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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 , .

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