrt\mu)''}{\sqrt\mu}-\frac{\omega^2}{V_s^2(z)}\right)U=-\xi^2U, \] \[ \left.(U'-hU)\right| _{z=0}=0, \] \[ \left.U\right| _{z\to\infty}=0, \] where \(h=\left.\frac{\left(\sqrt{\mu(z)}\right)''}{2\sqrt{\mu(z)}}\right| _{z=0}\). Here, \(\mu(z)\) is the Lamé coefficient, \(V_s(z)\equiv\sqrt{\frac{\mu(z)}{\rho(z)}}\) is the velocity of the shear waves, \(\rho(z)\) is the density of the medium, \(\xi^2(\omega)\) is the spectral parameter. If \(\mu\in C^2(0,\infty)\), \(\mu(z)>0\), \(\rho(z)>0\) and \(\mu(z)\), \(\rho(z)\) are constant for \(z\) sufficiently large, it is proved that Love waves exist if and only if there exists \(z^*>0\) such that \(V_s(z^*)<V_s(\infty)\).
scientific article; zbMATH DE number 5128117