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

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 1 2 u2 t (x, t) + 1 2Tu2 x(x, t) dx; (the first term is the kinetic energy due to the motion of the string, and the second term isthe potential energy created by the displacement of the string away from its equilibriumposition.For the displacement u(x, t) given by Eq. (20)that is, for the solution of the stringproblem with zero initial velocityshow thatE(t) = 2T4Ln=1n2c2n. (ii)Note that the right side of Eq. (ii) does not depend on t. Thus the total energy E is aconstant and therefore is conserved during the motion of the string.Hint: Use Parsevals equation ( of Section 10.4 and of Section10.3), and recall that a2 = T/.