Utility of the Method of Steepest Descents in Analysis of Traveling Wave Instabilities.
AIR FORCE INST OF TECH WRIGHT-PATTERSON AFB OHIO SCHOOL OF ENGINEERING
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The report analyzes some simple traveling wave instabilities which are mathematically similar to the instabilities found in electron beam tubes. The differential equation describing the system is solved by applying the Laplace transform over time and the Fourier transform over space. The resulting equation in transformed variables is solved algebraically and the inverse double transform obtained to yield the desired solution in space and time. The calculus of residues is used to obtain the inverse Fourier transform. The inverse Laplace transform cannot always be obtained exactly however. When this is the case, the method of steepest descents is used to obtain a closed form approximation to the inverse Laplace transform which will asymptotically approach the inverse Laplace transform for late times. The method of steepest descents not only provides an asymptotic approximation in closed form, but does so in a manner that facilitates a physical interpretation of the results obtained. Modified author abstract
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