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# Part I. On the Breaking of Nonlinear Dispersive Waves. Part II. Variational Principles in Continuum Mechanics

## Citation

Seliger, Robert Lewis (1968) Part I. On the Breaking of Nonlinear Dispersive Waves. Part II. Variational Principles in Continuum Mechanics. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/K2AZ-JX87. https://resolver.caltech.edu/CaltechTHESIS:04032013-094741062

## Abstract

A model equation for water waves has been suggested by Whitham to study, qualitatively at least, the different kinds of breaking. This is an integro-differential equation which combines a typical nonlinear convection term with an integral for the dispersive effects and is of independent mathematical interest. For an approximate kernel of the form e^(-b|x|) it is shown first that solitary waves have a maximum height with sharp crests and secondly that waves which are sufficiently asymmetric break into "bores." The second part applies to a wide class of bounded kernels, but the kernel giving the correct dispersion effects of water waves has a square root singularity and the present argument does not go through. Nevertheless the possibility of the two kinds of breaking in such integro-differential equations is demonstrated.

Difficulties arise in finding variational principles for continuum mechanics problems in the Eulerian (field) description. The reason is found to be that continuum equations in the original field variables lack a mathematical "self-adjointness" property which is necessary for Euler equations. This is a feature of the Eulerian description and occurs in non-dissipative problems which have variational principles for their Lagrangian description. To overcome this difficulty a "potential representation" approach is used which consists of transforming to new (Eulerian) variables whose equations are self-adjoint. The transformations to the velocity potential or stream function in fluids or the scaler and vector potentials in electromagnetism often lead to variational principles in this way. As yet no general procedure is available for finding suitable transformations. Existing variational principles for the inviscid fluid equations in the Eulerian description are reviewed and some ideas on the form of the appropriate transformations and Lagrangians for fluid problems are obtained. These ideas are developed in a series of examples which include finding variational principles for Rossby waves and for the internal waves of a stratified fluid.

Item Type: Thesis (Dissertation (Ph.D.)) Applied Mathematics California Institute of Technology Engineering and Applied Science Applied Mathematics Public (worldwide access) Whitham, Gerald Beresford Unknown, Unknown 10 November 1967 Thesis title varies slightly in 1968 Commencement Program: "I. On the Breaking of Nonlinear Dispersive Waves. II. Variational Principles in Continuum Mechanics." CaltechTHESIS:04032013-094741062 https://resolver.caltech.edu/CaltechTHESIS:04032013-094741062 10.7907/K2AZ-JX87 No commercial reproduction, distribution, display or performance rights in this work are provided. 7575 CaltechTHESIS Dan Anguka 03 Apr 2013 17:07 16 Jun 2021 22:54

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