Wave Equation/Examples/Harmonic Wave

Examples of Use of the Wave Equation
Let $\phi$ be a harmonic wave which is propagated along the $x$-axis in the positive direction with constant velocity $c$ and without change of shape.

From Equation of Harmonic Wave, the disturbance of $\phi$ at point $x$ and time $t$ can be expressed using the equation:
 * $(1): \quad \map \phi {x, t} = a \map \cos {2 \pi \paren {\dfrac x \lambda - \dfrac t \tau} }$

where:
 * $x$ denotes the distance from the origin along the $x$-axis
 * $t$ denotes the time
 * $\lambda$ is the wavelength of $\phi$
 * $\tau$ is the period of $\phi$.

This equation satisfies the wave equation.

Proof
The wave equation is expressible as:
 * $\dfrac 1 {c^2} \dfrac {\partial^2 \phi} {\partial t^2} = \dfrac {\partial^2 \phi} {\partial x^2} + \dfrac {\partial^2 \phi} {\partial y^2} + \dfrac {\partial^2 \phi} {\partial z^2}$

We have by partial differentiation:

and:

As $y$ and $z$ do not appear in $(1)$, the partial derivative of $(1)$ $y$ and $z$ is identically zero.

Hence we have:
 * $\dfrac {\partial^2 \phi} {\partial t^2} = c^2 \paren {\dfrac {\partial^2 \phi} {\partial x^2} + \dfrac {\partial^2 \phi} {\partial y^2} + \dfrac {\partial^2 \phi} {\partial z^2} }$

and the result follows.