Surface Gravity Waves
Contents
Surface Gravity Waves¶
We will open with waves on the surface of a fluid, bounded from above by another fluid (think the ocean and atmosphere), in a small-amplitude regime so that we can linearize the equations of motion about a static configuration with a flat interface. As with any physical small oscillations, waves arise from a restoring force. For surface waves the sources of a restoring force are gravity and surface tension. Waves whose dynamics are dominated by gravity are called surface gravity waves, while those with dynamics dominated by surface tension are called capillary waves.
(These are distinct from gravitational waves of the kind measured by LIGO!)
Setup in 2 dimensions¶
For simplicity we will work in 2 dimensions corresponding to a horizontal and a vertical direction. We start by considering a fluid of constant density and finite depth, as shown here:
We will take the atmospheric pressure to be constant at the fluid interface (pretty good approximation for
ocean waves). We consider small amplitude waves about the basic state
We will assume that the perturbations are in some sense small, in particular that
We are going to look for solutions which oscillate in
Laplace’s equation then gives us
Next, we impose the boundary condition at
There are two obvious limits here. In the first,
Another limit is “shallow water”, in which
Solutions and their properties¶
Given the above, we have found that oscillatory solutions take the form:
where
Note the nontrivial phase relations. At the crests (at fixed
With this we can test he validity of the linear regime. This will hold when the advective term
In other words
Lagrangian picture¶
It is worth understanding what individual particles are doing in this flow. For this, we would like to solve the equations:
These are horribly nonlinear, even though the underlying equations are linear. However, we are
assuming
Now expanding the Lagrangian particle equations in powers of
With a bit of algebra, we find that to this order,
This is the equation for an ellipse. In the deep water limit, with
Thus, even though the=is “monochromatic wave” wave may be moving at the phase velocity, to lowest order the particles are just moving in circles. which get smaller at depth.
It is worth at least mentioning the effects of the next order corrections. At this point the above equations are
not in fact technically adequate as at order
together with the boundary conditions. We would expand
Since \({\vec v^{(0)} = 0\) we have the same equation as for
At the surface we have
Taking the time derivative of this equation, and imposing the boundary conditions, we get
Now let us work in the deep water limit, replacing the hyberbolic funvtions in
This still leaves us with solving the equations for
and similarly for
So we get a constant drift at this order, known as the Stokes drift, which is extremely important in understanding transport of particles by waves.