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**

**Example text**

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