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
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