Three-Dimensional Wave Equation

We have already seen that the one-dimensional plane wave solution, (7.1), satisfies the one-dimensional wave equation,

$$\frac{\partial^{\,2}\psi}{\partial t^{\,2}} = v^{\,2}\,\frac{\partial^{\,2}\psi}{\partial x^{\,2}},$$ (7.8)

where $v$ 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),

$$\frac{\partial^{\,2}\psi}{\partial t^{\,2}} = v^{\,2}\left(\frac{... ...al^{\,2}}{\partial y^{\,2}}+\frac{\partial^{\,2}}{\partial z^{\,2}}\right)\psi.$$ (7.9)