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
where represents wave displacement (pressure, rotation, dilatation, etc.) and V the velocity of the wave. Functions are solutions to this equation.
In spherical coordinates where r is the radius, the colatitude, and the longitude, the wave equation becomes:
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
which can be written in component form as
If , this gives an S-wave; if , 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.