A string along the x -axis is stretched by a force F tangential to the string. It has mass M per unit length and moves in the transverse direction with displacement y(x,t). (a) Consider a small string element of length Ax with kinetic energy AT = 2 *M*2*Ax and potential energy AU = 1/2 *F*y2*Ax. Write down the Lagrangian function for the whole string, and determine the action integral. (b) Consider a small variation dy(x,t) of the displacement, which vanishes at the times t = to, t and at the string ends x = 0, I. Determine the variation of the action integral and hence derive the Euler- Lagrange equation for the string. (c) Convert the Euler-Lagrange equation into a PDE for y(x,t). (d) Verify by direct substitution that y(x,t) = f(x-ct) is a solution, where c = sqrt(F/M). Give a physical interpretation of this solution.