where *Q *is the volume flux and *R *is the Bernoulli constant. The velocity *u*α can be

expanded as a Fourier series:

∑ jkB

cos( *jk*ξ)

(A5)

where *N *is the number of Fourier components, *B*j are the Fourier coefficients, and

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*/2*N *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 )*u*m + α + *h*3uξξ,*m *+ *Q *= 0

(A7)

3

1 2

(A8)

2

The above 2*N*+2 equations involve 2*N *+ 5 unknowns ηj (*j *= 0, ..., *N*), *B*j (*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)

= 0

(A10)

η0 + η N + 2∑ η j

= 0

(A11)

noting that *C *= *-B*0 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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