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

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In order to determine the function u(x, t) we obtain the boundary-value problems: (a) when the ends of the rod are rigidly fixed, then Utt = a^Uxx if «(0,/) = «(/,/) = 0 «(Λ:, 0) /(χ), ut{x, 0) = F{x) 0

87. One end of a horizontal rod is rigidly fixed a n d the other end is free. At the initial time ί = 0 a mass Q = Mg strikes the free end of the rod with a velocity VQ, directed along the axis of the rod, and remains in contact with it until t = ÍQ. Find the longi tudinal vibrations of the rod for t > 0. 88. Solve the preceding problem for a rod, both ends of which are free. 89. Solve problem 87, assuming that the rod has the form of a truncated cone. 90. Solve problem 88 for a rod having the shape of a truncated cone.

Formulate the problem on the torsion vibrations of a cylin der, similar to problem 41 on the electrical vibrations in a conductor, taking the function characterizing the electrical vibrations, first as the voltage and then as the strength of the current. Estabhsh the necessary and sufficient conditions that the first problem is similar to the second. § 2. Method of Travelling Waves (D^Alembert's Method) The general solution u = w(x, t) of the wave equation "if = a^u^x (1) may be represented in the formf 0 = φ^(χ-αί)+φ^χ+αί), where φι(ζ) and ^ai^) arc arbitrary functions, φι{χ—α() (2) is a for ward wave, propagating in the positive direction with respect to t It is sometimes more convenient to make use of other equivalent forms of representing the solution in the form of a travelling wave, for example, or COLLECTION OF PROBLEMS ON MATHEMATICAL PHYSICS [52 the X-axis with a velocity a, and φ<^{χ^α{) is a backward wave, propagating with the same velocity in the negative direction t .