Download e-book for iPad: A Collection of Problems on Mathematical Physics by B. M. Budak, A. A. Samarskii, A. N. Tikhonov, I. N. Sneddon,

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

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A semi-infinite flexible rod 0 < x < +oo with a free end at X = 0 is perturbed at time ί = 0 by longitudinal displacements, the profile of which+ is depicted in Fig. 10. Find at what points and when for ί > 0 the displacement reaches its greatest value. What is the value of this greatest displacement? 63. An impulse / is transmitted at a point χ = XQ io Ά semiinfinite string with a fixed end at the initial time ί 0 by means of a transverse blow. Find the deflections 0 of points of the string from positions of equilibrium for ί > 0 if the initial deflec­ tions u{x, 0) = 0, and the initial velocities at points χ φ x^ also equal zero.

F(t) = - f - + Σ («/I COS ηωΐ+βη sin ηωί), 0 < / < + oo. 141. A continuously distributed force with hnear density Φ(χ, t) = 0o(x)sin ωt is applied to a string 0 < χ < / with rigidly fixed ends at time t = 0. Find the vibrations of the string for zero initial conditions assuming that the medium produces a resistance 147] Π. EQUATIONS OF HYPERBOLIC TYPE 35 proportional to the velocity. Find the steady-state vibrations, re­ presenting the principal part of the solution for ί + 0 0 . ) Note. The steady-state oscillations have the frequency of the constraining force; oscillations with other frequencies are damped out.

43 II. EQUATIONS OF HYPERBOLIC TYPE 192] is valid if the function / ( x , 0 is extended over the negative semiaxis of X so that the function is o d d with respect t o x. 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

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