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set7-f7 - amplitude spectrum ˜ f f y = 1 √ 2 π Z...

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Astro 405/505, fall semester 2007 Homework, 7th set, return before Friday, October 26, 4pm. Don’t forget to give your name. Problem 13: Collisionless shocks In class we studied shocks as solutions of the hydrodynamical equations. These are valid on scales much larger the mean free path of particles, so in reality shocks will have a Fnite thickness, and we speak of collisionless shocks. a) Discuss whether or not you expect a di±erence in shock thickness for neutral gas, as opposed to ionized gas, in the upstream region. b) ²or an ionized gas, do you expect a di±erence in the mean free paths of ions and electrons, respectively? If so, what may be the consequences? Problem 14: Fourier transformation In class we derived the general solution of the low-amplitude acoustic wave equation as 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
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Unformatted text preview: 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 ²ourier 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 ²ourier 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 ω and that the frequency is a weak function of k , so we may write ω ( k ) ’ ω + d ω d k ± ± ± ± ( k-k ) + . . . Calculate the propagation velocity of the wave packet, called the group velocity....
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