Short-Wave Behavior of Long-Wave Equations
Publication: Journal of Waterway, Port, Coastal, and Ocean Engineering
Volume 124, Issue 5
Abstract
The stability of nonlinear dispersive-wave equations near the short wave limit is examined analytically and computationally. Equations of first and second order are analyzed based on their linear dispersion relations. Computational tests are performed on the nonlinear version of the equations, which confirm the theoretical estimates. Equations are derived based on approximations of the water-wave Hamiltonian, and are shown to possess different stability properties depending on the order of the Hamiltonian expansion in terms of a small parameter. It is shown that the smoothest solution is achieved by the regularized form of the equations, which, however, lead to excessively dissipative computational results. Consistent results are obtained by the second-order, Hamiltonian-based equations. All equations are solved by a Fourier pseudo-spectral method, which permits direct comparison of the various equation forms based on identical initial conditions that asymptotically reach the short wave limit.
Get full access to this article
View all available purchase options and get full access to this article.
References
1.
Abbott, M. B., Petersen, H. M., and Skovgard, O. (1978). “On the numerical modeling of short waves in shallow waters.”J. Hydr. Res., 3, 173—203.
2.
Abdollahi, J. (1994). “Stable equations for nonlinear dispersive waves,” PhD thesis, Univ. of Michigan, Ann Arbor, Mich.
3.
Benjamin, T. B., and Olver, P. J. (1982). “Hamiltonian structure, symmetries and conservation laws for water waves.”J. Fluid Mech., 125, 137—185.
4.
Benjamin, T. B. (1984). “Impulse, flow force and variational principles.”IMA J. Appl. Mathematics, 32(1-3), 3—68.
5.
Boussinesq, J. (1872). “Thèorie des ondes et des remous qui se propagent le long d'un canal rectangulaire horizontal, en communiquant au liquide contenu dans ce canal de vitesses sensiblement parreilles de la surface au fond.”J. de Mathematiques Pures et Appliquées Ser. 2, 17, 55—108 (in French).
6.
Boyd, J. P. (1989). Chebyshev and Fourier spectral methods. Springer-Verlag, Berlin.
7.
Broer, L. J. F. (1974). “On the Hamiltonian theory of surface waves.”Appl. Scientific Res., 30, 430—446.
8.
Broer, L. J. F., Van Groesen, E. W. C., and Timmers, J. M. W. (1976). “Stable model equations for long water waves.”Appl. Scientific Res., 32, 619—636.
9.
Channel, P. J., and Scovell, J. C. (1990). “Symplectic integration of Hamiltonian systems.”Nonlinearity, 3, 231—259.
10.
Craig, W., and Groves, M. D. (1994). “Hamiltonian long-wave approximations to the water-wave problem.”Wave Motion, 19, (4), 367—389.
11.
Craig, W., and Sulem, C. (1993). “Numerical simulation of gravity waves.”J. Computational Phys., 108(1), 73—83.
12.
Dingemans, M. W. (1973). “Water waves over an uneven bottom–A discussion of long-wave equations.”Rep. R 729-II, Delft Hydr. Lab., Delft, The Netherlands.
13.
Katopodes, N. D., and Abdollahi, J. (1993). “Stable forms of Boussinesq-type equations.”ASME-ASCE-SES Meeting, ASME, New York, N.Y., 720.
14.
Katopodes, N. D., and Dingemans, M. W. (1988). “Stable equations for surface wave propagation on an uneven bottom.”Rep. Prepared for Delft Hydr. Lab.
15.
Katopodes, N. D., and Dingemans, M. W. (1989). “Hamiltonian approach to surface wave propagation.”Hydrocomp, Int. Assn. for Hydr. Res., 137—147.
16.
Kirby, J. T., and Wei, G. (1994). “Derivation and properties of a fully nonlinear model for weakly dispersive waves.”Proc., IAHR Symp.: Waves–Phys. and Numer. Modeling, Int. Assn. for Hydr. Res., 386—395.
17.
Madsen, O. S., and Mei, C. C. (1969). “Dispersive long waves of finite amplitude over an uneven bottom.”MIT Rep. 117, Massachusetts Inst. of Technol., Cambridge, Mass.
18.
Madsen, P. A., Murray, R., and Sorensen, O. R. (1991). “A new form of the Boussinesq equations with improved linear dispersion characteristics.”Coast. Engrg., 15, 371—388.
19.
Miles, J. W. (1977). “On Hamilton's principle for surface waves.”J. Fluid Mech., 83, 153—158.
20.
Miles, J. W. (1981). “Hamiltonian formulations for surface waves.”Appl. Scientific Res., 37, 103—110.
21.
Mooiman, J., and Verboom, G. K. (1992). “New Boussinesq model based on a positive definite Hamiltonian.”Proc., 9th Int. Conf. on Computational Methods in Water Resour., 513—527.
22.
Nwogu, O. (1993). “An alternative form of the Boussinesq equations for neashore wave propagation.”J. Wtrwy., Port, Coast., and Oc. Engrg., ASCE, 119, 618—638.
23.
Peregrine, D. H. (1966). “Calculations of the development of an undular bore.”J. Fluid Mech., 25(Part 2), 321—330.
24.
Pullin, D. I. (1992). “Contour dynamics methods.”Annu. Rev. Fluid Mech., 24, 89—115.
25.
Roseau, M. (1976). Asymptotic wave theory. North-Holland/American Elsevier, Amsterdam, The Netherlands.
26.
Sanders, B. F., Katopodes, N. D., and Boyd, J. P. (1998). “Spectral modeling of nonlinear dispersive waves.”J. Hydr. Engrg., ASCE, 124(1), 2—12.
27.
Schaper, H., and Zielke, W. (1984). “A numerical solution of Boussinesq type equations.”Proc., 19th Conf. on Coast. Engrg.
28.
Van Der Houwen, P. J., Mooiman, J., and Wubs, F. W. (1991). “Numerical analysis of time dependent Boussinesq models.”Int. J. Numer. Methods in Fluids, 13(10), 1235—1250.
29.
Whitham, G. B. (1974). Linear and nonlinear waves. Wiley-Interscience, New York, N.Y.
30.
Witting, J. M. (1984). “A unified model for the evolution of nonlinear water waves.”J. Computational Phys., 56, 203—236.
31.
Zakharov, V. E. (1974). “The Hamiltonian formalism for waves in nonlinear media having dispersion.”Izv. VUZ. Radiofizika, 17(4), 326—343.
Information & Authors
Information
Published In
Journal of Waterway, Port, Coastal, and Ocean Engineering
Volume 124 • Issue 5 • September 1998
Pages: 238 - 247
Copyright
Copyright © 1998 American Society of Civil Engineers.
History
Published online: Sep 1, 1998
Published in print: Sep 1998
Authors
Metrics & Citations
Metrics
Citations
Download citation
If you have the appropriate software installed, you can download article citation data to the citation manager of your choice. Simply select your manager software from the list below and click Download.