Stokes wave theory

Assume that an infinite series of plane, uniform waves travels through the fluid in the positive S-direction. The z-coordinate is chosen to be positive in the vertical direction, so the gravity potential is $G=g\left(z-z_0\right)$, where $z_0$ is an arbitrary datum.

Assume that the fluid is inviscid and incompressible. The fluid particle velocities are derivable from a flow potential

Equilibrium is

where $\rho$ is the fluid density and p is the pressure. Writing $\partial \mathbf{v}/\partial t$ in terms of the flow potential then gives

Integrating with respect to $\mathbf{x}$ (note that $\rho$ is constant since the fluid is incompressible) gives the Bernoulli equation

where $F\left(t\right)$ is an arbitrary function (which for convenience is set to zero) and $p_0$ is the atmospheric pressure. Substituting in the gravity potential, this is

where $z_s$ is the undisturbed surface level. From this equation the total pressure at a point below the instantaneous fluid surface is

Hence, the total pressure is the air pressure plus the hydrostatic pressure plus the dynamic pressure, $p_{dyn}$, where $p_{dyn}$ is given by

Let $\eta$ be the elevation of the free surface above this level. At the free surface the Bernoulli equation is

assuming the pressure at the surface is negligible.

Assuming the waves are uniform, of wavelength $\lambda$ and period $\tau$, and that they travel in the positive S-direction means that the solution as a function of S and t must appear in terms of a phase angle

where $\overline{c}=\frac{\lambda }{\tau }$ is the wave celerity. This means that, for any function in the solution,

Thus, at the free surface boundary

and the Bernoulli equation at the free surface is

or

A further boundary condition at the free surface is that the fluid particle velocity relative to the wave celerity must be tangential to the slope of the wave:

At the seabed $\left(z=z_b\right)$, there is no fluid motion in the vertical direction:

The problem now consists of finding a potential function, $\varphi$, that satisfies Equation 1—the boundary condition at the seabed—as well as the boundary conditions at the surface—Equation 2 and Equation 3.

Stokes proposed a power series solution to this problem, and Skjelbreia and Hendrickson (1960) have obtained that solution to fifth-order. The potential function is assumed to be

where $\beta =2\pi /\lambda$, the $A_{ij}$ are constants that depend on the ratio of water depth to wavelength $\left(z_s-z_b\right)/\lambda$, and $\mu$ is a parameter. The wave profile, $\eta \left(\theta \right)$, is assumed to be

where the $B_{ij}$ are constants for a given water depth and wavelength. Finally, it is assumed that

and that

Skjelbreia and Hendrickson obtain the 18 constants $A_{ij}, B_{ij}$, $C_i$, and $c_0$ from matching terms in equal powers of $\mu$ and $\cos \theta$ in the free surface boundary conditions, Equation 2 and Equation 3. They give the constants as functions of $s=\sinh \left[\beta \left(z_s-z_b\right)\right], c=\cosh \left[\beta \left(z_s-z_b\right)\right],$ as

Skjelbreia and Hendrickson (1960) have a factor +2592 multiplying $c^8$ in the equation for $C_2$. This was corrected to −2592 by Nishimura et al. (1970).

They then obtain equations for $\beta \left(=2\pi /\lambda \right)$ and $\mu$. The wave height is

so Equation 5 gives

Also, the form assumed for the wave celerity gives

Given the wave period, wave height, and water depth, Equation 7 and Equation 8 must be solved simultaneously for the wavelength, $\lambda$, and the parameter $\mu$. This is done with a Newton method, using the Airy (linear) wave solution as an initial guess.