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Nonlinear dispersive wave problems


Luke, Jon Christian (1966) Nonlinear dispersive wave problems. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/EFW2-CC09.


The nonlinear partial differential equations for dispersive waves have special solutions representing uniform wavetrains. An expansion procedure is developed for slowly varying wavetrains, in which full nonlinearity is retained but in which the scale of the nonuniformity introduces a small parameter. The first order results agree with the results that Whitham obtained by averaging methods. The perturbation method provides a detailed description and deeper understanding, as well as a consistent development to higher approximations. This method for treating partial differential equations is analogous to the "multiple time scale" methods for ordinary differential equations in nonlinear vibration theory. It may also be regarded as a generalization of geometrical optics to nonlinear problems.

To apply the expansion method to the classical water wave problem, it is crucial to find an appropriate variational principle. It was found in the present investigation that a Lagrangian function equal to the pressure yields the full set of equations of motion for the problem. After this result is derived, the Lagrangian is compared with the more usual expression formed from kinetic minus potential energy. The water wave problem is then examined by means of the expansion procedure.

Item Type:Thesis (Dissertation (Ph.D.))
Subject Keywords:Mathematics
Degree Grantor:California Institute of Technology
Division:Physics, Mathematics and Astronomy
Major Option:Mathematics
Thesis Availability:Public (worldwide access)
Research Advisor(s):
  • Whitham, Gerald Beresford
Thesis Committee:
  • Unknown, Unknown
Defense Date:4 April 1966
Record Number:CaltechTHESIS:04032013-104428364
Persistent URL:
Default Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:7577
Deposited By: Dan Anguka
Deposited On:03 Apr 2013 19:52
Last Modified:21 Dec 2019 02:32

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