
Velocity potential

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,
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 velocity potential, then is also a velocity potential for , where 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.
The Laplacian of a velocity potential is equal to the divergence of the corresponding flow. Hence if a velocity potential satisfies Laplace equation, the flow is incompressible.
Unlike a stream function, a velocity potential can exist in three-dimensional flow.
Usage in acoustics.
In theoretical acoustics, 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 does not 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