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 00 subject to the boundary conditions ди (0,t)=0 and u(5,t)=0 for t>0 and the initial conditions и (х,0) - 0 , for 0l=10 4) There will be no constant term in HRFCS of o(x) (i.e., the estimated coefficient ao will be zero)
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 00 subject to the boundary conditions ди (0,t)=0 and u(5,t)=0 for t>0 and the initial conditions и (х,0) - 0 , for 0l=10 4) There will be no constant term in HRFCS of o(x) (i.e., the estimated coefficient ao will be zero)
Calculus For The Life Sciences
2nd Edition
ISBN:9780321964038
Author:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Publisher:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Chapter14: Discrete Dynamical Systems
Section14.3: Determining Stability
Problem 15E
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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)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F1ac7e570-d904-47c2-b4e7-fbd4c3551513%2Fdb13f468-a916-494e-82c5-de7c7a13e130%2Fl4qbfxo_processed.jpeg&w=3840&q=75)
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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