Solved: This problem illustrates that the eigenvalue parameter sometimes appears in the

Chapter 11, Problem 25

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This problem illustrates that the eigenvalue parameter sometimes appears in the boundary conditions as well as in the differential equation. Consider the longitudinal vibrations of a uniform 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 is u(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. From the theory of elasticity it can be shown that the force exerted by the bar on the mass is given by EAux(L, t). Hence the boundary condition is EAux(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 differential equations X + X = 0, (iv) T + (E/)T = 0. (v) (b) Show that the boundary conditions are X(0) = 0, X (L) LX(L) = 0, (vi) where = m/AL is a dimensionless parameter that gives the ratio of the end mass to the mass of the bar. Hint: Use the differential equation forT(t)in simplifying the boundary condition at x = L. (c) Determine the form of the eigenfunctions and the equation satisfied by the real eigenvalues of Eqs. (iv) and (vi). (d) Find the first two eigenvalues 1 and 2 if = 0.5