On the criteria of existence of eigenfrequencies corresponding to Love waves (Q868020)

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 \(V_s(z^*)<V_s(\infty)\).