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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 fluid dynamics, when a fluid occupies a simply-connected region and is irrotational. In such a case,
 * $$\nabla \times \mathbf{u} =0,$$

where $$ \mathbf{u} $$ denotes the flow velocity of the fluid. As a result, $$ \mathbf{u} $$ can be represented as the gradient of a scalar function $$\Phi\;$$:
 * $$ \mathbf{u} = \nabla \Phi\;$$,

$$\Phi\;$$ is known as a velocity potential for $$\mathbf{u}$$.

A velocity potential is not unique. If $$a\;$$ is a constant, or a function solely of the temporal variable, then $$\Phi+a(t)\;$$ is also a velocity potential for $$\mathbf{u}\;$$. Conversely, if $$\Psi\;$$ is a velocity potential for $$\mathbf{u}\;$$ then $$\Psi=\Phi+b\;$$ for some constant, or a function solely of the temporal variable $$b(t)\;$$. 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 $$\nabla \times (\nabla \times \mathbf{u}) $$ 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
In theoretical acoustics, it is often desirable to work with the acoustic wave equation of the velocity potential $$\Phi\;$$ instead of pressure $$p\;$$ and/or particle velocity $$\mathbf{u}\;$$.
 * $$ \nabla ^2 \Phi - {1 \over c^2} { \partial^2 \Phi \over  \partial t ^2 } = 0  $$

Solving the wave equation for either $$p\;$$ field or $$\mathbf{u}\;$$ field doesn't necessarily provide a simple answer for the other field. On the other hand, when $$\Phi\;$$ is solved for, not only is $$\mathbf{u}\;$$ found as given above, but $$p\;$$ is also easily found as
 * $$ p = -\rho {\partial \over \partial t}\Phi $$.