Appendix A
Fourier Series Solutions of
Boussinesq Equations
As the height of surface waves in intermediate and shallow-water depths
increases, the wave profile changes from a sinusoidal shape to an asymmetric one
with peaked crests and broad shallow troughs. The Boussinesq equations are
nonlinear and are able to describe the change in wave shape, provided the wave
height and period are within nonlinear and dispersive limits of the equations.
However, it is important that the velocity and flux time-histories imposed along
the wave generation boundaries of the numerical model be consistent with the
equations that are solved within the computational domain. In BOUSS-2D, the
derive nonlinear boundary conditions for periodic waves in water of constant
depth. The one-dimensional form of the weakly nonlinear form of the Boussinesq
equations (Equations 4 to 6) for water of constant depth can be written as:
1
ηt + [(h + η)uα ]x + α + h3uα,xxx
= 0
(A1)
3
()
1 2
uα,t + gη x +
u α x +αh 2 uα,xxt =0
(A2)
2
For periodic waves, the partial differential equations can be transformed into a
set of coupled nonlinear ordinary differential equations in terms of coordinate
system, ξ = x C t, moving at the phase speed of the waves, C, and integrated
once to yield:
1
(h + η)uα + α + h3uα,ξξ + Q = 0
(A3)
3
()
1 2
gη+
u α +αh 2 u α,ξξ -R=0
(A4)
2
1
References cited in Appendices A-E are listed in the References at the end of the main text.
A1
Appendix A Fourier Series Solutions of Boussinesq Equations
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