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Basic solution of the elastic wave equation

The elastic wave equation is

\begin{displaymath}\rho \vec{\ddot{u}}=\nabla P[\vec{u}\/]
\end{displaymath} (1)

where $\vec{u}$ is the particle displacement and P the stress tensor calculated from $\vec{u}$ and the elastic moduli. As is well known, the displacement can be decomposed into a rotation-free and a divergence-free part:

\begin{displaymath}\vec{u}=\nabla \Phi + \nabla \times \vec{A}
\end{displaymath} (2)

where $\Phi$ is the ``compressional'' potential and $\vec{A}$ the ``shear'' potential. $\Phi$ and $\vec{A}$ are uncoupled in an unlimited homogeneous medium. Purely radial displacements such as produced by a spherical source in a homogeneous medium with the P-wave velocity c can be described by $\Phi$ alone:

\begin{displaymath}\vec{u}=\nabla \Phi \qquad \textrm{with} \qquad \Phi(r,t)=-\frac{1}{r} \, f(t-\frac{r}{c})
\end{displaymath} (3)

From this we get the radial displacement

 \begin{displaymath}u_r=\frac{1}{r^2} \, f(t-\frac{r}{c}) + \frac{1}{rc} \,f'(t-\frac{r}{c})
\end{displaymath} (4)

Erhard Wielandt