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 A: Mathematical, Physical and Engineering Sciences |
| Volume | 437 |
| Issue number | 1901 |
| DOIs | |
| State | Published - Jan 1 1992 |
ASJC Scopus subject areas
- Mathematics(all)
- Engineering(all)
- Physics and Astronomy(all)