A method for constructing solutions of homogeneous partial differential equations

Localized waves

Rod Donnelly, Richard W Ziolkowski

Research output: Contribution to journalArticle

52 Citations (Scopus)

Abstract

We introduce a method for constructing solutions of homogeneous partial differential equations. This method can be used to construct the usual, well-known, separable solutions of the wave equation, but it also easily gives the non-separable localized wave solutions. These solutions exhibit a degree of focusing about the propagation axis that is dependent on a free parameter, and have many important potential applications. The method is based on constructing the space-time Fourier transform of a function so that it satisfies the transformed partial differential equation. We also apply the method to construct localized wave solutions of the wave equation in a lossy infinite medium, and of the Klein-Gordon equation. The localized wave solutions of these three equations differ somewhat, and we discuss these differences. A discussion of the properties of the localized waves, and of experiments to launch them, is included in the Appendix.

Original languageEnglish (US)
Pages (from-to)673-692
Number of pages20
JournalProceedings of The Royal Society of London, Series A: Mathematical and Physical Sciences
Volume437
Issue number1901
DOIs
StatePublished - 1992

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Homogeneous differential equation
partial differential equations
Partial differential equations
Partial differential equation
Wave equations
wave equations
Wave equation
Klein-Gordon equation
Klein-Gordon Equation
Nonseparable
Fourier transform
Fourier transforms
Space-time
Propagation
propagation
Dependent
Experiment
Experiments

ASJC Scopus subject areas

  • Mathematics(all)
  • Engineering(all)
  • Physics and Astronomy(all)

Cite this

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