Internal generation of waves for extended Boussinesq equations

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C. Lee et al.rCoastal Engineering 42 2001 155–162
waves which pass through the wave generation point
cause no serious numerical distortion while the incident
waves are generated at the point. The numerical
results of internal generation of sinusoidal waves,
with different water depths ranging from shallow to
deep waters, show that the energy transport approach
gives wave amplitudes properly. However, the mass
transport approach gives wave amplitudes different
from the desired ones by the ratio of phase to energy
velocities. The technique of internal generation of
waves shows the capability of generating nonlinear
cnoidal waves as well as linear sinusoidal waves.
Although the numerical experiment supports that the
energy transport is a proper approach to internally
generating waves in the extended Boussinesq equations,
a theoretical investigation would be much
valuable.
Acknowledgements
The first author wishes to acknowledge the financial
support of the Korea Science and Engineering
Foundation Ž no. 2000-1-31100-001-3.
during the visit
of the University of Delaware. The second author
wishes to acknowledge the financial support of the
Korea Science and Engineering Foundation Žno.
1999-2-311-005-3 ..
References
Boussinesq, J.V., 1872. Theorie ´ des ondes et des remous qui se
propaent le long d’un canal rectangulaire horizontal, en communiquant
au liquide contenu dans ce canal des vitesses
sensiblement pareilles de la surface au dond. J. Math. Pures
Appliq. 17, 55–108.
Chen, Y., Liu, P.L.-F., 1995. Modified Boussinesq equations and
associated parabolic models for water wave propagation. J.
Fluid Mech. 288, 351–388.
Chen, Q., Madsen, P.A., Schaffer, ¨ H.A., Basco, D.R., 1998.
Wave–current interaction based on an enhanced Boussinesq
approach. Coastal Eng. 33, 11–39.
Copeland, G.J.M., 1985. A practical alternative to the Amild-slopeB
wave equation. Coastal Eng. 9, 125–149.
Larsen, J., Dancy, H., 1983. Open boundaries in short wave
simulations—a new approach. Coastal Eng. 7, 285–297.
Lee, C., Suh, K.D., 1998. Internal generation of waves for timedependent
mild-slope equations. Coastal Eng. 34, 35–57.
Madsen, P.A., Larsen, J., 1987. An efficient finite-difference
approach to the mild-slope equation. Coastal Eng. 11, 329–351.
Madsen, P.A., Schaffer, ¨ H.A., 1998. Higher-order Boussinesq-type
equations for surface gravity waves: derivation and analysis.
Philos. Trans. R. Soc. London A 356, 3123–3184.
Madsen, P.A., Sorensen, O.R., 1992. A new form of the Boussinesq
equations with improved linear dispersion characteristics:
Part 2. A slowly varying bathymetry. Coastal Eng. 18, 183–
204.
Madsen, P.A., Murray, R., Sorensen, O.R., 1991. A new form of
the Boussinesq equations with improved linear dispersion
characteristics. Coastal Eng. 15, 371–388.
Nwogu, O., 1993. Alternative form of Boussinesq equations for
nearshore wave propagation. J. Waterw., Port, Coastal Ocean
Eng. 119, 618–638.
Peregrine, D.H., 1967. Long waves on a beach. J. Fluid Mech. 27,
815–827.
Press, W.H., Teukolsky, S.A., Vetterling, W.T., Flannery, B.P.,
1992. Numerical Recipes in Fortran: The Art of Scientific
Computing. 2nd edn. Cambridge Univ. Press, pp. 42–47.
Radder, A.C., Dingemans, M.W., 1985. Canonical equations for
almost periodic, weakly nonlinear gravity waves. Wave Motion
7, 473–485.
Suh, K.D., Lee, C., Park, W.S., 1997. Time-dependent equations
for wave propagation on rapidly varying topography. Coastal
Eng. 32, 91–117.
Wei, G., Kirby, J.T., Grilli, S.T., Subramanya, R., 1995. A fully
nonlinear Boussinesq model for surface waves: Part 1. Highly
nonlinear unsteady waves. J. Fluid Mech. 294, 71–92.
Wei, G., Kirby, J.T., Sinha, A., 1999. Generation of waves in
Boussinesq models using a source function method. Coastal
Eng. 36, 271–299.
Yoon, S.B., Lee, J.I., Lee, J.K., Chae, J.W., 1996. Boundary
treatment techniques of numerical model for the prediction of
harbor oscillation. J. Korean Soc. Civ. Eng. 16, 53–62, ŽIn
Korean ..