# Dictionary:Wave equation

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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

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

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

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

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

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

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

which can be written in component form as

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

If ${\displaystyle \nabla \cdot \Psi =0}$, this gives an S-wave; if ${\displaystyle \nabla \times \Psi =0}$, 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.