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) |
|---|---|
| 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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- Engineering(all)
- Physics and Astronomy(all)
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A method for constructing solutions of homogeneous partial differential equations : Localized waves. / Donnelly, Rod; Ziolkowski, Richard W.
In: Proceedings of The Royal Society of London, Series A: Mathematical and Physical Sciences, Vol. 437, No. 1901, 1992, p. 673-692.Research output: Contribution to journal › Article
}
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.
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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 -