Hostname: page-component-848d4c4894-xfwgj Total loading time: 0 Render date: 2024-06-19T19:43:23.201Z Has data issue: false hasContentIssue false

Physical interpretation of the mathematical theory of wave generation by wind

Published online by Cambridge University Press:  28 March 2006

M. J. Lighthill
Affiliation:
Royal Aircraft Establishment, Farnborough, Hants

Abstract

In the light of accumulated evidence in favour of Miles's mathematical theory of wave generation by wind, the author has thought it desirable to translate the theory into the form of a physical argument, which goes as follows. Travelling water waves in a wind produce, to a first approximation, airflow undulations with pressures least over crests and greatest over troughs. Hence, just above the ‘critical height’ where the airflow component in the direction of propagation equals the wave velocity, air after slowly overtaking a crest is turned back by the higher pressure near the trough, moves down to a lower level and back towards the crest. Similarly, behind crests, an upward movement at the critical height occurs. Quantitatively, the vertical velocity v is such that the ‘vortex force’ − ρων (where ρ is density and ω vorticity) balances the sinusoidal pressure gradient. Furthermore, since in turbulent boundary layers vorticity decreases with height, any downflow produces a local vorticity defect, and upflow a local vorticity excess, and hence the vortex force varies about a negative mean at the critical height (although at other levels, where air moves sinusoidally, with vertical displacement and velocity 90° out of phase, the mean vortex force is zero). The negative total mean force extracts wind energy, and transfers it to the wave, enabling it to grow exponentially. For pressure gradients adequate to initiate substantial energy transfer, the critical height must be fairly small compared with the wavelength, and waves can grow whenever their velocity and direction satisfies this condition, a conclusion supported by measurements (Longuet-Higgins 1962) of the directional spectrum of wind-generated waves.

Type
Research Article
Copyright
© 1962 Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Brooke Benjamin, T. 1959 J. Fluid Mech. 6, 161.
Francis, J. R. D. 1954 Phil. Mag. (7), 45, 695.
Francis, J. R. D. 1956 Phil. Mag. (8), 1, 685.
Jeffreys, H. 1925 Proc. Roy. Soc. A, 110, 341.
Kelvin, Lord 1871 Mathematical and Physical Papers, Vol. IV, pp. 7685. Cambridge University Press (1910).
Lamb, H. 1932 Hydrodynamic, 6th ed. Cambridge University Press.
Lighthill, M. J. 1957 J. Fluid Mech. 3, 113.
Lighthill, M. J. 1958 An Introduction to Fourier Analysis and Generalised Functions. Cambridge University Press.
Lin, C. C. 1955 The Theory of Hydrodynamic Stability. Cambridge University Press.
Longuet-Higgins, M. S. 1962 Proc. Roy. Soc. A, 265, 286.
Longuet-Higgins, M. S., Cartwright, D. E. & Smith, N. D. 1962 Proc. Conference on Ocean Wave Spectra, Easton, U.S.A. Nat. Acad. Sci. (Washington).
Miles, J. W. 1957 J. Fluid Mech. 3, 185.
Miles, J. W. 1959a J. Fluid Mech. 6, 568.
Miles, J. W. 1959b J. Fluid Mech. 6, 583.
Miles, J. W. 1960 J. Fluid Mech. 7, 469.
Phillips, O. M. 1957 J. Fluid Mech. 2, 417.
Phillips, O. M. 1958 J. Fluid Mech. 4, 426.
Phillips, O. M. 1959 J. Fluid Mech. 5, 177.
Phillips, O. M. 1960 J. Fluid Mech. 9, 193.
Prandtl, L. 1918 Nachr. Ges. Wiss. Göttingen, Math.-phys. Kl. (1918), 151.
Stewart, R. W. 1961 J. Fluid Mech. 10, 189.
Taylor, G. I. 1915 Phil. Trans. A, 215, 1.
Ursell, F. 1956 Surveys in Mechanics, pp. 21649. Cambridge University Press.
Willmarth, W. W. 1959 Nat. Aero. Space Admin. Mem. no. 3-17-59W.