set4-f5 - Astro 405/505, fall semester 2005 Homework, 4th...

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Astro 405/505, fall semester 2005 Homework, 4th set, return before Friday, November 4, 4pm. Don’t forget to give your name. Problem 7: Fourier transformation In class we derived the general solution of the low-amplitude acoustic wave equation as (11.9) 2 ρ 1 ∂t 2 = c 2 s, 0 2 ρ 1 ∂x 2 ρ 1 = f ( x - c s, 0 t ) + g ( x + c s, 0 t ) where f and g are arbitrary functions that can be written as the superposition of waves with amplitude spectrum ˜ f f ( y ) = 1 2 π Z -∞ dk ˜ f ( k ) exp( ı k y ) ˜ f ( k ) = 1 2 π Z -∞ dy f ( y ) exp( - ı k y ) The integral transformation that links f and ˜ f is called Fourier transformation. Solve the low-amplitude acoustic wave equation for a sound speed that depends on the fre- quency ω or wavenumber k . Can I still write the solution as a Fourier integral depending on y = x ± c s t ? The wave packets f and g now consist of waves that propagate with di±erent phase velocities ω/k . Assume a wave packet that contains only waves with frequencies close to
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set4-f5 - Astro 405/505, fall semester 2005 Homework, 4th...

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