# Do water waves moving from deep water to shallow water always have higher amplitude?

I've seen many images like the following one that the height (amplitude) increases as the water becomes shallower.

However, I also found some images showing that the amplitude remains unchanged.

So I would like to ask if it is always true that when a wave in a ripple tank moves from a region of deep water to a region of shallow water, it's amplitude increases. Thanks!

• And also why does the numbers of waves increases in the shallow water than the deep water? – Sanmveg saini Feb 24 '18 at 4:25
• @Sanmvegsaini I think it's as the wave speed decreases when the water becomes shallower, the wavelength becomes shorter too (with the same frequency). As such, the number of waves increases in the shallow water than the deep water. – He Yifei 何一非 Feb 24 '18 at 5:17

You can answer this for certain cases using conservation of energy. Consider a slowly varying wave train entering shallow water of depth $h$. Let the amplitude of the waves be $a$.

Conservation of energy tells us $$\frac{\partial E}{\partial t} +\frac{\partial}{\partial x}(c_g E) =0$$

where $c_g$ is the group velocity, given by $\sqrt{gh}$ in shallow water, while $E=\frac{1}{2} g a^2$. Assume that the wave field is stationary (time invariant), then we have

$$c_g E = \sqrt{gh}\frac{g}{2} a^2 =\gamma_0$$ for some constant $\gamma_0$, which implies the wave amplitude relates to the water depth as

$$a\sim h^{-1/4}.$$

Hence, as the waves enter shallow water, the height of the waves increase.

Some of your images are outside of this asymptotic regime. For a step, like you show, some energy is reflected and some transmitted (and under some situations some remains bound to the step), and a more detailed treatment of the problem must be given.