In the problem cu = 0, u|t=0 = φ, ut |t=0 = ψ, take the data φ(x) ≡ 0 and ψ(x) = 1 for |x| < a and ψ(x) = 0 for |x| ≥ a. Find the values of u for all (t, x) and indicate them on the (t, x)-plane. Sketch the string profile (u versus x) at each of the successive instants t = a/2c, a/c, 3a/2c, 2a/c, and 5a/c. You may take c = 1 in this problem. [General strategy: Calculate u(x, t) = 1 2c Z x+ct x−ct ψ(s) ds = 1 2c {length of (x − ct, x + ct) ∩ (−a, a)}. Then u(x, a/2c) = (1/2c){length of (x − a/2, x + a/2) ∩ (−a. a)}. This takes on different values for |x| < a/2, for a/2 < x < 3a/2, and for x > 3a/2 as the characteristic picture on t, x-plane shows. Continue in this manner for each case.]