# 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 **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

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