PX285 - J7- surface waves

moving to the frame of reference of the wave so that bernoulli's principle is applicable
from mass conservation:

\begin{matrix} u_{1} (h + \frac{ψ}{2}) = u_{2} (h - \frac{ψ}{2}) \\ (1) & \frac{ψ}{2} (u_{1} + u_{2}) = h (u_{2} - u_{1}) \end{matrix}

\begin{matrix} ρ g (h + \frac{ψ}{2}) + ρ \frac{u_{1}^{2}}{2} = ρ g (h - \frac{ψ}{2}) + ρ \frac{u_{2}^{2}}{2} \\ g ψ = \frac{u_{2}^{2} - u_{1}^{2}}{2} = \frac{(u_{2} - u_{1}) (u_{2} + u_{1})}{2} \end{matrix}

\begin{matrix} g ψ = \frac{u_{2}^{2} - u_{1}^{2}}{2} = \frac{(u_{2} + u_{1})^{2} ψ}{4 h} \\ g h = {(\frac{u_{2} + u_{1}}{2})}^{2} = v_{p h a s e}^{2} \end{matrix}

v_{p h a s e} = \sqrt{g h}