### ASA 127th Meeting M.I.T. 1994 June 6-10

## 5pSA10. On the nonlinear waves in a slender rod.

**S. A. Rybak
**

**
Yu. I. Skrynnikov
**

**
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*N. N. Andreev Acoust. Inst. of Russian Acad. of Sci., Shvernik str. 4,
117036, Moscow, Russia
*

*
*
To describe the propagation of nonlinear longitudinal strain waves in a
rod that curves into an arc of radius R use is made of the nonlinear
Klein--Gordon equation in the form u[sub tt]-c[sup 2]u[sub xx]+(c[sup 2]/R[sup
2])u=((beta)/2(rho))(u[sup 2])[sub xx]. Here, (beta) and (rho) are the
nonlinearity parameter and the density of the material, and u is the x
derivative (the x axis is directed along the rod) of the longitudinal component
of the displacement vector, i.e., the longitudinal strain; c is the velocity of
linear longitudinal waves. The steady-state exact solution in the form of
solitary wave was found. The wave has two symmetrically located vertical edges.
The tails of the wave obtained, i.e., the parts that decay infinitely in either
direction, have the same structure as the tails of a Korteweg--de Vries
soliton. As in the case of the Korteweg--de Vries soliton, the amplitude A and
the width (Delta) of the wave depend on its velocity. However, a different
relation exists between the amplitude and the width: (Delta)~A[sup 1/2].