Why do waves slow down in shallow water?

Many surfers, and watermen in general, have the notion that waves slow down as they approach coastal water. This process is crucial in wave dynamics especially because it plays a big role in wave breaking. Remember that the wave period remains constant along wave propagation in coastal water. Being the wave celerity simply the ratio between wavelength and period, as the celerity decreases, wavelength must decrease as well.

This decrease in wave speed and wavelength leads to an increase of the wave steepness and eventually wave breaking, as the wave profile turns unstable in shallow water. Although the notion of the change in celerity is quite well-known, the reason why this happens is still misunderstood. Moreover, some information found nowadays on internet websites is misleading. This post tries to clarify this concept with no (or minimal) use of equations.

False notions

• First false notion to dismantle: the bottom of the wave slows down more than the top of the wave. Not true. I see that lots of books and websites are misleading about this. Outside the surf zone, velocity within the water column (I mean vertical water column) is in phase with and proportional to the surface elevation. This means that under a wave crest we find the peak of onshore velocity and under the wave trough the peak of offshore velocity, at any water depth. Including the bottom.
• Second false notion to dismantle: waves slow down because of bottom friction that eventually causes wave breaking in shallow water. Completely wrong. A water waves is essentially an oscillatory motion in which inertial and pressure forces balance themselves. The equation of motion of water waves indicates that inertial (read velocity) and pressure forces are order of magnitude larger than friction. That is why we can neglect friction locally. Bottom friction can dissipate wave energy only when it act over a long distance of many wavelengths. But it cannot slow down a wave and certainly it plays no role in wave breaking.

Waves slow down as they approach the shoreline. Photo credits: Antonio Muglia

Dispersion relationship

So the question remains: why do waves slow down in shallow water? If you ask to experts, they will probably reply: “Because dispersion relation says so!”. Dispersion relation proceeds from the equation of mass and momentum conservation with some assumptions and boundary conditions that I’m not going to list here. I want just to mention that one assumption is the absence of viscosity, meaning that friction plays no role. Dispersion relation links wave celerity with water depth and wave period. And in fact it shows that as waves propagate into shallower depths, their celerity decreases.

The dispersion relation is part of the so-called linear wave theory (first derived in the 19th century) that describes a wide range of wave processes. Even modern wave models used for wave forecast nowadays rely on many linear wave concepts. This is also the case of the WaveWatchIII model that is operational at swellbeat.com. But what is actually hidden behind the linear wave theory and its dispersion relation regarding this change in wave celerity?

It’s all about balance

To answer to this question in a clear and undemanding way, I will try to evaluate inertial and pressure forces in shallow water. The conservation of momentum states that inertial and pressure forces must balance themselves, they are equal. Pressure forces are given by 0.5\rho g h^2, where \rho is the water density, g is the gravity acceleration and h is the depth. Inertial forces are equal to 0.5\rho h c^2 , being c the celerity. From here you get the well-known velocity propagation of a perturbation in shallow water which is c=\sqrt{gh}. So, being the wave a perturbation, its celerity decreases as water depth decreases, not because of friction, but because inertial forces must balance pressure forces. In other words, it is the reduction of the pressure force as waves approach the shore that ultimately causes the reduction of wave celerity.