Stokes wave theory

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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 $G = g (z - z_{0})$ , 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

\begin{matrix} (1) & ▽^{2} ϕ = 0 \end{matrix}

v = - \frac{\partial ϕ}{\partial x} .

Equilibrium is

ρ (\frac{\partial v}{\partial t} + \frac{\partial v}{\partial x} \cdot v) = - ρ \frac{\partial G}{\partial x} - \frac{\partial p}{\partial x},

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

ρ (\frac{\partial^{2} ϕ}{\partial x \partial t} - \frac{\partial v}{\partial x} \cdot v) = ρ \frac{\partial G}{\partial x} + \frac{\partial p}{\partial x} .

Integrating with respect to $x$ (note that $ρ$ is constant since the fluid is incompressible) gives the Bernoulli equation

\frac{\partial ϕ}{\partial t} - \frac{1}{2} v \cdot v = G + \frac{p - p_{0}}{ρ} + g F (t),

where $F (t)$ 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

\frac{\partial ϕ}{\partial t} - \frac{1}{2} v \cdot v = g (z - z_{s}) + \frac{p - p_{0}}{ρ},

where $z_{s}$ is the undisturbed surface level. From this equation the total pressure at a point below the instantaneous fluid surface is

p = p_{0} + ρ g (z_{s} - z) + p_{d y n} .

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

p_{d y n} = ρ (\frac{\partial ϕ}{\partial t} - \frac{1}{2} v \cdot v) .

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

{(\frac{\partial ϕ}{\partial t} - \frac{1}{2} v \cdot v) |}_{z = η + z_{s}} = g η,

assuming the pressure at the surface is negligible.

Assuming the waves are uniform, of wavelength $λ$ and period $τ$ , 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

θ = 2 π (\frac{S}{λ} - \frac{t}{τ} + \frac{α}{360}) = \frac{2 π}{λ} (S - \bar{c} t + \frac{λ α}{360}),

where $\bar{c} = \frac{λ}{τ}$ is the wave celerity. This means that, for any function in the solution,

\frac{\partial}{\partial t} = - \bar{c} \frac{\partial}{\partial s} .

Thus, at the free surface boundary

\frac{\partial ϕ}{\partial t} = - \bar{c} \frac{\partial ϕ}{\partial s} = \bar{c} v_{s},

and the Bernoulli equation at the free surface is

{(- \bar{c} v_{s} + 1 / 2 (v \cdot v)) |}_{z = η + z_{S}} = - g η

\begin{matrix} (2) & {(v_{z}^{2} + {(v_{s} - \bar{c})}^{2}) |}_{z = η + z_{S}} = {\bar{c}}^{2} - z g η . \end{matrix}

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:

\begin{matrix} (3) & {\frac{v_{z}}{v_{s} - \bar{c}} |}_{z = η + z_{s}} = \frac{\partial η}{\partial s} . \end{matrix}

At the seabed $(z = z_{b})$ , there is no fluid motion in the vertical direction:

- v_{z} = {\frac{\partial ϕ}{\partial z} |}_{z = z_{b}} = 0.

The problem now consists of finding a potential function, $ϕ$ , 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

\begin{matrix} (4) & \begin{matrix} \frac{β ϕ}{\bar{c}} = \frac{2 π ϕ}{λ \bar{c}} & ​ = (μ A_{11} + μ^{3} A_{13} + μ^{5} A_{15}) \cosh [β (z - z_{b})] \sin θ \\ - (μ^{2} A_{22} + μ^{4} A_{24}) \cosh [2 β (z - z_{b})] \sin 2 θ \\ + (μ^{3} A_{33} + μ^{5} A_{35}) \cosh [3 β (z - z_{b})] \sin 3 θ \\ - μ^{4} A_{44} \cosh [4 β (z - z_{b})] \sin 4 θ \\ + μ^{5} A_{55} \cosh [5 β (z - z_{b})] \sin 5 θ, \end{matrix} \end{matrix}

where $β = 2 π / λ$ , the $A_{i j}$ are constants that depend on the ratio of water depth to wavelength $(z_{s} - z_{b}) / λ$ , and $μ$ is a parameter. The wave profile, $η (θ)$ , is assumed to be

\begin{matrix} (5) & \begin{matrix} β η = ​ & - μ \cos θ \\ + (μ^{2} B_{22} + μ^{4} B_{24}) \cos 2 θ \\ - (μ^{3} B_{33} + μ^{5} B_{35}) \cos 3 θ \\ + μ^{4} B_{44} \cos 4 θ \\ - μ^{5} B_{55} \cos 5 θ, \end{matrix} \end{matrix}

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

\begin{matrix} (6) & β {\bar{c}}^{2} = c_{0}^{2} (1 + μ^{2} C_{1} + μ^{4} C_{2}) \end{matrix}

and that

β F = μ^{2} C_{3} + μ^{4} C_{4} .

