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).
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).
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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Write clearly how to express final answer in terms of signum function.
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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