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

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**Additional info for A Collection of Problems on Mathematical Physics**

**Example text**

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.