hwk7 - Burgers equation and is given as follows ∂n ∂t n...

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Physics 762 Homework #7 Spring ’11 Dr. Drake 1. Consider the equations for a nonlinear sound wave in a plasma with no magnetic field. Take the limit in which k ± k D so that dispersive terms can be neglected but include a constant viscosity term in the momentum equation. Take the system to be isothermal. (a) Normalize the equations to get rid of all constants. (b) Transform to a frame of reference moving with the sound speed and de- velop an ordering in which dissipation balances wave steepening and the nonlinearity is weak. (c) Using this ordering derive a nonlinear equation for the system. This is
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Unformatted text preview: Burgers equation and is given as follows: ∂n ∂t + n ∂n ∂x-1 2 ∂ 2 n ∂x 2 = 0 (1) (d) Assume that this equation has a steady state solution which propagates at some velocity v . Derive this solution and show that it takes the form of a shock. Find a relationship between the velocity of the shock and the density on either side of the shock. Hint: Write the solution in terms of the densities, n + and n-, on either side the the shock. (e) Solve explicitly for the width of the shock in dimensional units....
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