1. 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 along the

string is given by

y(1,0) = Ar (L-1).

(a) Sketch a graph of y(x, 0).

(b) The initial displacement y(x, 0) is now represented as a sum of harmonics,

Yn (2) = sin (177) .

y(x,0) = Σanyn (T),

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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