Answer Happy

2. Consider a circular membrane of radius a as shown below: a Similar to the situation considered in the lecture, let u be the displacement of the membrane from an equilibrium position and suppose that we are dealing with radially symmetric situation. However, instead of fixing the boundary, suppose that we allow the boundary to move, so that we now have a, ulr-a = 0. a) Using the method of separation of variables, show that the mode solution is uo = Aot + Bo when the separation constant is zero, where A, and B, are constants. Meanwhile, the mode solutions are Un = = (An sin(^„t) + B cos(^nt))J. (Bar) when the separation constant is negative, where n are positive integers, A and B₂ are constants, λ = cpn/a, and ßn are the nth non-zero zeroes of Jo(x). Prime indicates a derivative with respect to x. [Note: You may ignore positive separation constant, which leads to trivial solution u = 0.] b) Show that are also the zeroes of J₁ (x), and hence show that SªrJo ( * rJo (Bar) dr = 0 0. [Hint: Use one or more of the derivative formulae.]