Internal generation of waves for extended Boussinesq equations

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C. Lee et al.rCoastal Engineering 42 2001 155–162
superiority of the Boussinesq equations to other wave
equations has been strengthened. Two types of the
extended Boussinesq equations have been developed.
The first type was developed by adding correction
terms to the momentum equations in order to get the
linear dispersion relation close to the exact one
Ž Madsen et al., 1991; Madsen and Sorensen, 1992 ..
The second type was developed by using the horizontal
velocities at a certain level instead of the
depth-averaged velocities in order to get the linear
dispersion relation close to the exact one ŽNwogu,
1993 .. The model of Nwogu, which is accurate to
OŽŽ kh . 2 , arh ., was developed in a different form
with the velocity potential instead of the particle
velocity by Chen and Liu Ž 1995 .. It was extended to
OŽŽ kh . 2 , Ž arh. 3 Ž kh. 2 . by Wei et al. Ž 1995 ., and
further to OŽŽ kh . 4 , Ž arh. 5 Ž kh. 4 . by Madsen and
Schaffer ¨ Ž 1998 .. It was also extended to consider the
case of waves on a current with OŽŽ kh . 2 , arh.
by
Chen et al. Ž 1998 ..
The hyperbolic-type wave equations may be used
to solve a lot of engineering problems such as design
wave height or harbor resonance because the behavior
of water waves is predicted in real time and the
reflection conditions at coastal structures can be
specified at will. The solution process of the hyperbolic-type
wave equations starts with a calm state of
surface elevations and continues until the wave energy,
which is made at offshore boundaries, propagates
to the region of interest. The wave energy
would reflect against the coastal structures and propagate
back to the offshore boundaries. The open
boundary conditions may be specified perfectly if the
wave energy which is reflected back to the offshore
boundaries is transmitted without any re-reflection
back to the computational domain. This is possible
by generating waves internally at a wave generation
line which is placed inside the computational domain
while a sponge layer is placed at offshore boundaries
Ž Larsen and Dancy, 1983 .. Then the wave which is
generated at the wavemaker would propagate in both
directions from the wave generation line and some
wave energy which is reflected against the structure
would pass through the wave generation line without
any numerical distortion, and further be absorbed in
the sponge layer at the offshore boundaries.
There are two different viewpoints related to the
internal wave generation. One is the viewpoint of
mass transport that the water mass which is generated
at the wave generation line transports in phase
velocity. The viewpoint is applied in the Boussinesq
equations of Peregrine Ž Larsen and Dancy, 1983.
and
the mild-slope equations of Copeland Ž 1985. ŽMad-
sen and Larsen, 1987; Yoon et al., 1996 .. The other
is the viewpoint of energy transport that the wave
energy which is generated at the wave generation
line transports in energy velocity. This viewpoint is
applied in the mild-slope equations of Radder and
Dingemans Ž 1985. and Copeland ŽSuh et al., 1997;
Lee and Suh, 1998 .. Lee and Suh found that application
of the mass transport approach to the mild-slope
equations of Radder and Dingemans yields improper
wave energy. They also found that, in the case of
equations of Copeland, two viewpoints of the mass
transport and energy transport give the same results
because the energy velocity is the phase velocity.
Further, they implied that, in the case of equations of
Peregrine, the mass transport approach might be
applied properly because the phase velocity in shallow
water is almost the same as the energy velocity.
In other words, it would be problematic to use the
viewpoint of mass transport in deeper water.
In this study, it will be found whether the mass
transport or energy transport approach yields a proper
wave energy for internally generating waves in the
Boussinesq equations of Nwogu. In Section 2, the
approaches of the mass transport and energy transport
are described and the phase and energy velocities
are derived. In Section 3, in a horizontally
one-dimensional domain, a finite difference method
is used to discretize the equations of Nwogu. Then,
the cnoidal waves and sinusoidal waves are generated
internally with two different approaches. Finally,
concluding remarks are made in Section 4.
2. Mass transport and energy transport
The extended Boussinesq equations of Nwogu
Ž 1993.
are written as:
Eh z 2 a h 2
q=P Ž hqh. u q=P y h= Ž =Pu.
Et 2 6
ž /
½ž /
h
q z q h= =P Ž hu. s0 Ž 1.
2
a 5