# Velocity potential

{{ safesubst:#invoke:Unsubst||$N=Refimprove |date=__DATE__ |$B= {{#invoke:Message box|ambox}} }} A velocity potential is a scalar potential used in potential flow theory. It was introduced by Joseph-Louis Lagrange in 1788.

It is used in continuum mechanics, when a continuum occupies a simply-connected region and is irrotational. In such a case,

$\nabla \times \mathbf {u} =0,$ $\mathbf {u} =\nabla \Phi \;$ ,

A velocity potential is not unique. If $a\;$ is a constant, or a function solely of the temporal variable, then $\Phi +a(t)\;$ is also a velocity potential for $\mathbf {u} \;$ . Conversely, if $\Psi \;$ is a velocity potential for $\mathbf {u} \;$ then $\Psi =\Phi +b\;$ for some constant, or a function solely of the temporal variable $b(t)\;$ . In other words, velocity potentials are unique up to a constant, or a function solely of the temporal variable.

If a velocity potential satisfies Laplace equation, the flow is incompressible ; one can check this statement by, for instance, developing $\nabla \times (\nabla \times \mathbf {u} )$ and using, thanks to the Clairaut-Schwarz's theorem, the commutation between the gradient and the laplacian operators.

Unlike a stream function, a velocity potential can exist in three-dimensional flow.

## Usage in Acoustics

$\nabla ^{2}\Phi -{1 \over c^{2}}{\partial ^{2}\Phi \over \partial t^{2}}=0$ $p=-\rho {\partial \over \partial t}\Phi$ .