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 + f(t) is also a velocity potential for u, where f(t) is a scalar function of time and can be constant.
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.
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.
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