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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EQUATIONS OF HYPERBOLIC TYPE 3] 3 . The axis Ox is directed along the longitudinal axis of the cylinder, and the angle of deflection of a cross-section with abscissa χ is denoted by θ(χ, /), the ends of the cylinder being defined by the abscissae jc = 0 and χ = /. We obtain the following boundary-value problems for the function 0 ( x , / ) : (a) in the case of rigidly fixed ends = 0

6 53. An infinite string is excited by a locahzed initial having the form of a quadratic parabola (Fig. 7). 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.

7). 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.