# 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,

${\displaystyle \nabla \times \mathbf {u} =0\,,}$

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

${\displaystyle \mathbf {u} =\nabla \Phi \ ={\frac {\partial \Phi }{\partial x}}\mathbf {i} +{\frac {\partial \Phi }{\partial y}}\mathbf {j} +{\frac {\partial \Phi }{\partial z}}\mathbf {k} \,.}$

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.

${\displaystyle \nabla ^{2}\Phi -{\frac {1}{c^{2}}}{\frac {\partial ^{2}\Phi }{\partial t^{2}}}=0}$

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

${\displaystyle p=-\rho {\frac {\partial \Phi }{\partial t}}\,.}$

## Notes

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