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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.
If a velocity potential satisfies Laplace equation, the flow is incompressible; one can check this statement by, for instance, developing ∇ × (∇ × 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
Solving the wave equation for either p field or u field does not necessarily provide a simple answer for the other field. 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
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