a global sense over the entire record. The method also does not guarantee a sym-

metric directional distribution even when the target distribution is symmetric.

The velocity and flux boundary conditions along the wave generation lines

are obtained from the surface elevation time-histories using the linear transfer

function approach described previously in Equations 40 to 44.

In applications where there is significant wave reflection from bathymetric

features or structures within the computational domain, it is desirable to absorb

the waves that propagate back to the wave generation boundary to prevent a

buildup of wave energy inside the domain. This can be achieved by modifying

the boundary conditions along the wave generation boundary to simultaneously

generate and absorb reflected waves (e.g., Van Dongeren and Svendsen 1997). A

different approach proposed by Larsen and Dancy (1983) is to generate the

waves inside the computational domain and absorb reflected waves in a damping

layer placed behind the generation boundary. This approach has been adopted in

BOUSS-2D with the governing equations modified to allow for the generation of

waves inside the computational domain.

Consider the generation of waves along a horizontal line by a distribution of

sources that extend from the seabed (*z *= -*h*) to the free surface (*z *= η). The veloc-

ity potential associated with the fluid motion satisfies the Laplace equation

everywhere in the fluid except for generation line (*x *= *x*g) where there is a point

source of fluid mass. The governing equation for the fluid motion can thus be

written as:

∇2φ = *q*( *y*, *z*, *t *)δ( *x *- *x*g )

(51)

where *q*(*y,z,t*) is the volume flux density. Assuming that the water depth is

constant along the generation line, a modified form of the second-order Taylor

series expansion of the velocity potential about an arbitrary elevation *z *= *z*α in the

water column (Equation 1) can be written as:

1

( zα + *h *) - ( z + *h *) ∇2φα - *q*δ( *x *- *x*g )

2

2

φ = φα +

(52)

2

The horizontal fluid velocities are obtained from the velocity potential as:

1

( zα + *h *) - ( z + *h*) ∇ ∇2φα - *q*δ( *x *- *x*g )

2

2

(53)

2

On a rectangular grid with a finite grid spacing ∆*x*, the delta function can be

replaced with 2/∆*x*. To generate waves with a given velocity profile *u*o(*y,z,t*), the

mass and momentum equations along the grid generation line and adjacent

velocity points can be written as:

22

Chapter 3 Numerical Solution

Integrated Publishing, Inc. |