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

Find the transverse displacement y of a particle at x = 1.66 m at time t = 0.150 s.

y(x,t) = A sin [(kx ± wt) + Q] expresses the displacement of a harmonic wave as a function of x and t in terms of the wave parameters k and co. Write the equivalent expressions that contain the following pairs of parameters instead of k and w: (a) k and v, (b) λ and f (c) 2 and T, (d) λ and v, and (e) fand v