String. Show the following. (a) Substitution of x Il7fX

Chapter 12, Problem 12.3

(choose chapter or problem)

String. Show the following. (a) Substitution of x Il7fX (17) II(X, t) = L GnU) sin L n=l (L = length of the string) into the wave equation (I) governing free vibrations leads to [see (lO'~)J (18) 2_ Gn + An G - 0, (b) Forced vibrations of the string under an external force P(x, t) per unit length acting normal to the string are governed by the PDE (19) (c) For a sinusoidal force P = Ap sin wt we obtain (20) p ex n7fX = A sin wt = L knCt) sin L ' p n~l _ {(4A11l7f) sin wt k,,(t) - o (11 odd) (n even). Substituting (17) and (20) into (19) gives 2 _ 2A ,,' Gn + An Gn - - (l - cos 1l7f) sm wt. f (d) (Resonance) Show that if An = w, thenA - -- (1 - cos 1l7f)T cos WT.n7fW(e) (Reduction of boundary conditions) Show thata problem (1)-(3) with more complicated boundaryconditions 11(0, t) = 0, u(L, t) = h(t), can be reducedto a problem for a new function v satisfying conditionsv(O, t) = v(L, f) = 0, vex. 0) = fl(x), Vt(x, 0) = gl(X)but a nonhomogeneous wave equation. Him: SetII = V + I\" and determine w suitably.