By B. M. Budak, A. A. Samarskii, A. N. Tikhonov, I. N. Sneddon, M. Stark and S. Ulam (Auth.)

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**Sample text**

An infinite string is excited by a locahzed initial having the form of a quadratic parabola (Fig. 7). F i n d : (a) describing the profile of the string for t > 0, and (b) representing the law of motion of an arbitrary point string for t > 0. deflection formulae, formulae, χ of the t See [7], pages 39-54 and 57-68. Use of solutions in the form (2) for steady-state problems, where / is a geometric coordinate, will be given in chap ter V. t Here and in later problems a means the wave velocity appearing in equa tion (1) Utt = a^Uxx.

106. Find the vibrations of a flexible rod with free ends, which has received a longitudinal impulse / at one end at t = 0. 107. Solve the preceding problem for the case where the end to which the impulse is not applied, is fixed. 108. One end of a rod is fixed elastically, and the other end is free. Find the longitudinal vibrations of the rod for arbitrary initial conditions. t See [7], pages 147-150. t For the excitation of a string by a supple convex hammer see problem 152. 122] II. EQUATIONS OF HYPERBOLIC TYPE 31 109.

Solve problem 136 assuming that the vibrations occur in a medium with a resistance proportional t o the velocity. Find the steady-state vibrations representing the main part of the solution for t - > + 0 0 . 143. f. ), a n d the end χ = 0 is insulated. Find the steady-state vibrations representing the main part of the solution for ί -> + 0 0 , 144. f. , is applied at the end Λ: = / of the conductor at time t = 0, a n d the end x = 0 is earthed. Find the steady-state vibrations. 145. Find the steady-state vibrations of pressure at the end x = / of a tube 0 < x < /, if a damping cap exists at this end (cf.