I know that we can find the speed of the wave in shallow water by:
c^2 = gh
but how do we derive it?
We derive this using Bernoulli theorem and continuity equation.
In the reference frame which is moving along with the wave:
where [tex]V[/tex] is the speed of the wave, [tex]\delta V[/tex] is a drop of the speed in the water where its level grows from the normal [tex]h[/tex] to [tex]h+a[/tex]
We suppose that [tex]h<\lambda[/tex] where [tex]\lambda[/tex] is a wavelength.
Form the second equation one has
[tex]h \delta V = a V[/tex] (*)
([tex]a \delta V[/tex] is very very small). Then from the fiirst we get [[tex](\delta V)^2[/tex] is also very small, so we ignore it]:
[tex]V \delta V = ga[/tex] and with (*) one has
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