Download A Collection of Problems on Mathematical Physics by B. M. Budak, A. A. Samarskii, A. N. Tikhonov, I. N. Sneddon, PDF

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

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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.

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