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The type of wave motion which most people are familiar with are waves that occur on the free surface of water. For example, the ripples that occur when a small rock is dropped into the water or the waves that can be seen breaking on a beach (Pinery Provincial Park on Lake Huron, below).

Pinery Provincial Park on Lake Huron

In this type of wave motion the restoring force is gravity (sometimes surface tension needs to be considered as well, and in some situations surface tension can dominate gravity) and therefore these waves are known as surface gravity waves. We would like to create a mathematical formulation of this problem. To begin we will make some simplifying assumptions:
  1. Inviscid: nu = 0

  2. Constant density: rho = const => grad u = 0

  3. Irrotational: vortcity = 0

  4. The mean depth of the fluid is constant: H = const

  5. The fluid is invariant with y (the in-page direction): derivative with respect to y -> 0
Surface wave diagram
 
The displacement of the free surface will be denoted by,
 
eq1
 
Thus, our surface is found at,
 
eq2
 
From the irrotational assumption we can deduce that,
 
eq3
 
Then using our incompressibility (constant density) assumption we get our first governing equation,
 
eq4
 
 

To get the remainder of the equations we now need to consider some boundary conditions. First we will look at the bottom boundary. The bottom is solid and hence there is no flow normal to this boundary. Mathematically this gives us w=0 at z =-H , assuming we have a flat bottom. This is the kinematic bottom condition.

We have considered the bottom boundary so next we will look at the surface. At the surface we want to ensure that particles that start off on the surface always remain on the surface. This condition is known as the kinematic surface condition. Following a particle which is on the surface the rate of change of the difference of the z position and the displacement of the surface is zero,


eq5

Mathematically, it gives the following condition, note that this condition is nonlinear however it does assume there is no wave breaking:

eq6

The final boundary condition is the balance of forces on at the free surface. Since the fluid is inviscid, the forces at the surface are the pressures above and below the surface (a surface force from the stress tensor) and the surface tension (a line force that occurs only at the surface, and is chemical in nature).

Surface tension diagram

After some work the balance of forces gives us the following equation for pressure at the surface:

eq7

The atmospheric pressure is often taken to be approximately zero (relative to the pressure in the water). Since, we have inviscid and irrotational flow we can use Bernoulli's Equation,

eq8

and without loss of generality we can take B(t) = 0 this is a tricky point that you should think about) which gives our last boundary condition, the dynamic surface condition.

eq9

To summarize what we have deduced so far:

eq10

The above problem is fully nonlinear and still remains unsolved. The next thing to consider are waves that have a relatively small amplitude, a << 1. For this case we will set,

eq11

and,

eq12

Since, a​2​ << a we can use this to linearize our problem, to first order this gives us the linearized wave equations:

eq13

Solving the linearized wave equation

To solve the linearized wave equation we start by looking for normal mode solutions of the from,

eq14

It is sufficient to only consider solutions of this form since any continuous function can be written as a sum of sines and cosines.

From this starting point and using the kinematic boundary condition at the bottom and the dynamic boundary condition at the surface one can deduce that we should look for solutions of the following form,

eq15

Using Laplace's, equation and the kinematic boundary conditions we can determine that our solution takes on the form:
 
eq16
 
Now using the dynamic boundary condition at the free surface we determine a dispersion relation for our problem,
 
eq17
 
This gives us a relation between the frequency, omega and the wavenumber, k. Also, from this we can determine the phase speed, c, which is the ratio of the frequency to the wavenumber.
 
eq18

Special limits

Looking at the dispersion relation we deduce some special limits and look at the types of waves these limits produce. Since surface tension only shows up in one of the terms we can look at the weighting of the gravity term, gk, versus the surface tension term, sigma k​3​ / p.

If surface tension dominates over gravity, i.e.. sigma >> rho g / k2​ , then we have waves whose restoring force is mainly due to surface tension. These waves are known as capillary waves.

Under normal situations when do capillary waves occur?

At an air water interface at temperatures of about 20 degrees Celsius we have the following values for our parameters, sigma = 0.0728 N/m and p = 1000 kg/m3​. Which gives k​2​ >> 1.35x105​ m-2​  for capillary waves or lambda << 1.7cm ~ 2cm. This tells us for waves with wave length much less then 2cm gravity is negligible and surface tension is the dominating force, giving us capillary waves.

Now if we assume that we have waves with wavelength much longer than 2cm we can neglect surface tension, giving the following dispersion relation:

eq18

If kH << 1 then we have waves with a long wave length relative to the depth of the water, this is known as the shallow water limit (or sometimes long gravity waves). In this case the dispersion relationship can be approximated by w​2​ = gHk​2​ which gives c = sqrt(gH) for the phase speed. Note that the phase speed is independent of the wave length, this makes these waves non-dispersive. The picture below shows the pathlines of particles to a first order approximation.
 
p5
 
If kH >> 1 the wavelength is short relative to the depth of the water. This limit is known as the deep water limit (or sometimes referred to as short gravity waves). The dispersion relation and the phase speed can be written as follows, w​2​ = gk and c = sqrt (g/k) The propagation speed of the waves in this case depends on the wavelength, making waves in this limit dispersive. The picture below shows the pathlines of particles to a first order approximation.
 
p6

Stokes' Drift

When looking at waves at the beach you can notice that the waves do not transport things (like seaweed of drift wood) at the same speed at which the crests and troughs of the waves propagate. However, objects still appear to be moving in the direction of wave propagation, albeit a lot slower. This effect is called Stokes' Drift. It is due to second order (nonlinear) effects. Looking at the pathlines of a particle one can see that the pathline for one period is not a closed loop nor does it follow the shape of the wave.

Stokes particle

Wave refraction

Another phenomena that can be noticed when at the beach is that waves appear to arrive parallel to the shore. This phenomena is known as wave refraction. Assume we have a depth that decreases as we approach the shore, i.e. H = H(x). We can now look at how this affects the phase speed of our gravity wave in shallow water:

eq19

From this relation we can see that the phase speed of the waves decreases as the depth decreases. This will cause waves approaching the shore at an angle to rotate parallel to the shore as they approach.

Group velocity

Group velocity is a concept that is initially quite difficult to understand. Mathematically, it is defined as,

eq20

Physically, it is the speed at which energy travels. If there is no energy then no waves can be present and thus the group speed tells us how far a wave packet propagates in a given amount of time. Using the above definition for the group velocity, we come up with the following expression for the group velocity in surface gravity waves:

eq21