linear transfer function given by:

ω

(35)

=

α

1

2

- (*kh*)2

2

2 6

lines perpendicular to the y-axis with cosθ replaced by sinθ in Equation 33.

For large amplitude waves [*H *= *O*(*h*)], higher harmonic components (2ω,

3ω, etc.) are generated due to the nonlinear terms in the governing equations.

These waves are often called bound waves since they are attached to the primary

wave and travel at its phase speed, *C *= ω/*k*, instead of the phase speed of a free

wave at the corresponding frequency. The wave shape also changes from the

sinusoidal shape assumed in Equation 32 to an asymmetric one with peaked

crests and broad shallow troughs. If linear wave conditions are imposed at the

boundaries, the numerical model will generate free higher harmonic components

with the same magnitude but 180 deg out of phase with the bound waves at the

wavemaker to satisfy the linear boundary condition. The presence of bound and

free higher frequency waves that travel at different speeds will lead to a spatially

nonhomogenous wave field with the wave height and shape changing continu-

ously over the computational domain.

The Fourier approximation method of Rienecker and Fenton (1981) has been

used to solve the weakly nonlinear form of the Boussinesq equations and develop

nonlinear boundary conditions for the generation of large-amplitude regular

waves in shallow water. The partial differential Equations 4 to 6 are initially

transformed into a set of coupled nonlinear ordinary differential equations in

terms of a moving coordinate system, ξ = *x * *C t*. The velocity variable *u*α is

expanded as a Fourier series and substituted into the governing equations, which

are evaluated at a finite number of collocation points over half a wavelength to

yield a system of nonlinear algebraic equations. A Newton-Raphson iterative

procedure is used to solve the nonlinear equations for the unknown values of the

free surface displacement at the collocation points, the Fourier coefficients, the

wave number, and the phase speed. Details of the technique are provided in

Appendix A.

conditions are typically expressed in the form of a wave spectrum, which

describes the frequency distribution of wave energy. Different parametric shapes

have been proposed for wave spectra including:

deep water, which is defined in terms of the wind speed.

spectrum but is defined in terms of the significant wave height and peak

period.

17

Chapter 3 Numerical Solution

Integrated Publishing, Inc. |