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.
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.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 Bernoulli equation for irrotational and unsteady flow – as
