The total energy E(t) of the vibrating string is given as a function of time by E(t) = _

Chapter 10, Problem 22

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The total energy E(t) of the vibrating string is given as a function of time by E(t) = _ L 0 12 u2t ( x, t) + 1 2 Tu2x ( x, t)_dx; (44) the first term is the kinetic energy due to the motion of the string, and the second term is the potential energy created by the displacement of the string away from its equilibrium position. For the displacement u( x, t) given by equation (20)---that is, for the solution of the string problem with zero initial velocity---show that E(t) = 2T 4L _ n=1 n2c2 n. (45) Note that the right-hand side of equation (45) does not depend on t. Thus the total energy E is a constant and therefore is conserved during the motion of the string. Hint: Use Parsevals equation ( of Section 10.4 and of Section 10.3), and recall that a2 = T/.