The Bernoulli equation for unsteady potential flow is used in the theory of ocean surface waves and acoustics. For an irrotational flow, the flow velocity can be described as the gradient  of a velocity potential . In that case, and for a constant density , the momentum equations of the Euler equations can be integrated to:
which is a Bernoulli equation valid also for unsteady—or time dependent—flows. Here  denotes the partial derivative of the velocity potential  with respect to time , and  is the flow speed. The function  depends only on time and not on position in the fluid. As a result, the Bernoulli equation at some moment  applies in the whole fluid domain. This is also true for the special case of a steady irrotational flow, in which case  and  are constants so equation () can be applied in every point of the fluid domain. Further  can be made equal to zero by incorporating it into the velocity potential using the transformation:
resulting in:
Note that the relation of the potential to the flow velocity is unaffected by this transformation: .
The Bernoulli equation for unsteady potential flow also appears to play a central role in Luke's variational principle, a variational description of free-surface flows using the Lagrangian mechanics.