Weakly Nonlinear Wave Theory for Periodic Waves (Stokes Expansion)

Chapter 3.1 Weakly Nonlinear Wave Theory for Periodic Waves (Stokes Expansion)

Introduction

The solution for Stokes waves is valid in deep or intermediate water depth. It is assumed that the wave steepness is much smaller than one.

[math]\displaystyle{ \varepsilon = a k \ll 1 \, }[/math] [math]\displaystyle{ k h = O (1) \, }[/math]

where [math]\displaystyle{ k \, }[/math] is the wavenumber and [math]\displaystyle{ h \, }[/math] is the water depth which is assumed constant.

• Nondimensional Variables

[math]\displaystyle{ X = x k, \quad Z = z k, \quad Y = y k, \, }[/math] [math]\displaystyle{ \bar{t} = t \sigma, \quad \bar{\eta} = \eta / a, \, }[/math] [math]\displaystyle{ \Phi = \frac{\sigma\phi}{a g}, \quad \bar{C} = \frac{C}{a g}, \quad D = \frac{g k}{\sigma^2}, \, }[/math]

where [math]\displaystyle{ x, \ z, \ t, \ h, \ \phi \, }[/math] and [math]\displaystyle{ C \, }[/math] are dimensional variables and [math]\displaystyle{ X, \ Z, \ \bar{t}, \ \bar{\eta}\, \ \Phi \, }[/math] and [math]\displaystyle{ \bar{C} \, }[/math] are corresponding nondimensional variables.

Nondimensional Governing Equation & Boundary Conditions

[math]\displaystyle{ \frac{\partial^2\theta}{\partial X^2} + \frac{\partial^2\phi}{\partial Y^2} + \frac{\partial^2\theta}{\partial Z^2} = 0 \, }[/math]		[math]\displaystyle{ - k h \lt Z \lt \varepsilon \bar{\eta} \, }[/math]	[math]\displaystyle{ (3.1.1) \, }[/math]
[math]\displaystyle{ \frac{\partial\Phi}{\partial Z} = 0 \, }[/math]	at	[math]\displaystyle{ Z = - k h \, }[/math]	[math]\displaystyle{ (3.1.2) \, }[/math]
[math]\displaystyle{ \frac{\partial\bar{\eta}}{\partial\bar{t}} + \varepsilon D \nabla_h \Phi \cdot \nabla_h \bar{\eta} = D \frac{\partial\Phi}{\partial Z} }[/math]	at	[math]\displaystyle{ Z = \varepsilon \bar{\eta} \, }[/math]	[math]\displaystyle{ (3.1.3) \, }[/math]
[math]\displaystyle{ \frac{\partial\Phi}{\partial\bar{t}} + \frac{1}{2} \varepsilon D [ \nabla \Phi ]^2 = \bar{C} \, }[/math]	at	[math]\displaystyle{ Z = \varepsilon \bar{\eta} \, }[/math]	[math]\displaystyle{ (3.1.4) \, }[/math]

where [math]\displaystyle{ \nabla \, }[/math] and [math]\displaystyle{ \nabla_h \, }[/math] stand for gradient and horizontal gradient, respectively.

Perturbation (Stokes Expansion)

Assuming the wave train is weakly nonlinear [math]\displaystyle{ ( \varepsilon = a k \ll 1 ) \, }[/math], its potential and elevation can be perturbed in the order of [math]\displaystyle{ \varepsilon \, }[/math].

[math]\displaystyle{ \Phi = \Phi^{(1)} + \varepsilon\Phi^{(2)} + \varepsilon^2\Phi^{(3)} + \ldots + \varepsilon^{j-1}\Phi^{(j)} + \ldots }[/math] [math]\displaystyle{ \bar{\eta} = {\bar{\eta}}^{(1)} + \varepsilon {\bar{\eta}}^{(2)} + \varepsilon^2 {\bar{\eta}}^{(3)} + \ldots + \varepsilon^{j-1} {\bar{\eta}}^{(j)} + \dots }[/math] [math]\displaystyle{ \bar{C} = {\bar{C}}^{(1)} + \varepsilon {\bar{C}}^{(2)} + \varepsilon^2 {\bar{C}}^{(3)} + \ldots + \varepsilon^{j-1} {\bar{C}}^{(j)} + \dots }[/math]

Hierachy Equations

Using the Taylor expansion, the free-surface boundary conditions (Equations(3.1.3) and (3.1.4) are expanded at the still water level [math]\displaystyle{ (Z=0) \, }[/math]. Then we substitute perturbation forms of potential and elevation into the Laplace Equation, bottom and free-surface boundary conditions. The equations are sorted and grouped according to the order in wave steepness [math]\displaystyle{ \varepsilon^{(j)} \, }[/math]. The governing equations for [math]\displaystyle{ j-th \, }[/math] order solutions is given by: