sol4-f5 - Astro 405/505 fall semester 2005 Homework 4th set...

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Astro 405/505, fall semester 2005 Homework, 4th set, solutions Problem 7: Fourier transformation If the sound speed depends on the frequency ω or wavenumber k , I can not write the solution as a Fourier integral depending on y = x ± c s t , because y now depends on k . However, individual waves are still partial solutions to the wave equation, ρ 1 = C exp ( ı k x - ı ω ( k ) t ) if ω ( k ) = c s k where C is a constant that be chosen to satisfy boundary conditions, as is easily shown by inserting this partial solution into the wave equation. In fact we have a multitude of solutions, for k can be chosen freely as long as ω satisfies the dispersion relation ω ( k ) = c s k . Neglecting the trivial homogeneous solutions, that are constant or linear in time and space, the general solution is now the complete superposition of all partial solutions, ρ 1 ( x, t ) = Z -∞ dk ˜ ρ 1 ( k ) exp ( ı k x - ı ω ( k ) t ) ω 0 which, again, is not a Fourier integral, because the wave packet no longer propagates on the characteristic x = ± c s, 0 t .
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