1. Consider a uniform string of length 6. Its vertical displacement o. = u(m,t) above its equilibrium at position a: and time t obeys the equation uttzc2um, 00, Where c > 0 is a constant. (a) (b) (3 marks) What is the physical interpretation of the constant c ? (4 marks) Show that, for every pair of sufciently smooth functions F and G of one variable, Mm, t) = F(:1: + ct) + G(:c ct) (*) solves the partial differential equation. For the remainder of the question, it Will be assumed that the string is tied down at the ends, so that u(0,t) =u(,t) = 0, t> 0. The string is pulled upward at the middle, raised to a height h and then released from rest so that u(:1:,0): U0(CL') :: %:E if0 0, as a sine series in m, with tdependent coefcients. iii. (5 marks) Find F and G such that the vertical displacement is given, for 0 0, by Equation (*)