Three-Dimensional Wave Equation
We have already seen that the one-dimensional plane wave solution, (7.1), satisfies the one-dimensional wave equation,
![$\displaystyle \frac{\partial^{\,2}\psi}{\partial t^{\,2}} = v^{\,2}\,\frac{\partial^{\,2}\psi}{\partial x^{\,2}},$](img1174.png) |
(7.8) |
where
is the characteristic wave speed of the medium through which the wave propagates. (See Section 6.3.)
Likewise, the three-dimensional plane wave solution, (7.5), satisfies the three-dimensional
wave equation (see Exercise 1),
![$\displaystyle \frac{\partial^{\,2}\psi}{\partial t^{\,2}} = v^{\,2}\left(\frac{...
...al^{\,2}}{\partial y^{\,2}}+\frac{\partial^{\,2}}{\partial z^{\,2}}\right)\psi.$](img1846.png) |
(7.9) |