The vertical displacement u(x, t) of an infinitely long string is determined from the initial-value problem a ^ 2 partial^ 2 u partial x^ 2 = partial^ 2 u partial t^ 2 , - ∞ < x < ∞ , t > 0 u(x, 0) = f(x) partial u partial t | f = 0 = g(x) . (13) e This problem can be solved without separating variables. (a) Show that the wave equation can be put into the form du/and = 0 by means of the substitutions xi = x at and η = x - at. (b) Integrate the partial differential equation in part (a), first with respect to η and then with respect to xi to show that u(x, t) = F(x at) G(x - at) where F and Gare arbitrary twice differentiable functions, is a solution of the wave equation. Use this solution and the given initial conditions to show that F(x) = 1/2 * f(x) 1/(2a) * integrate g(s) ds from x_{0} to x c G(x) = 1/2 * f(x) - 1/(2a) * integrate g(s) ds from x_{0} to x - c and where x_{0} is arbitrary and c is a constant of integration. (c) Use the results in part (b) to show that u(x, t) = 1/2 * [f(x at) f(x - at)] 1/(2a) * integrate g(s) ds from x - at to x at Note that when the initial velocity g(x) = 0 we obtain u(x, t) = 1/2 * [f(x at) f(x - at)] (14) This last solution can be interpreted as a traveling waves, one moving to the right (that is, 1/2 * f(x - at) ) and one moving superposition to the left (1/2 * f(x at)) Both waves travel with speed and hat