Along solid wall or fully reflecting boundaries, the horizontal velocity

normal to the boundary must be zero over the entire water depth, i.e.,

=

-*h*< *z *<η

(29)

0

where *n *is the normal vector to the boundary. This condition is satisfied in the

depth-integrated equations by specifying that both the volume flux density and

velocity normal to the boundary are zero, i.e.,

=

(30)

0

0

(31)

Since the equations are solved on a staggered grid, the boundary conditions are

or *v*α = *v*f = 0 along wall boundaries perpendicular to the y-axis.

specified. The time-histories may correspond to regular or irregular, unidirec-

tional or multidirectional waves.

terms of a wave height, *H*, wave period, *T*, and direction of propagation, θ. The

water-surface elevation for small-amplitude waves (*H *<< *h*) may be written as:

cos (kx cos θ + *ky *sin θ - ω*t *)

η( *x*, *y*, *t *)

=

(32)

2

where ω = 2π/*T *is the angular frequency, *k *is the wave number, and θ is the

direction of wave propagation relative to the positive x-axis. The boundary

conditions along a wave generation line perpendicular to the x-axis may be

obtained from the linearized form of the continuity equation (Equation 5) for

water of constant depth as:

=

(33)

α2 1

=

(*h *+ η) - *h *

- (*kh*) *u*α ( *x*g , *y*g , *t *)

2

(34)

2

2 6

16

Chapter 3 Numerical Solution

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