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

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**Extra resources for A Collection of Problems on Mathematical Physics**

**Example text**

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.