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Is it true for shallow waves that the speed is related to gravity and depth by $c^2 = gh$? This cannot be true for all $h$ can it? What are the constraints and assumptions of this equation, apart from the obvious conservation of mass and incompressibility of fluids. Does the size of the container of the fluid make a difference. How can we assign a reasonable range for $h$ given other variables?

Is there some other equation to use for great depths?

Qmechanic
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The dispersion relation for gravity waves is given by ([https://en.wikipedia.org/wiki/Dispersion_(water_waves)][Wiki]) $$ \omega=\sqrt{gk\tanh (kh)}. $$

Recall that group velocity is defined by $v_g=\frac{\partial \omega}{\partial k}$. So, if we assume that $h$ is small, i.e. $\tanh(kh) \simeq kh$ then $v_g\simeq gh$. So, for small $h$ this is true.

ZeroTheHero
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Timasq
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