# Why does the speed of a wave travelling in shallow water increase in deep water?

Exactly as the title says.

Why does the speed of a wave travelling in shallow water increase in deep water ?

• Does this help? en.wikipedia.org/wiki/Waves_and_shallow_water – Farcher Sep 9 '17 at 10:31
• How much math do you allow? – user154997 Sep 11 '17 at 10:34
• @Luc J. Bourhis enough to make it clear – MSh Sep 11 '17 at 11:05

Irrotational inviscid linear surface gravity waves have the following phase velocity:

$$c=\sqrt{\frac{g}{k}\tanh kh},$$ where $c$ is the phase velocity, $g$ the acceleration due to gravity, $k$ the wavenumber and $h$ the water depth.

For fixed $k$ then we note that $\tanh hk$ monotonically increases, hence the speed is greater for larger $h$.

Note, this is missing some of what's going on, as the wavelength changes as waves change water depth (but the frequency in the absence of breaking, forcing, wave current interactions or nonlinear interactions is conserved). Perhaps a more interesting limit is what happens when waves have very large wavelengths (like tsunamis) so that $k$ is small and $\tanh kh \approx kh$ and the phase velocity goes as $\sqrt{gh}$ (i.e. shallow water waves are non dispersive - the phase velocity does not depend on the wavelength). In the middle of the ocean the water depth is large, so these waves can travel at speeds of around 500 mph.

More details can be found, for instance, here: What determines the speed of waves in water?

Simpler answer. Assuming no energy loss, conservation of energy. Wave height stores the energy as potential energy. As a wave enters deeper water the height and potential energy decrease. Therefore the speed of the wave must increase. Potential energy converted to kinetic energy.

A great answer was given regarding the phase velocity of the wave so I'm not going to delve more into that part.

I think though, that due to the nature of the waves we should take into account friction inside the body of water. Specifically, I believe that small water depth plays the role of a water breaker, slowing down significantly the speed of the waves. The effects of that are present at larger water depths as well but are minuscule, and do not make a difference on the wave speed at the surface.

Here is a rather simple answer that requires basic relations only.

$$c=\frac{\lambda}{T}$$

where $c$ is the speed of the wave, $\lambda$ is the wavelength of the wave, and T the wave period.

We also have $$\frac{\lambda^2}{gh}=T^2$$

where $g$ is the gravitational constant, and $h$ the height of the wave. In deep water, $h$ is greater, so $T$ is lower, and this would mean $c$ in the previous is higher.

I think it is related with pressure. With increasing depth (deep water), pressure increases, so the force with which it moves must increase as pressure is directly proportional to force. Here force is the velocity. According to equation, velocity is equal to frequency multiplied with the wavelength. Therefore its wavelength increases, with increased velocity. Conversly is shallow water, the pressure decreases due to less depth, velocity decreases, hence its wavelength decreases.