Shallow Depth occurs when the wavelength is much longer than the water depth (which is assumed constant). It removes the depth dependence in the equations of motion. The theory is sometimes referred to as Long Wave Theory.
Derivation of the Linear Equations
The linear shallow water equations can be derived by a Taylor series expansion about the bottom surface. The equations for finite depth in the time domain are
where is the velocity potential and is the surface displacement.
Since the depth is shallow we can perform a Taylor series expansion about the potential at the bottom surface and obtain (retaining only three terms)
We now see that (under this approximation)
Therefore, the change which occurs is that
This means that we are not required to solve Laplace's equation in the fluid, and leads to great simplifications.
The equations are therefore
where the potential now depends only on . In three-dimensions the formula is the same but we replace the second derivative by the Laplacian.
Equations for Variable Depth
For the case when the depth varies the equations become
It is standard to non-dimensionalise so that gravity , and the equations then become
subject to the initial conditions
where is the displacement, is the string density and is the variable depth (note that we are unifying the variable density string and the wave equation in variable depth because the mathematical treatment is identical). The equations in two dimensions are