# 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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Find the vibrations of the rod. 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.

A transverse initial velocity VQ = const, is imparted to an infinite string over a section — c < Λ: < c; outside this section the initial velocity equals zero. Find formulae, describing the law of motion of points of the string with different abscissae for t > 0, and plot the positions of the string for the times where k = 2, 4, 6. 56. At the initial time ί = 0 an infinite string receives a trans­ verse blow at the point χ = XQ, transmitting an impulse / t o the string. Find the deflection w(x, t) of points of the string from positions of equihbrium for ί > 0 assuming that the initial displacements of other points of the string and their initial velocities equal zero.

Vibrations with Inhomogeneous Media and Other Conditions Leading to Equations with Variable Coe£Bcients; Calculations with Concentrated Forces and Masses 164. Investigate the longitudinal vibrations of an inhomogeneous rod 0 < X < / of constant cross-section, obtained by joining two homogeneous rods at χ = XQ, if (a) the volume density and coefficient of elasticity are respect­ ively equal to \p, 0 < x < X o , ^(^> = b , x„