By B. M. Budak, A. A. Samarskii, A. N. Tikhonov, I. N. Sneddon, M. Stark and S. Ulam (Auth.)
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Additional info for A Collection of Problems on Mathematical Physics
Solve the preceding problem, assuming that the vibrations are produced by a transverse blow at the point χ = XQ, trans mitting an impulse / to the rod. 2. Free Vibrations in a Resistant Medium 117-122. In problems 97, 101, 103, 105, 108, 111, vibrations of strings and rods in a non-resistant m e d i u m were considered. We assume now that in these problems the medium produces a re- 32 COLLECTION OF PROBLEMS ON MATHEMATICAL PHYSICS [123 sistance proportional to the velocity, we then obtain problems 117, 118, 119, 120, 121 and 122 respectively.
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.
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. 91. Find the longitudinal vibrations of a rod with zero initial conditions, if one of its ends is fixed or free and the other moves according to a given law; consider the case where (a) the right-hand end is fixed, (b) the left-hand end is fixed, (c) the right-hand end is free. 92. Find the pressure vibrations at the end x = 0 of a tube for ί > 0, if it is equal to zero at the end χ = /, and the input of liquid at the end χ = 0 is a known function of time.
A Collection of Problems on Mathematical Physics by B. M. Budak, A. A. Samarskii, A. N. Tikhonov, I. N. Sneddon, M. Stark and S. Ulam (Auth.)