2. Consider the first order quasilinear partial differential equation utt ; (u? )x = 0, -00 1/2, r, if x 1/2. (3) (a) [4 marks] Determine the speed for a shock with left and right states u = u_ and u = #+, respectively. Specify a condition on u- and u+ for which the shock is causal. (b) [12 marks] Show that the shocks present in the initial data (2) are causal. Hence use the method of characteristics to determine the solution of the initial value problem (1)-(3), for all t 2 0. You should also determine the trajectory of the shocks. (c) [9 marks] Now consider the initial condition u(x,0) = > g(x - 2k), k=-00 for (1), where g is as defined in (3). (i) Sketch the initial condition. (ii) Determine the solution for 0