Dictionary:Wave equation

An equation that relates the spatial and time dependence of a disturbance which can propagate as a wave. In rectangular coordinates x, y, z, it is

$\nabla ^{2}\psi ={\frac {\partial ^{2}\psi }{\partial x^{2}}}+{\frac {\partial ^{2}\psi }{\partial y^{2}}}+{\frac {\partial ^{2}\psi }{\partial z^{2}}}=\left({\frac {1}{V^{2}}}\right){\frac {\partial ^{2}\psi }{\partial t^{2}}}$,

where $\psi$ represents wave displacement (pressure, rotation, dilatation, etc.) and V the velocity of the wave. Functions $f(lx+my+nz\pm Vt)$ are solutions to this equation.

In spherical coordinates where r is the radius, $\theta$ the colatitude, and $\phi$the longitude, the wave equation becomes:

$\left({\frac {1}{V^{2}}}\right){\frac {\partial ^{2}\Psi }{\partial t^{2}}}=\left({\frac {1}{r^{2}}}\right)\left[\left({\frac {\partial }{\partial r}}\right)\left(r^{2}{\frac {\partial \Psi }{\partial r}}\right)+\left({\frac {1}{sin{\theta }}}\right)\left({\frac {\partial }{\partial \theta }}\right)\left(sin{\theta }{\frac {\partial \Psi }{\partial \theta }}\right)+\left({\frac {1}{sin^{2}{\theta }}}\right){\frac {\partial ^{2}\Psi }{\partial \phi ^{2}}}\right]$

The foregoing are forms of the scalar wave equation These forms do not provide for the conversion of P-waves to S-waves nor vice-versa.

The vector wave equation is more general; for isotropic media it is

$\left(2\mu +\lambda \right)\nabla (\nabla \cdot \psi )-\mu \nabla \times (\nabla \times \psi )=\rho {\frac {\partial ^{2}\psi }{\partial t^{2}}}$,

which can be written in component form as

$\mu \nabla ^{2}\Psi _{x}+(\mu +\lambda ){\frac {\partial }{\partial x}}\left({\frac {\partial \Psi _{x}}{\partial x}}+{\frac {\partial \Psi _{y}}{\partial y}}+{\frac {\partial \Psi _{z}}{\partial z}}\right)=\rho {\frac {\partial ^{2}\Psi _{x}}{\partial t^{2}}}$.

If $\nabla \cdot \Psi =0$, this gives an S-wave; if $\nabla \times \Psi =0$, a P-wave. The wave equation in polar anisotropic (transversely isotropic) media is given in Figure T-13.

For a derivation, and the relation of the wave equation to the equation of motion, see the main page: Wave equation.