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„