This problem illustrates that the eigenvalue parameter sometimes appears in the boundaryconditions as well as in the differential equation. Consider the longitudinal vibrations of auniform straight elastic bar of length L. It can be shown that the axial displacement u(x, t)satisfies the partial differential equation(E/)uxx = utt; 0 < x < L, t > 0, (i)where E is Youngs modulus and is the mass per unit volume. If the end x = 0 is fixed,then the boundary condition there isu(0, t) = 0, t > 0. (ii)Suppose that the end x = L is rigidly attached to a mass m but is otherwise unrestrained.We can obtain the boundary condition here by writing Newtons law for the mass. Fromthe theory of elasticity, it can be shown that the force exerted by the bar on the mass isgiven by EAux(L, t). Hence the boundary condition isEAux(L, t) + mutt(L, t) = 0, t > 0. (iii)(a) Assume that u(x, t) = X(x)T(t), and show that X(x) and T(t) satisfy the differentialequationsX+ X = 0, (iv)T+ (E/)T = 0. (v)(b) Show that the boundary conditions areX(0) = 0, X(L) LX(L) = 0, (vi)where = m/AL is a dimensionless parameter that gives the ratio of the end mass tothe mass of the bar.Hint:Use the differential equation for T(t)in simplifying the boundary condition at x = L.(c) Determine the form of the eigenfunctions and the equation satisfied by the realeigenvalues of Eqs. (iv) and (vi).(d) Find the first two eigenvalues 1 and 2 if = 0.5.
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