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,

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

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

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.

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,^[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.

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

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