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The nonlinear oscillations of a string

A. G. Mackie

1968
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Quarterly of Applied Mathematics
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This note is intended to shed further light on an observation by Rosen [1] concerning papers by Zabusky [2] and Kruskal and Zabusky [3] in which he shows that the nonlinear vibrations of a string considered in these papers are formally equivalent to solutions of the one-dimensional equations of motion of a polytropic gas in homentropic flow. We show here that the equivalence is more than formal and that the equation (4) below governing the motion of a string considered in [2] and [3] is exactly
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... and [3] is exactly the same as that governing the motion of a gas with a certain equation of state, provided Lagrangian co-ordinates are used to describe the motion of the gas. By having the equivalence linked in this way through the Lagrangian representation, Rosen's observation is motivated physically and may in fact be generalized. We consider a gas in one-dimensional motion with p, the pressure, a given function of p, the density. Let x be the length co-ordinate measured in the given direction and let y(x, t) be the longitudinal displacement at time t of a particle of gas which at t = 0 was at position x. Let us also suppose for convenience that p = p0, p = p0 for all x when t = 0, the suffix zero denoting a constant state. Then it is easy to show that the continuity equation gives p(l + dy/dx) = p0 (1> and that the equation of motion is Pa(d2y/dt2) = -dp/dx. If c = dp /dp, then the equation of motion is = (pV/pl)(a2y/dx2) from (1) , or alternatively d2y/dt2 = /(I + dy/dx){d2y/dx2), . (3) since c is a function of p. For a polytropic gas in which p p1, p2c pT+1 and d2y/dt2 = K( 1 + dy/dxy-\d2y/dx2), where K is some positive constant. Trivial scaling changes of variables now enable the equation to be reduced to that studied in [2] and [3], namely d2y/dt2 = (1 + <(dy/dx)Y(d2y/dx2). This approach also leads to a generalization of Rosen's analogy. For a gas which is nonpolytropic but which has an acceptable equation of state, the acoustic impedance pc is a decreasing function of 1/p. Accordingly, from (1) and (2), we may consider strings with an equation of motion of the general form (3) provided /'CO and still be able to analyze their vibrations by means of the conventional apparatus of one-dimensional gas dynamics.

doi:10.1090/qam/99881
fatcat:4sb4t3wgpfhmnhlqfhntpekiuy