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1. Let u(x; t) solve the wave equation utt = c2uxx in x  0 with Robin boundary data ut + ux = r(t) at x = 0 Here is a constant not equal to c and r(t) is a given function. Show that the boundary values at x = 0 for the standard Riemann variables R = ut  cux are related by R+(0; t) = 2cr(t) 􀀀 (c + )R􀀀(0; t) c 􀀀 (Hint: start by expressing ut and ux in terms of R.) Use previous equation to nd R+(x; t) everywhere for the homogeneous initial-value problem in which u(x; 0) = ut(x; 0) = 0. (Hint: make a sketch of the situation and use invariance of R􀀀 along left-going char- acteristics to argue that R􀀀 = 0 for all x  t and t  0.) Show that this is consistent with u(x; t) = c c 􀀀 Z t􀀀x=c 0 r( )d if t > x=c and u(x; t) = 0 otherwise 2. One-dimensional sound waves may be described by a density disturbance eld 0(x; t) and a velocity eld u0(x; t) that in linear theory satisfy this coupled system of rst-order PDEs: 0 t + 0u0 x = 0 and 0u0 t + c20 x = 0: Here c > 0 is the sound speed and 0 is the constant background density of the resting air, say. (a) Show that the non-dimensional variables  = 0=0 and u = u0=c satisfy t + cux = 0 and ut + cx = 0: (b) Show that both u and  satisfy the wave equation with wave speed c. (c) Show that if Riemann variables R+(x; t) and R􀀀(x; t) and chosen as R = u   then (@t  c@x)R = 0 holds (d) Find the functions u(x; t) and (x; t) that solve the initial-value problem u(x; 0) = A(x) and (x; 0) = 2A(x) where A(x) is an arbitrary function. 1

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Applied Differential Equations
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