# Velocity potential

A velocity potential is a scalar potential used in potential flow theory. It was introduced by Joseph-Louis Lagrange in 1788.[1]

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

where denotes the flow velocity. As a result, can be represented as the gradient of a scalar function :

is known as a velocity potential for .

A velocity potential is not unique. If is a constant, or a function solely of the temporal variable, then is also a velocity potential for . Conversely, if is a velocity potential for then for some constant, or a function solely of the temporal variable . 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 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

In theoretical acoustics,[2] it is often desirable to work with the acoustic wave equation of the velocity potential instead of pressure and/or particle velocity .

Solving the wave equation for either field or field doesn't necessarily provide a simple answer for the other field. On the other hand, when is solved for, not only is found as given above, but is also easily found – from the (linearised) Bernoulli equation for irrotational and unsteady flow – as

.

## Notes

1. Anderson John (1998). A History of Aerodynamics. Cambridge University Press. ISBN 0521669553.
2. Pierce A.D. (1994). Acoustics: An Introduction to Its Physical Principles and Applications. Acoustical Society of America. ISBN 0883186128.