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 x with kinetic energy T = *M*²*x and potential energy U = *F*y'²*x. Write down the Lagrangian function for the whole string, and determine the action integral.

(b) Consider a small variation y(x,t) of the displacement, which vanishes at the times t = t₀, t₁ and at the string ends x = 0, l. 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(xct) is a solution, where c = sqrt(F/M). Give a physical interpretation of this solution.