Clarification on Shallow Water Wave Equation

[poilop]

I know that we can find the speed of the wave in shallow water by:
c^2 = gh
but how do we derive it?

 

[physicsworks]

We derive this using Bernoulli theorem and continuity equation.
In the reference frame which is moving along with the wave:
Bernoulli theorem:
\frac{V^2}{2}+gh=\frac{(V-\delta V)}{2}+g(h+a)
continuity equation:
Vh=(V-\delta V)(h+a)
where V is the speed of the wave, \delta V is a drop of the speed in the water where its level grows from the normal h to h+a
We suppose that h<\lambda where \lambda is a wavelength.
Form the second equation one has
h \delta V = a V (*)
(a \delta V is very very small). Then from the fiirst we get [(\delta V)^2 is also very small, so we ignore it]:
V \delta V = ga and with (*) one has
V^2=gh

 

[poilop]

awesome,
thank you