test2_samples - Problems for the 1-D Wave Equation 18.303...

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Problems for the 1-D Wave Equation Matthew J. Hancock Fall 2005 1 Problem 1 (i) Suppose that an &in±nite string²has an initial displacement u ( x; 0) = f ( x ) = 8 > < > : x + 1 ; & 1 ± x ± 0 1 & 2 x; 0 ± x ± 1 = 2 0 ; x < & 1 and x > 1 = 2 and zero initial velocity u t ( x; 0) = 0 . Write down the solution of the wave equation u tt = u xx with ICs u ( x; 0) = f ( x ) and u t ( x; 0) = 0 using D³Alembert³s formula. Illustrate the nature of the solution by sketching the ux -pro±les y = u ( x; t ) of the string displacement for t = 0 ; 1 = 2 ; 1 ; 3 = 2 . (ii) Repeat the procedure in (i) for a string that has zero initial displacement but is given an initial velocity u t ( x; 0) = g ( x ) = 8 > < > : & 1 ; & 1 ± x < 0 1 ; 0 ± x ± 1 0 ; x < & 1 and x > 1 2 Problem 2 (i) For an in±nite string (i.e. we don³t worry about boundary conditions), what initial conditions would give rise to a purely forward wave? Express your answer in terms of the 1
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3 Question 1 [20 points total] Suppose you shake the end of a rope of dimensionless length 1 at a certain frequency ! .
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This note was uploaded on 01/06/2011 for the course MATH 3333 taught by Professor Staff during the Spring '08 term at University of Houston.

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test2_samples - Problems for the 1-D Wave Equation 18.303...

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