I'm supposed to find the equations and boundary conditions for the velocity potential. The only things which I have given are that we have a stream of depth h, which flows at a constant speed U in the direction of x.
A hint was given with: pay particular attention to the choice of the value of the integration constant in Bernoulli’s equation
Therefore I'm pretty sure that I'm supposed to integrate Bernoulli's equation:
${\phi}_{t}+\frac{1}{2}|\nabla\phi|^2 +\frac{p}{\rho}+ gz= f(t)$
Honestly, I don't even know where to start. I'm not sure if I should treat it as a differential equation and if yes how.