Skjelbreia and Hendrickson obtain the 18 constants $A_{i j}, B_{i j}$ , $C_{i}$ , and $c_{0}$ from matching terms in equal powers of $μ$ and $\cos θ$ in the free surface boundary conditions, Equation 2 and Equation 3. They give the constants as functions of $s = \sinh [β (z_{s} - z_{b})], c = \cosh [β (z_{s} - z_{b})],$ as

\begin{matrix} c_{0}^{2} & ​ = g s / c, \\ A_{11} & ​ = 1 / s, \end{matrix}

\begin{matrix} A_{13} & ​ = - c^{2} \frac{5 c^{2} + 1}{8 s^{5}}, \\ A_{15} & ​ = - \frac{1184 c^{1} - 1440 c^{8} - 1992 c^{6} + 2641 c^{4} - 249 c^{2} + 18}{1536 s^{11}}, \\ A_{22} & ​ = \frac{3}{8 s^{4}}, \\ A_{24} & ​ = \frac{192 c^{8} - 424 c^{6} - 312 c^{4} + 480 c^{2} - 17}{768 s^{10}}, \\ A_{33} & ​ = \frac{13 - 4 c^{2}}{64 s^{7}}, \\ A_{35} & ​ = \frac{512 c^{12} + 4224 c^{10} - 6800 c^{8} - 12808 c^{6} + 167704 c^{4} - 3154 c^{2} + 107}{4096 s^{13} (6 c^{2} - 1)}, \\ A_{44} & ​ = \frac{80 c^{6} - 816 c^{4} + 1338 c^{2} - 197}{1536 s^{10} (6 c^{2} - 1)}, \\ A_{55} & ​ = - \frac{2880 c^{10} - 72480 c^{8} + 324000 c^{6} - 432000 c^{4} + 163470 c^{2} - 16245}{61440 s^{11} (6 c^{2} - 1) (8 c^{4} - 11 c^{2} + 3)}, \\ B_{22} & ​ = c \frac{2 c^{2} + 1}{4 s^{3}}, \\ B_{24} & ​ = c \frac{272 c^{8} - 504 c^{6} - 192 c^{4} + 322 c^{2} + 21}{384 s^{9}}, \\ B_{33} & ​ = 3 \frac{8 c^{6} + 1}{64 s^{6}}, \\ B_{35} & ​ = \frac{88128 c^{14} - 208224 c^{12} + 70848 c^{10} + 54000 c^{8} - 21816 c^{6} + 6264 c^{4} - 54 c^{2} - 81}{12288 s^{12} (6 c^{2} - 1)}, \\ B_{44} & ​ = c \frac{768 c^{10} - 448 c^{8} - 48 c^{6} + 48 c^{4} + 106 c^{2} - 21}{384 s^{9} (6 c^{2} - 1)}, \\ B_{55} & ​ = \frac{192000 c^{16} - 262720 c^{14} + 83680 c^{12} + 20160 c^{10} - 7280 c^{8}}{12288 s^{10} (6 c^{2} - 1) (8 c^{4} - 11 c^{2} + 3)} \\ + \frac{7160 c^{6} - 1800 c^{4} - 1050 c^{2} + 225}{12288 s^{10} (6 c^{2} - 1) (8 c^{4} - 11 c^{2} + 3)}, \\ C_{1} & ​ = \frac{8 c^{4} - 8 c^{2} + 9}{8 s^{4}}, \\ C_{2} & ​ = \frac{3840 c^{12} - 4096 c^{10} - 2592 c^{8} - 1008 c^{6} + 5944 c^{4} - 1830 c^{2} + 147}{512 s^{10} (6 c^{2} - 1)} . \end{matrix}

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 $β ( = 2 π / λ)$ and $μ$ . The wave height is

H = η_{crest} - η_{trough} = {η |}_{θ = π} - {η |}_{θ = 0},

so Equation 5 gives

\begin{matrix} (7) & H = \frac{λ}{π} [μ + μ^{3} B_{33} + μ^{5} (B_{35} + B_{55})] . \end{matrix}

Also, the form assumed for the wave celerity gives

\begin{matrix} (8) & β {\bar{c}}^{2} = \frac{2 π λ}{τ^{2}} = c_{0}^{2} (1 + μ^{2} C_{1} + μ^{4} C_{2}) . \end{matrix}

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

Stokes wave theory

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