Marine Hydrodynamics/Small Amplitude Wave Theory

Introduction
Real water waves are exceedingly complex, however it is fortunate that a great number of observations can be explained on the basis of the small amplitude wave theory which makes many assumptions, but simplifies the mathematics involved considerably. The problem that is to be solved is essentially a boundary value problem. In order to solve this problem, we need to find the differential equation, and the conditions at the boundaries. We first begin by formulating the problem as a boundary value problem.

Differential equation
We are given that water waves form in some water body. What we need to find out are the essential properties of these waves and what parameters govern them. In short we need to develop a theory of water waves. The first step towards this end will be to find out what properties hold good in the water body while there are waves on its free surface.

Let us assume that the fluid is incompressible and only irrotational motion takes place. Basic hydrodynamics tells us that a velocity potential will exist which will satisfy the continuity equation. Therefore, the following equation will hold.


 * $$\nabla^2 \phi = 0$$

In three dimensions, this is the Laplace equation and is written as


 * $$\frac{\partial ^2 \phi}{\partial x^2} + \frac{\partial ^2 \phi}{\partial y^2} + \frac{\partial ^2 \phi}{\partial z^2} = 0$$

The Laplace equation is linear and therefore any solutions can be combined linearly to yield more solutions. Furthermore, there are many solutions to the equation and not all are valid for our problem. We will pick our solution depending on the boundary conditions.

Bottom Boundary Conditions
At the bottom, we basically see that the kinematic conditions must be satisfied. What this means is that at any interface, there must be no flow across the interface. The fluid cannot flow across the bottom. It is this limitation that we need to express in the form of an equation to form the bottom boundary condition. If we imagine that we are fixed with respect to the surface, the total derivative of the water surface will be zero, since we move with the surface. Thus on any surface,


 * $$\frac{DF(x,y,z,t)}{Dt}=0$$

Simplifying this equation leads us to the result that


 * $$ \vec{u}\cdot\vec{n}=\frac{\partial F/ \partial t}{|\nabla F|}$$ on the bottom surface.

Let us now imagine that we have a bottom described by $$z = h(x,y)$$. The above formulation if the condition leads us to the simple result that


 * $$u\frac{\partial h}{\partial x} + v\frac{\partial h}{\partial y} + w = 0$$

Free Surface Boundary Conditions
As far as the free surface conditions are concerned, there are two basic conditions that must be met. First, the kinematic condition must be taken up as described in the case of the development of the bottom boundary condition and then we also need to take up the dynamic free surface condition.

Kinematic Free Surface Condition
As described above, the kinematic condition is given by


 * $$\frac{DF(x,y,z,t)}{Dt}=w$$

If the surface is defined by $$z = \eta(x,y,t)$$, then


 * $$w = \left. \frac{\partial \eta}{\partial t} + u\frac{\partial \eta}{\partial x} + v\frac{\partial \eta}{\partial y} \right\|_{z=\eta(x,y,t)}$$

Dynamic Free Surface Condition
The water surface is not a fixed surface, nor is it a surface that can resist pressure variations. Thus the pressure on the surface is taken constant and the following equation holds on the surface.


 * $$-\frac{\partial \phi}{\partial t} + \frac{1}{2} (u^2 + v^2 + w^2) +gz = 0 $$

Lateral Conditions
These conditions depend on the specifics of the problem. It may require modeling of a beach or a wave maker. We will ignore this part of the problem for now and do an analysis having no lateral limits. We will study some lateral conditions later.

Other Conditions
Apart from the aforementioned boundary conditions, from the physical nature of the problem we see that the solution must be periodic in both space and time.

Solution
If we are looking for solutions that are waves, assume that the surface is $$ \eta(x,y,t) = a \cos(kx + ly - \omega t) $$. Given this choice of $$\eta$$, the boundary conditions suggest that the potential should be in the form


 * $$ \phi(x,y,z,t) = f(z) \sin(kx+ly-\omega t) $$

After applying the dynamic free surface boundary condition, we arrive at the dispersion relation


 * $$ \omega^2 = gk \tanh(kH) $$

or equivalently,


 * $$ c = \sqrt{g/k \tanh(kH)} $$

Special Cases
Shallow Water Limit

Deep Water Limit