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11. (a) A heavy chain of length L hanging in equilibrium from a fixed point o has line density p(x) at a point of the chain a distance r from O (sor is measured downwards). Show that when the chain makes small transverse oscillations in a vertical plane, the horizontal displacement A(x, t) of a point on the chain and the tension T(x) in the string at that point satisfy the equations a2A aA d. ar2 What boundary condition must T(2) satisfy at the free end (1 = L)? (b) The transverse displacement A(,y,t) of a taut membrane ("drum') is described by the two-dimensional wave equation dT = 29 IPA = P at2 + Pgdc @2A at2 a2 A a2 + ax? dy2 ). (i) Find the characteristic frequencies for a square drum fixed along its edges at r = 0, 1 = a, y = 0 and y = a. (ii) Show that there is a mode of oscillation with frequency 58c/a such that the line r = y remains at rest throughout the motion (so Alr, x, t) = 0, Vx € [0, a) and Vt). (ii) Find the next frequency, in increasing magnitude, with a mode of oscillation with the same property.