where Q is the volume flux and R is the Bernoulli constant. The velocity uα can be
expanded as a Fourier series:
N
∑ jkB
u α =B0 +
cos( jkξ)
(A5)
j
j =1
where N is the number of Fourier components, Bj are the Fourier coefficients, and
k is the wave number. To solve the problem numerically, the free surface eleva-
tion is discretized into N+1 equally spaced points over half a wavelength, i.e.,
ηm =η(ξ m ),m=0,1,K,N
(A6)
where ξm = mL/2N and L is the wavelength. Equations A3 to A4 are then evalu-
ated at points over half a wavelength to yield a system of nonlinear algebraic
equations:
1
(h + ηm )um + α + h3uξξ,m + Q = 0
(A7)
3
1 2
gηm + um + αh2uξξ,m - R = 0
(A8)
2
The above 2N+2 equations involve 2N + 5 unknowns ηj (j = 0, ..., N), Bj (j =
0, ..., N), k, Q, R, so three additional equations are needed. These can be obtained
from the wave height, H, wave period, T, the mean water level as:
η0 - ηN - H
= 0
(A9)
kCT - 2π
= 0
(A10)
N -1
η0 + η N + 2∑ η j
= 0
(A11)
j =1
noting that C = -B0 for a zero-mean Eulerian velocity. A Newton-Raphson pro-
cedure (see Rienecker and Fenton 1981) is used to solve the system of equations
(A7 to A11) for the unknown values of the free surface displacement at the
collocation points, the Fourier coefficients, the wave number, the phase speed,
and the constants Q and R.
A2
Appendix A Fourier Series Solutions of Boussinesq Equations
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