By B. M. Budak, A. A. Samarskii, A. N. Tikhonov, I. N. Sneddon, M. Stark and S. Ulam (Auth.)
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Extra resources for A Collection of Problems on Mathematical Physics
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.
106. Find the vibrations of a flexible rod with free ends, which has received a longitudinal impulse / at one end at t = 0. 107. 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 , 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.
1] II. EQUATIONS OF HYPERBOLIC TYPE 173 If the ends of the rod are rigidly fixed, then the boundary conditions are obvious. If the ends of the rod are free or fixed elastically, then the boundary conditions may be derived from Newton's second law for boundary elements. Let us consider, for example, the case where the end Λ: = / is fixed elastically. From the left, the remaining part of the rod acts on the boundary element (/—ΔΛ:, /), adjoining this end, with a force -ESUXU-AXJ), from the right, a ñexible support with a forcet -KU(L, T).
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.)