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

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

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87. One end of a horizontal rod is rigidly fixed a n d the other end is free. At the initial time ί = 0 a mass Q = Mg strikes the free end of the rod with a velocity VQ, directed along the axis of the rod, and remains in contact with it until t = ÍQ. Find the longi­ tudinal vibrations of the rod for t > 0. 88. Solve the preceding problem for a rod, both ends of which are free. 89. Solve problem 87, assuming that the rod has the form of a truncated cone. 90. Solve problem 88 for a rod having the shape of a truncated cone.

1. Problems for an Infinite String 52. An infinite string is excited by a locaHzed initial deflection, shown in Fig. 6. Plot (trace) the position of the string for the times ί = kcl^a, where /c = 0, 1, 2, 3, 5. FIG. 6 53. 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.

A similar expression is valid for a boundary condition of the form 22 COLLECTION OF PROBLEMS ON MATHEMATICAL PHYSICS [62 / to the string over the section 0 < x < 2/, the profile of the distri­ bution of velocity, obtained by the blow, having at time t = 0 the form of a half-wave sinusoidal with base 0 < x < 2/. Find the formulae, describing the law of motion of points of the string with different abscissae χ for t > 0. 21 31 FIG. 41 5L 61 71 χ 10 62. A semi-infinite flexible rod 0 < x < +oo with a free end at X = 0 is perturbed at time ί = 0 by longitudinal displacements, the profile of which+ is depicted in Fig.

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