Love wave

Basic theory

The conservation of linear momentum of a linear elastic material can be written as:

${\displaystyle \mathbf{\nabla }\cdot (\mathsf{C}:\mathbf{\nabla }\mathbf{u})=\rho \ddot{\mathbf{u}}}$

where ${\displaystyle \mathbf{u}}$ is the displacement vector and ${\displaystyle \mathsf{C}}$ is the stiffness tensor. Love waves are a special solution (${\displaystyle \mathbf{u}}$) that satisfy this system of equations. We typically use a Cartesian coordinate system (${\displaystyle x,y,z}$) to describe Love waves.

Consider an isotropic linear elastic medium in which the elastic properties are functions of only the ${\displaystyle z}$ coordinate, i.e., the Lamé parameters and the mass density can be expressed as ${\displaystyle \lambda (z),\mu (z),\rho (z)}$. Displacements ${\displaystyle (u,v,w)}$ produced by Love waves as a function of time (${\displaystyle t}$) have the form

${\displaystyle u(x,y,z,t)=0,v(x,y,z,t)=\hat{v}(x,z,t),w(x,y,z,t)=0.}$

These are therefore antiplane shear waves perpendicular to the ${\displaystyle (x,z)}$ plane. The function ${\displaystyle \hat{v}(x,z,t)}$ can be expressed as the superposition of harmonic waves with varying wave numbers (${\displaystyle k}$) and frequencies (${\displaystyle \omega }$). Consider a single harmonic wave, i.e.,

${\displaystyle \hat{v}(x,z,t)=V(k,z,\omega )\exp [i(kx-\omega t)]}$

where ${\displaystyle i}$ is the imaginary unit, i.e. ${\displaystyle i^2=-1}$. The stresses caused by these displacements are

${\displaystyle {\sigma }_{xx}=0,{\sigma }_{yy}=0,{\sigma }_{zz}=0,{\tau }_{zx}=0,{\tau }_{yz}=\mu (z)\frac{dV}{dz}\exp [i(kx-\omega t)],{\tau }_{xy}=ik\mu (z)V(k,z,\omega )\exp [i(kx-\omega t)].}$

If we substitute the assumed displacements into the equations for the conservation of momentum, we get a simplified equation

${\displaystyle \frac{d}{dz}\left[\mu (z)\frac{dV}{dz}\right]=[k^2\mu (z)-{\omega }^2\rho (z)]V(k,z,\omega ).}$

The boundary conditions for a Love wave are that the surface tractions at the free surface ${\displaystyle (z=0)}$ must be zero. Another requirement is that the stress component ${\displaystyle {\tau }_{yz}}$ in a layer medium must be continuous at the interfaces of the layers. To convert the second order differential equation in ${\displaystyle V}$ into two first order equations, we express this stress component in the form

${\displaystyle {\tau }_{yz}=T(k,z,\omega )\exp [i(kx-\omega t)]}$

to get the first order conservation of momentum equations

${\displaystyle \frac{d}{dz}\left[\begin{array}{c}V\\ T\end{array}\right]=\left[\begin{array}{cc}0& 1/\mu (z)\\ k^2\mu (z)-{\omega }^2\rho (z)& 0\end{array}\right]\left[\begin{array}{c}V\\ T\end{array}\right].}$

The above equations describe an eigenvalue problem whose solution eigenfunctions can be found by a number of numerical methods. Another common, and powerful, approach is the propagator matrix method (also called the matricant approach).