The diffraction of water waves by submerged obstacles in shallow water generally requires the use of a nonlinear theory since both dispersive and nonlinear effects are important. In this work, wave diffraction is studied in a numerical wave tank using the Level I Green-Naghdi (GN) equations. Cnoidal waves are generated numerically by a wave maker situated at one end of a two-dimensional numerical wave tank. At the downwave end of the tank, an open-boundary condition is implemented to simulate a wave-absorbing beach, and thus to reduce reflections. The GN equations are solved in the time-domain by employing a finite-difference method. The numerical method is applied to diffraction of cnoidal waves by a submerged shelf, or a sand bar, of considerable height relative to water depth. The predicted results are compared with the available experimental data which indicate the importance of nonlinearity for the shallow-water conditions.

1.
Demirbilek, Z., and Webster, W. C., 1992, “Application of the Green-Naghdi Theory of Fluid Sheets to Shallow-Water Wave Problems,” Technical Report No. CERC-92-11, U.S. Army Corps of Engineers, Waterways Experiment Station, Vicksburg, MS.
2.
Ertekin, R. C., 1984, “Soliton Generation by Moving Disturbances in Shallow Water: Theory, Computation and Experiment,” Ph.D. thesis, University of California at Berkeley, CA.
3.
Ertekin, R. C., 1988, “Nonlinear Shallow-Water Waves: The Green-Naghdi Equations,” Proceedings, Pacific Congress on Marine Science and Technology, PACON, Honolulu, Hawaii, pp. OST6/42-OST6/52.
4.
Ertekin
R. C.
,
Liu
Y. Z.
, and
Padmanabhan
B.
,
1994
, “
Interaction of Incoming Waves With a Steady Intake-Pipe Flow
,”
ASME JOURNAL OF OFFSHORE MECHANICS AND ARCTIC ENGINEERING
, Vol.
116
, pp.
214
220
.
5.
Ertekin, R. C., Webster, W. C., and Wehausen, J. V., 1984, “Ship Generated Solitons,” Proceedings, 15th Symposium on Naval Hydrodynamics, Hamburg, Germany, pp. 347–361.
6.
Ertekin
R. C.
,
Webster
W. C.
, and
Wehausen
J. V.
,
1986
, “
Waves Caused by a Moving Disturbance in a Shallow Channel of Finite Width
,”
Journal of Fluid Mechanics
, Vol.
169
, pp.
275
292
.
7.
Ertekin, R. C., and Wehausen, J. V., 1986, “Ship Generated Solitons,” Proceedings, 16th Symposium on Naval Hydrodynamics, Berkeley, pp. 167–184.
8.
Green
A. E.
, and
Naghdi
P. M.
,
1976
, “
Directed fluid sheets
,”
Proceedings Royal Society, London
, Series A, Vol.
116
, pp.
447
473
.
9.
Green
A. E.
, and
Naghdi
P. M.
,
1984
, “
A Direct Theory of Viscous Fluid Flow in Channels
,”
Arc. Rat. Mech. Anal.
Vol.
86
(
1
), pp.
39
63
.
10.
Miles
J.
, and
Salmon
R.
,
1985
, “
Weakly Dispersive Nonlinear Gravity Waves
,”
Journal of Fluid Mechanics
, Vol.
157
, pp.
519
531
.
11.
Naghdi, P. M., 1972, The Theory of Shells and Plates, S. Flugges’s Handbuch der Physik, Vol. VIa/3, pp. 425–640, Berlin, Germany.
12.
Neill, D. R., and Ertekin, R. C., 1997, “Diffraction of Solitary Waves by a Vertical Cylinder: Green-Naghdi and Boussinesq-Equations,” Proceedings, 16th International Conference on Offshore Mechanics and Arctic Engineering, ASME OMAE Vol. I-B, Yokohama, Japan, pp. 63–71.
13.
Ohyama
T.
,
Kioka
W.
, and
Tada
A.
,
1995
, “
Applicability of Numerical Models to Nonlinear Dispersive Waves
,”
Coastal Engineering
Vol.
24
, pp.
275
296
.
14.
Peregrine
D. H.
,
1967
, “
Long Waves on a Beach
,”
Journal of Fluid Mechanics
, Vol.
27
, pp.
815
827
.
15.
Sarpkaya, T., and Isaacson, M., 1981, Mechanics of Wave Forces on Offshore Structures, Van Nostrand Reinhold Co., New York, NY.
16.
Shields
J. J.
, and
Webster
W. C.
,
1988
, “
On Direct Methods in Water-Wave Theory
,”
Journal of Fluid Mechanics
, Vol.
197
, pp.
171
199
.
17.
Sun, X., 1991, “Some Theoretical and Numerical Studies on Two-Dimensional Cnoidal-Wave Diffraction Problems,” Master’s thesis, Department of Ocean Engineering, University of Hawaii at Manoa, Honolulu, HI.
18.
Ursell
F.
,
1953
, “
The Long Wave Paradox in the Theory of Gravity Waves
,”
Proceedings, Cambridge Philosophical Society
, Vol.
49
, pp.
685
694
.
19.
Whitham
G. B.
,
1967
, “
Variational Methods and Applications to Water Waves
,”
Proceedings, Roy. Society of London
, Series A, Vol.
299
, pp.
6
25
.
20.
Wu
T. Y.
,
1981
, “
Long Waves in Ocean and Coastal Waters
,”
Journal of Engineering Mechanics
, ASCE, Vol.
107
, pp.
501
522
.
This content is only available via PDF.
You do not currently have access to this content.