Download A Collection of Problems on Mathematical Physics by B. M. Budak, A. A. Samarskii, A. N. Tikhonov, I. N. Sneddon, PDF

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

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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.

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