# Definition:Wave Equation

## Equation

The **wave equation** is a second order PDE of the form:

\(\ds \dfrac 1 {c^2} \dfrac {\partial^2 \phi} {\partial t^2}\) | \(=\) | \(\ds \dfrac {\partial^2 \phi} {\partial x^2} + \dfrac {\partial^2 \phi} {\partial y^2} + \dfrac {\partial^2 \phi} {\partial z^2}\) | ||||||||||||

\(\ds \) | \(=\) | \(\ds \nabla^2 \phi\) | where $\nabla^2$ is the Laplacian operator |

## Also presented as

The **wave equation** is also seen presented in the form:

\(\ds \dfrac {\partial^2 \phi} {\partial t^2}\) | \(=\) | \(\ds c^2 \paren {\dfrac {\partial^2 \phi} {\partial x^2} + \dfrac {\partial^2 \phi} {\partial y^2} + \dfrac {\partial^2 \phi} {\partial z^2} }\) | ||||||||||||

\(\ds \) | \(=\) | \(\ds c^2 \nabla^2 \phi\) | where $\nabla^2$ is the Laplacian operator |

## Also known as

The **wave equation** is referred to as the **equation of wave motion** by some authors, ostensibly so as not to confuse it with certain entities in the field of quantum physics.

However, this confusion may not in fact be actual.

## Examples

### Wave with Constant Velocity

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

From Equation of Wave with Constant Velocity, the **disturbance** of $\phi$ at point $x$ and time $t$ can be expressed using the equation:

- $(1): \quad \map \phi {x, t} = \map f {x - c t}$

where:

This equation satisfies the wave equation.

### Harmonic Wave

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.

## Also see

- Results about
**the wave equation**can be found**here**.

## Sources

- 1955: C.A. Coulson:
*Waves*(7th ed.) ... (previous) ... (next): Chapter $\text {I}$: The Equation of Wave Motion: $\S 5$: $(14)$ - 1972: George F. Simmons:
*Differential Equations*... (previous) ... (next): $1$: The Nature of Differential Equations: $\S 1$: Introduction - 1989: Ephraim J. Borowski and Jonathan M. Borwein:
*Dictionary of Mathematics*... (previous) ... (next):**wave equation**