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A guitar string obeys the linear wave equation with wave-speed u, and is fixed at x = 0 and r= L. It is pulled out into a parabolic shape, and then released at time t = 0, so that immediately afterwards the transverse displacement at a position a along the string is given by y(x,0) = Ax (L-x). (a) Sketch a graph of y(x, 0). (b) The initial displacement y(x, 0) is now represented as a sum of harmonics, y(x,0) = Σanyn(x), ¹("7ª). Yn(x) = sin n=1 For the above form of y(x, 0), find an expression for the Fourier coefficients an. (c) Hence, and assuming that the string is released from rest, write down an expres- sion for y(x, t), the transverse displacement profile of the string at an arbitrary

A guitar string obeys the linear wave equation with wave-speed u, and is fixed at x = 0 and r= L. It is pulled out into a parabolic shape, and then released at time t = 0, so that immediately afterwards the transverse displacement at a position a along the string is given by y(x,0) = Ax (L-x). (a) Sketch a graph of y(x, 0). (b) The initial displacement y(x, 0) is now represented as a sum of harmonics, y(x,0) = Σanyn(x), ¹("7ª). Yn(x) = sin n=1 For the above form of y(x, 0), find an expression for the Fourier coefficients an. (c) Hence, and assuming that the string is released from rest, write down an expres- sion for y(x, t), the transverse displacement profile of the string at an arbitrary

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A guitar string obeys the linear wave equation with wave-speed u, and is fixed at x = 0 and x = L. It is pulled out into a parabolic shape, and then released at time t = 0, so that immediately afterwards the transverse displacement at a position a along the string is given by y(x,0) = Ax (L- x). (a) Sketch a graph of y(x, 0). (b) The initial displacement y(x, 0) is now represented as a sum of harmonics, (TTT). y(x,0) = anyn(x), Yn (x) = sin n=1 For the above form of y(x, 0), find an expression for the Fourier coefficients an (c) Hence, and assuming that the string is released from rest, write down an expres- sion for y(x, t), the transverse displacement profile of the string at an arbitrary later time.
Transcribed Image Text:A guitar string obeys the linear wave equation with wave-speed u, and is fixed at x = 0 and x = L. It is pulled out into a parabolic shape, and then released at time t = 0, so that immediately afterwards the transverse displacement at a position a along the string is given by y(x,0) = Ax (L- x). (a) Sketch a graph of y(x, 0). (b) The initial displacement y(x, 0) is now represented as a sum of harmonics, (TTT). y(x,0) = anyn(x), Yn (x) = sin n=1 For the above form of y(x, 0), find an expression for the Fourier coefficients an (c) Hence, and assuming that the string is released from rest, write down an expres- sion for y(x, t), the transverse displacement profile of the string at an arbitrary later time.
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