New PDF release: 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.)

ISBN-10: 008010021X

ISBN-13: 9780080100210

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Travels along the hne. Consider the case where the end of the transmission Une is earthed (a) through a lumped resistance RQ, (b) through a lumped capacity CQ, (c) through a lumped inductance LQ. Estabhsh under what conditions in case (a) the reflected wave is absent ("complete absorption") a n d under what conditions the ampHtude of the reflected wave is half the amphtude of the incident wave. 72. f. Ε is applied to the end x = 0 of a semiinfinite distortionless transmission hne over a sufficiently long in­ terval of time, so that a steady distribution of voltage and current intensity is estabUshed in the Une.

Solve the preceding problem for the case where the end to which the impulse is not applied, is fixed. 108. One end of a rod is fixed elastically, and the other end is free. Find the longitudinal vibrations of the rod for arbitrary initial conditions. t See [7], pages 147-150. t For the excitation of a string by a supple convex hammer see problem 152. 122] II. EQUATIONS OF HYPERBOLIC TYPE 31 109. One end of a rod (x = I) is fixed elastically, and a longi­ tudinal force FQ = const, is applied to the other end (x = 0).

67. Initial longitudinal displacements are imparted to the sec­ tions of a semi-infinite flexible rod with its end fixed elastically to a support u{x, 0) = sin^ if 0 < x < / , 0 if / < x < + o o and initial velocities w,(x, 0) = 0. Find the longitudinal deflections w(x, t) of cross-sections of the rod for t > 0. 68. A semi-infinite vertical circular axle 0 < A : < + o o f o r i < 0 rotates with angular velocity ω = const. At time t = 0 its end X = 0 touches a horizontal supporting surface and is acted on by a twisting moment of a frictional force, proportional to the angular velocity of the end.

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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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