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The subsequent displacement u(x,f) at time t and a distance x from the left end is govemed by the wave equation a?u 1ôu ô 9 ôr? OKx<5, t>0

The subsequent displacement u(x,f) at time t and a distance x from the left end is govemed by the wave equation a?u 1ôu ô 9 ôr? OKx<5, t>0

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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1 A taut string of length 3 with ends constrained on the x-axis at x-0 and x-5 is released from its central position with initial velocity defined by f(x) = 2, for 0<x<5 The subsequent displacement u(x,f) at time f and a distance x from the left end is governed by the wave equation a?u 1 a?u 9 ôr? 0<x<5, t>0 subject to the boundary conditions ди (0,t)=0 and u(5,t)=0 for t>0 ôx and the initial conditions u(x,0)= 0 , for 0<x<5 ôu (x,0)= f (x) , for 0<x<5 ốt Find the expression for the subsequent displacement u(x,t) in the string. Hint: Use the auxiliary function g(x) which is extension of fx) into the interval from 0 to 10: fs(x)=2, (F(10-x)=-2, 5<x<10 0<x<5 g(x)= 8(x): Let ø(x) be an even periodic extension of g(x). Then, o(x) will have the following properties: 1) 9(x)=g(x)=f(x), 0<x<5 2) 9(x) can be expanded into HRFCS. 3) The period of ø(x) is T=21=20=>l=10 4) There will be no constant term in HRFCS of 9(x) (i.e., the estimated coefficient ao will be zero)
Transcribed Image Text:1 A taut string of length 3 with ends constrained on the x-axis at x-0 and x-5 is released from its central position with initial velocity defined by f(x) = 2, for 0<x<5 The subsequent displacement u(x,f) at time f and a distance x from the left end is governed by the wave equation a?u 1 a?u 9 ôr? 0<x<5, t>0 subject to the boundary conditions ди (0,t)=0 and u(5,t)=0 for t>0 ôx and the initial conditions u(x,0)= 0 , for 0<x<5 ôu (x,0)= f (x) , for 0<x<5 ốt Find the expression for the subsequent displacement u(x,t) in the string. Hint: Use the auxiliary function g(x) which is extension of fx) into the interval from 0 to 10: fs(x)=2, (F(10-x)=-2, 5<x<10 0<x<5 g(x)= 8(x): Let ø(x) be an even periodic extension of g(x). Then, o(x) will have the following properties: 1) 9(x)=g(x)=f(x), 0<x<5 2) 9(x) can be expanded into HRFCS. 3) The period of ø(x) is T=21=20=>l=10 4) There will be no constant term in HRFCS of 9(x) (i.e., the estimated coefficient ao will be zero)
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