# Velocity potential

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

Φ is known as a **velocity potential** for **u**.

A velocity potential is not unique. If Φ is a velocity potential, then *Φ* + *a*(*t*) is also a velocity potential for **u**, where *a*(*t*) is a scalar function of time and can be constant. 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 ∇ × (∇ × **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[edit]

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

Solving the wave equation for either p field or **u** field does not necessarily provide a simple answer for the other field. On the other hand, when Φ is solved for, not only is **u** found as given above, but p is also easily found—from the (linearised) Bernoulli equation for irrotational and unsteady flow—as

## Notes[edit]

**^**Anderson, John (1998).*A History of Aerodynamics*. Cambridge University Press. ISBN 978-0521669559.^{[page needed]}**^**Pierce, A. D. (1994).*Acoustics: An Introduction to Its Physical Principles and Applications*. Acoustical Society of America. ISBN 978-0883186121.^{[page needed]}

## See also[edit]

## External links[edit]

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