# Download A Collection of Problems on Mathematical Physics by B. M. Budak, A. A. Samarskii, A. N. Tikhonov, I. N. Sneddon, PDF

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

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Investigate the solution for E^-^O and for £"2 - > +00. 79. At the point χ = 0 of an infinite homogeneous string a concentrated mass Μ is attached, supported by a spring of rigidity k with negligibly small internal mass (Fig. 11). Find the deflection of the string u{x, t) for ί > 0, if the string is excited at time r = 0 by a transverse impulse I = MVQ, transmitted to the mass Μ and directed along the axis of the spring. 26 COLLECTION OF PROBLEMS ON MATHEMATICAL PHYSICS [80 80. The mass Μ of the preceding problem, in oscillating, ex­ periences a frictional resistance proportional to the velocity.

92] II. EQUATIONS OF HYPERBOLIC TYPE 27 84. Solve the problem of the longitudinal vibrations of a rod, one end of which {x = 0) is rigidly fixed, and the other end (x = /) is free, if the rod has an inhial extension u{x,0) = Ax, 0

Find the longitudinal vibrations of the rod for t >0, if the initial velocities a n d initial deflections of its points equal zero. 75. A semi-infinite horizontal tube of constant cross-section is filled at ί < 0 with a fluid at rest. Beginning at time ί = 0 a pressure p u m p with a compensating air cap is fitted t o its e n d t . Find the pressure a n d velocity of the fluid in the tube for t > 0. t See problems 5 and 6. 79] II. EQUATIONS OF HYPERBOLIC TYPE 25 76. Find the longitudinal vibrations of a semi-infinite rod with zero initial conditions, if at the times tk = kT, /c = 0 , 1, 2 , .