How do I solve for w and p in an incompressible flow using Euler's equation?

[Yalldoor]

I'm stumped on a HW question that I just can't seem to proceed on.

Homework Statement

An incompressible ( rho = constant ) flow in 2 dimensions [x = (x,z)], with F = (0,-g), satisfies Euler's equation. For this flow, the velocity is u = (u₀,w(x)), where u₀ is a constant, with w = 0 on x = 0 and p = p₀ on z = 0. Find the solution for w and p, and show that it contains one free parameter.

The Attempt at a Solution

I've managed to get p(z) = p₀ - rho*g*z, though I don't really know where to go from there or if I've even done it right to begin with.

Any guidance would be much appreciated, thankyou.

 

[HallsofIvy]

Perhaps you should start by actually writing down Euler's equation and trying to solve it.

 

[Yalldoor]

That's what I did do and ended up with that p(z) I stated.

Anyhow, I've managed to make headway now. Thanks for your help.