## Abstract

At the end of the sixties, it was shown that pressure waves in a bubbly liquid obey the KdV equation, the nonlinear term coming from convective acceleration and the dispersive term from volume oscillations of the bubbles.

For a variable *u*, proportional to -*p*, where *p* denotes pressure, the appropriate KdV equation can be casted in the form *u* _{ t } -6*uu* _{ x } +*u* _{ xxx } = 0. The theory of this equation predicts that, under certain conditions, solitons evolve from an initial profile *u*(*x*, 0). In particular, it can be shown that the number *N* of those solitons can be found from solving the eigenvalue problem *ψ* _{ xx } -*u*(*x*,0)*ψ* = 0, with *ψ*(0) = 1 and *ψ*′(0) = 0. *N* is found from counting the zeros of the solution of this equation between *x* = 0 and *x* = *Q*, say, *Q* being determined by the shape of *u*(*x*,0). We took as an initial pressure profile a Shockwave, followed by an expansion wave. This can be realised in the laboratory and the problem, formulated above, can be solved exactly.

In this contribution the solution is outlined and it is shown from the experimental results that from the said initial disturbance, indeed solitons evolve in the predicated quantity.

## Key words

waves solitons## Preview

Unable to display preview. Download preview PDF.

## References

- 1.Hammack, J. L. and Segur, H.: The Korteweg-de Vries equation and water waves. Part 2, Comparison with experiments,
*J. Fluid Mech*.**65**(2), (1974), 289–314.MathSciNetzbMATHCrossRefGoogle Scholar - 2.Ince, E. L.:
*Ordinary Differential Equations*, Dover Publications, 1956.Google Scholar - 3.Korteweg, D. J. and de Vries, G.: On the change of form of long waves advancing in a rectangular canal and on a new type of long stationary waves,
*Phil. Mag*.**39**(240) (1895), 422.zbMATHGoogle Scholar - 4.Minnaert, M: Musical air bubbles and the sound of running water,
*Phil. Mag*.**16**(1933), 235.Google Scholar - 5.Nakoryakov, V. E., Pokusaev, B. C, and Shreiber, I. R.:
*Wave Propagation in Gas-Liquid Media*, Ch. 3, 1993.Google Scholar - 6.Noordzij, L.: Shock waves in bubble-liquid mixtures, in L. I. Sedov (ed.),
*Non-Steady Flow of Water at High Speeds*, Proc. IUTAM Symp., Nauka, Moscow, 1973.Google Scholar - 7.Noordzij, L. and Wijngaarden, L. van: Relaxation effects, caused by relative motion, on shock waves in gas-bubble/liquid mixtures,
*J. Fluid Mech*.**66**(1) (1974), 115–143.zbMATHCrossRefGoogle Scholar - 8.Watanabe, M. and Prosperetti, A.: Shock waves in dilute bubbly liquids,
*J. Fluid Mech*.**274**(1994), 349–381.MathSciNetzbMATHCrossRefGoogle Scholar - 9.Whitham, G. B.:
*Linear and Nonlinear Waves*, Wiley, New York, 1974.zbMATHGoogle Scholar - 10.Whittaker, E. T. and Watson, G. N.:
*A Course of Modern Analysis*, Cambridge University Press, Cambridge, 1952.Google Scholar - 11.Wijngaarden, L. van: On the equations of motion for mixtures of liquid and gas bubbles,
*J. Fluid Mech*.**33**(3) (1968), 465–474.zbMATHCrossRefGoogle Scholar - 12.Wijngaarden, L. van: One-dimensional flow of liquids containing small gas bubbles,
*Ann. Rev. Fluid Mech*.**4**(1972), 369–396.CrossRefGoogle Scholar