TY - JOUR

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

T2 - Localized waves

AU - Donnelly, Rod

AU - Ziolkowski, Richard

PY - 1992/1/1

Y1 - 1992/1/1

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

AN - SCOPUS:0039791284

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 -