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s6_95c

# s6_95c - ACM 95/100c Problem Set VI 0(2 pts Write down your...

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ACM 95/100c May 14, 2004 Problem Set VI 0.- (2 pts) Write down your grading-section number. 1.- (10 pts) Sketch the solutions to the wave equation: u ( x, t ) = 1 2 [ u ( x + ct, 0) + u ( x - ct, 0)] + 1 2 c x + ct x - ct u t ( τ, 0) for various values of t arising from the following initial conditions: (a) (5 pts) u ( x, 0) = 0 , u t ( x, 0) = sin ωx ( ω is a constant), and (b) (5 pts) u ( x, 0) = 0 , u t ( x, 0) = 1 for 0 < x < 1 - 1 for - 1 < x < 0 . 0 for | x | > 1 Note that in part (b) the velocity u t ( x, 0) is discontinuous; the appropriate interpretation of the problem is that we consider a sequence of initial conditions, described by continuous (smooth) functions that approach the given initial condition in the limit. In this sense it is possible to give meaning to discontinuous initial conditions. Note that in contrast to the heat equation which smoothes out discontinuities, solutions to the wave equation with discontinuous initial conditions retain their discontinuity for t > 0. 2.- (15 pts) (a) (5 pts) Consider the solution u ( x, t ) of the wave equation u tt = c 2 u xx

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s6_95c - ACM 95/100c Problem Set VI 0(2 pts Write down your...

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