Skip to main content
Log in

Stability of periodic waves of finite amplitude on the surface of a deep fluid

  • Published:
Journal of Applied Mechanics and Technical Physics Aims and scope


We study the stability of steady nonlinear waves on the surface of an infinitely deep fluid [1, 2]. In section 1, the equations of hydrodynamics for an ideal fluid with a free surface are transformed to canonical variables: the shape of the surface η(r, t) and the hydrodynamic potential ψ(r, t) at the surface are expressed in terms of these variables. By introducing canonical variables, we can consider the problem of the stability of surface waves as part of the more general problem of nonlinear waves in media with dispersion [3,4]. The resuits of the rest of the paper are also easily applicable to the general case.

In section 2, using a method similar to van der Pohl's method, we obtain simplified equations describing nonlinear waves in the small amplitude approximation. These equations are particularly simple if we assume that the wave packet is narrow. The equations have an exact solution which approximates a periodic wave of finite amplitude.

In section 3 we investigate the instability of periodic waves of finite amplitude. Instabilities of two types are found. The first type of instability is destructive instability, similar to the destructive instability of waves in a plasma [5, 6], In this type of instability, a pair of waves is simultaneously excited, the sum of the frequencies of which is a multiple of the frequency of the original wave. The most rapid destructive instability occurs for capillary waves and the slowest for gravitational waves. The second type of instability is the negative-pressure type, which arises because of the dependence of the nonlinear wave velocity on the amplitude; this results in an unbounded increase in the percentage modulation of the wave. This type of instability occurs for nonlinear waves through any media in which the sign of the second derivative in the dispersion law with respect to the wave number (d2ω/dk2) is different from the sign of the frequency shift due to the nonlinearity.

As announced by A. N. Litvak and V. I. Talanov [7], this type of instability was independently observed for nonlinear electromagnetic waves.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others


  1. H. Lamb, Hydrodynamics [Russian translation], OGIZ-Gostekhizdat, 1964.

  2. N. N. Moiseev, Surface Waves (introduction) [in Russian], Fizmatgiz, 1960.

  3. S. A. Akhmanov and R. V. Khokhlov, Problems in Nonlinear Optics [in Russian], Izd-vo AN SSSR, 1964.

  4. V. E. Zakharov, “A solvable model for weak turbulence,” PMTF [Journal of Applied Mechanics and Technical Physics], no. 1, p. 14, 1965.

    Google Scholar 

  5. V. N. Oraevskii and R. Z. Sagdeev, “On the stability of steady longitudinal oscillations of a plasma,” ZhTF, vol. 32, p. 1921, 1963.

    Google Scholar 

  6. V. N. Oraevskii, “The stability of nonlinear steady oscillations of a plasma,” Yadernyi sintez, vol. 4, no. 4, p. 263, 1964.

    Google Scholar 

  7. A. G. Litvak and V. I. Talanov, “The application of the parabolic equation to the calculation of fields in dispersing nonlinear media,” Izv. VUZ. Radiofizika, vol. 10, no. 4, p. 539, 1967.

    Google Scholar 

Download references

Author information

Authors and Affiliations


Additional information

The author wishes to thank L. V. Ovsyannikov and R. Z. Sagdeev for fruitful discussions.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Zakharov, V.E. Stability of periodic waves of finite amplitude on the surface of a deep fluid. J Appl Mech Tech Phys 9, 190–194 (1968).

Download citation

  • Issue Date:

  • DOI: