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Matakuliah Tahun : K0124 / Matematika Teknik II : 2006/2007 PERTEMUAN 15 VIBRATING STRING EQUATION (finding a solution) 1 A string of length L is strecthed between points (0, 0) and (L, 0) on the x axis. At time t = 0 it has a shape given by f(x), 0 < x< L and it is released from rest. 2 Find the displacement of the string at any later time y Y =(x, t) L 3 The equation of the vibrating string is 2 2 y 2 y a 2 t x 2 0< x < L, t > 0. where y(x, t) = displacement from x axis at time t. To solve this boundy-value problem , let y = XT as usual. " 2 " " 2 " XT a X T or T a T X X Then 4 Calling the separation constant 2 , we have T " 2 a 2T 0, X " 2 X 0 The result is m x m x m at 2 L y ( x, t ) f ( x) sin dx sin cos 0 L L L m 1 L which can be verified as the solution. 5 TERIMA KASIH 6