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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96. f. Ε = const, is applied to the other end (x = 0 ) at time t = 0 . Find the voUage v(x, t) at the end Λ: = / for ί > 0 for aU cases. § 3. Method of Separation of Variables In this section problems on vibrations of a finite section of a string with various boundary conditions are considered, and also analogous problems on vibrations from other fields of physics and engineering. t See the footnote to problem 58. 100] II. EQUATIONS OF HYPERBOLIC TYPE 29 1. Free Vibrations in a Non-resistant Medium^ 97.

Solve the boundary-value problem Utt = a^Uxx+f{x,t), w(x,0)=:0, —oo

F i n d : (a) describing the profile of the string for t > 0, and (b) representing the law of motion of an arbitrary point string for t > 0. deflection formulae, formulae, χ of the t See [7], pages 39-54 and 57-68. Use of solutions in the form (2) for steady-state problems, where / is a geometric coordinate, will be given in chap­ ter V. t Here and in later problems a means the wave velocity appearing in equa­ tion (1) Utt = a^Uxx. 56] EQUATIONS OF HYPERBOLIC TYPE 19 54. At time ί = 0 an infinite string is excited by an initial deflection, having the form described in Fig.

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