Definition:Wave Equation

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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:

$x$ denotes the displacement from the origin along the $x$-axis
$t$ denotes the time.

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 displacement 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

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