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