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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