### 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 language | English (US) |
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Pages (from-to) | 673-692 |

Number of pages | 20 |

Journal | Proceedings of The Royal Society of London, Series A: Mathematical and Physical Sciences |

Volume | 437 |

Issue number | 1901 |

DOIs | |

State | Published - 1992 |

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### ASJC Scopus subject areas

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

### Cite this

**A method for constructing solutions of homogeneous partial differential equations : Localized waves.** / Donnelly, Rod; Ziolkowski, Richard W.

Research output: Contribution to journal › Article

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TY - JOUR

T1 - A method for constructing solutions of homogeneous partial differential equations

T2 - Localized waves

AU - Donnelly, Rod

AU - Ziolkowski, Richard W

PY - 1992

Y1 - 1992

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=0039791284&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0039791284&partnerID=8YFLogxK

U2 - 10.1098/rspa.1992.0086

DO - 10.1098/rspa.1992.0086

M3 - Article

VL - 437

SP - 673

EP - 692

JO - Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences

JF - Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences

SN - 0962-8444

IS - 1901

ER -