Consider a semi-infinite elastic string that extends along the x-axis from x = 0 to infinity and let u(x, t) be its deflection about the equilibrium position. Suppose that the endpoint at x = 0 is free to move vertically, so that axu(0,t) = 0. At some time, denoted as t = 0, the string is suddenly hit such that the initial deflection and velocity can be idealised as u(x,0) = 0, du(x,0) = g(x) = 508(x - 20), respectively. Determine u(x, t) for t> 0 using a suitable type of Fourier transformation. Leave your final answer in terms of the signum function (see below). You may find the following formula useful: 0 sir sin(as) S ds = sgn(x): = TT zsgn(a), where sgn(a) is the signum (or sign) function that that defined as follows -1 for x < 0, 0 for x = 0, 1 for x > 0.
Consider a semi-infinite elastic string that extends along the x-axis from x = 0 to infinity and let u(x, t) be its deflection about the equilibrium position. Suppose that the endpoint at x = 0 is free to move vertically, so that axu(0,t) = 0. At some time, denoted as t = 0, the string is suddenly hit such that the initial deflection and velocity can be idealised as u(x,0) = 0, du(x,0) = g(x) = 508(x - 20), respectively. Determine u(x, t) for t> 0 using a suitable type of Fourier transformation. Leave your final answer in terms of the signum function (see below). You may find the following formula useful: 0 sir sin(as) S ds = sgn(x): = TT zsgn(a), where sgn(a) is the signum (or sign) function that that defined as follows -1 for x < 0, 0 for x = 0, 1 for x > 0.
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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Solve this question priority using Fourier Cosine Transform. If can’t solve with Fourier Cosine Transform, pls state clearly the reason and try with another Fourier transformation and why that is used instead of Fourier Cosine Transform. Write the steps clearly especially how to convert to signum function for the final answer.
Transcribed Image Text:Consider a semi-infinite elastic string that extends along the x-axis from x = 0 to infinity and let
u(x, t) be its deflection about the equilibrium position. Suppose that the endpoint at x = 0 is
free to move vertically, so that axu(0,t) = 0. At some time, denoted as t = 0, the string is
suddenly hit such that the initial deflection and velocity can be idealised as
u(x, 0) = 0,
respectively. Determine u(x, t) for t > 0 using a suitable type of Fourier transformation. Leave
your final answer in terms of the signum function (see below).
du(x,0) = g(x) = 508(x − 20),
-
You may find the following formula useful:
S
sin(as)
S
sgn(x) =
ds
=
TT
where sgn(a) is the signum (or sign) function that that defined as follows
-1 for x < 0,
0
for x = 0,
1
for x > 0.
sgn(a),
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