linear transfer function given by:
ω
(35)
Tu (ω)
=
α
1
2
kh 1 -
- (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 uα 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.
Irregular Unidirectional Waves. For nonperiodic waves, the incident wave
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:
a. The Pierson-Moskowitz (PM) spectrum for fully developed sea states in
deep water, which is defined in terms of the wind speed.
b. The Bretschneider (1959) spectrum, which has the same shape as the PM
spectrum but is defined in terms of the significant wave height and peak
period.
17
Chapter 3 Numerical Solution
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