Problem 6 (15 points) Let u(x, t) be a solution to the wave equation describing a vibrating string UttUrr = 0, x € R, t > 0. (i) Show that the energy density satisfies a certain conservation law E(x, t) = ½ (u²(x, t) + u²(x, t)) Et(x, t) + z(x, t) = 0. Find a flux function. There are many flux functions. You are asked to find one of them, not all. Remarks. • For the explanation of why & is the energy density of a vibrating string with a small amplitude, see the file "energy of a string" in Canvas → Files → Readings → Wave equation. (1) • This mathematical result shows that the energy density can be interpreted as a nonnegative physical quantity that moves in space as the string vibrates. (ii) Show that the total energy of the (infinite) string = [ε (x, t) da -∞ E(t) = is a conserved quantity, i.e., does not change in time. Hint: integrate the conservation law. Alternatively, you can directly differentiate E and then integrate by parts. The second approach might be a bit harder than the first one. (iii) Verify that the 'wave train' function u(x, t) = sin(xt) satisfies the wave equation (1). Compute the energy density E(x, t) and explain why it propagates with the same velocity as the wave train.

Fig: 1