1. The problem statement, all variables and given/known data A wire is fixed at both ends vibrating fundamentally. For what value of x (x position on the wire, with 0 being one edge and L being the other) is the potential energy per unit length has the maximum value? Known: Length of wire (L), Tension in wire (T), Mass of wire (m), Amplitude (A). 2. Relevant equations y(x,t) = A sin(kx) cos(wt) dU/dx ~= T (dy/dx)^2 /2 (dy/dx is a partial derivative) 3. The attempt at a solution dy/dx = Ak cos(kx) cos(wt) dU/dx = T (Ak)^2 cos(wt)^2 cos(kx)^2 d/dx(dU/dx) = -T (Ak)^2 cos(wt)^2 2k cos(kx) sin(kx) d/dx(dU/dx) = -T (Ak)^2 cos(wt)^2 k sin(2kx) Critical points exist at d/dx(dU/dx) = 0, so since everything else is a constant, sin(2kx) = 0, and thus 2kx = 0 (x = 0), 2kx = pi (x = L/2), and 2kx = 2pi (x = L) are all solutions. dU/dx = 0 at x = L/2, but when cos(wt)^2 > 0 then dU/dx is greater at x = 0, L than at x = L/2 The textbook says that the maximum potential energy per unit length occurs at the middle of the wire. My math says that it's at the two endpoints? Why? Also, I googled the problem and they are taking dy/dx at 0 to get the result. I don't understand this. Can anyone explain?