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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
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. 

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),
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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