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This paper provides a new analytical method to obtain Green's functions oflinear dispersive partial differential equations. The Euler-Bernoulli beamequation and the one-dimensional heat conduction equation (dissipationequation) under impulses in space and time are solved as examples. The complexinfinite-domain Green's function of the Euler-Bernoulli beam is derived. A newapproach is proposed to obtain the finite-domain Green's function from theinfinite-domain Green's function by the reflection and transmission analysis inthe complex Fourier transform domain. It is found that the solution obtained bythis approach converges much better at short response times compared with thatobtained by the traditional modal analysis. Besides, by applying the geometricsummation formula for matrix series, a new modal expansion solution requiringno calculation of each mode's inner product is derived, which analyticallyproves the wave-mode duality and simplifies the calculation. Thesemi-infinite-domain cases and the coupled-domain cases are also derived by thenewly developed method to show its validity and simplicity. It is found thatthe non-propagating waves also possess wave speed, and heat conduction can alsobe treated as propagating waves

journal of sound and vibration

Wave analysis in the complex Fourier transform domain: A new method to  obtain the Green's functions of dispersive linear partial differential  equations