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
L ί FIG. 12 94. f. Ε is appUed t o the end x = 0 of a distor tionless transmission line^, starting at time t = 0; the end χ = I is earthed. The initial voltage a n d initial current in the Une equal zero. Find the electrical vibrations in the line for ί > 0 a n d find at what time the current in the Une wiU differ by less than 1 0 per cent from the Umiting value (for t -> + oo). 95. Solve the preceding problem for the condition that the end Λ: = / is insulated. 96. f. Ε = const, is applied to the other end (x = 0 ) at time t = 0 .
Investigate the vibrations of a string with fixed ends x = 0 and X = /, excited by an initial deflection, depicted in Fig. 13 and evaluate the energy of the various harmonics. The initial velocities equal zero. FIG. 1 3 98. , applied at the point Xo of the string, perpendic ularly to the undisturbed position of the string. At the initial time ί = 0 the action of the force FQ ceases instantaneously. Find the vibrations of the string for t > 0. 99. The ends of a string are fixed, a n d the initial deflection has the form of a quadratic parabola, symmetrical with respect to the perpendicular to the mid-point of the string.
Solve the preceding problem in the case where the action of the force F = FQ continues only u p to a time ί = J > 0 . 162. Solve problem 1 6 0 in the case where F = FQ sin ωί. 163. The end χ = / of a rod is rigidly fixed, and the end χ = 0 is hinged. Find the transverse vibrations of the rod, produced by a uniformly distributed transverse force with hnear density /o sin ωί, apphed to the rod at time t = 0 . 4. Vibrations with Inhomogeneous Media and Other Conditions Leading to Equations with Variable Coe£Bcients; Calculations with Concentrated Forces and Masses 164.