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

ISBN-10: 008010021X

ISBN-13: 9780080100210

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11). Find the deflection of the string u{x, t) for ί > 0, if the string is excited at time r = 0 by a transverse impulse I = MVQ, transmitted to the mass Μ and directed along the axis of the spring. 26 COLLECTION OF PROBLEMS ON MATHEMATICAL PHYSICS [80 80. The mass Μ of the preceding problem, in oscillating, ex periences a frictional resistance proportional to the velocity. Find the reflected and transmitted waves, taking the wave u^(^x, t) = f(x—at) travelhng from the region x < 0 as the initial condition.

Solve the problem of propagation of electrical vibrations in an infinite conductor for the condition that GL=CR, (1) where G, L, C, R are the leakage conductance, self-inductance, capacity and resistance per unit length of the c o n d u c t o r t . The voltage and the current in the conductor at the initial time are given. 2. Problems for a Semi-infinite Region If only one end of a string* is far enough from the part of it under investigation so that a reflection from that end is n o t im p o r t a n t in the oscillations of this part, at least during the time interval being considered, then we arrive at the problem of the vibrations of a semi-infinite string 0 < x < + o o , where x = 0 corresponds to the " n e a r " end of the string.

192. igi are odd. and ^ ( . ) = Σ ^ ^ 44 COLLECTION OF PROBLEMS ON MATHEMATICAL PHYSICS [193 193. Prove that a representation of the solution of the b o u n d a r y value problem Utt = a'u^^+c^u+f(x, Akj^ t), = 0, w(jc,0) = 0 , 0 < X, ί < + 0 0 , 0<í<+cx), wXx,0) = 0, x = 0, 0

### 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.)

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