Mean Square Slope Research Articles

Distribution and Steepness of Ripples on Carrier Waves

The slopes of ripples and the profiles of their carrier waves were simultaneously measured in a wind-wave tank with winds of various velocities blowing over preexisting, long, regular surface waves. The results include the apportionment and the slope distributions (and therefore the mean-square slopes) of ripples located on various portions of the carrier-wave profile. At low wind velocities, the surface-tension governing regime of wind-wave interaction, the leeward face of the carrier wave was found to contain more ripples than the windward face. The parasitic capillaries are concentrated on the upper half of the leeward face, and move along the leeward face toward the trough of carrier waves as the wind velocity increases. At high wind velocities, the gravity governing regime of wind-wave interaction, the ripples become more evenly distributed on the leeward and on the windward faces. How ever, the ripples on the windward face are concentrated near the carrier-wave crest, and the ripples on the leeward face are concentrated near the carrier-wave trough. At all wind velocities, the rms slope of ripples on the windward face of the carrier waves is greater than that on the leeward face.

Transient Response of a Shear Beam to Correlated Random Boundary Excitation

The transient mean-square displacement, slope, and relative motion of a viscously damped shear beam subjected to correlated random boundary excitation is presented. The effects of various system parameters including the spectral characteristics of the excitation, the delay time between the beam support motion, and the beam damping have been investigated. Marked amplifications in the mean-square response are shown to occur for certain dimensionless time delays.

On the variation and growth of wave-slope spectra in the capillary-gravity range with increasing wind

A new laser device has been used to make direct wave-slope measurements in the capillary-gravity range. Owing to the design principles, the digital nature of the system and the use of a laser beam as a probe, the earlier problems of intensity variations and meniscus effects were avoided. Using this new technique, wave-slope spectra both down and across the channel were obtained for different wind conditions, along with corresponding mean-square slope values. Comparisons are made with existing data. The results indicate that a quasi-equilibrium state may exist for each wind speed and that it increases in intensity with increasing wind, which may imply an asymptotic nature for the equilibrium-range coefficientCaa. From the data, two significant frictional velocities, 17.5 and 31 cm/s respectively, are identified as critical values for different ranges of wave development.

Wave directional spectra from synthetic aperture observations of radio scatter

We have measured the directional distribution of waves produced by a quasi-stationary, homogeneous wind field. This was done by observing radio backscatter from the LORAN A transmitter on Wake Island. The LORAN frequency of 2 MHz is in Bragg resonance with 7 S ocean waves. A directional resolution of ±3° has been attained from a synthesized antenna formed by moving the receiver at constant speed along the airport taxiway. The directional distribution of the 7 S ocean waves as inferred from the variable scatter cross-section compared favorably with directional moments measured with a pitch-and-roll buoy. The directional wave energy is found to fall off monotonically with angle θ relative to the wind, in accord with a model cos S( 1 2 θ ) proposed by Longuet-Higgins, Cartwright and Smith, Ocean wave spectra, pp 111–136, Prentice-Hall (1963); s increases with wind speed and decreases with frequency, and can be plotted against a single parameter μ = μ ∗/cκ , where √ ( u ∗/ρ ) is wind stress, c is phase velocity, and κ = 0·4 is von Karman's constant. The radio measurements give an upper limit of 0·02 for the upwind/downwind ration in wave energy flux. The ratio of up-down to cross-wind components in mean-square slope computed for the cos S( 1 2 θ ) model is compared to the ratio inferred from measurements of sun glitter. The (second-order) mean-square pressure fluctuations at great depth are computed from our estimates of oppositely traveling wave energy.

Sea-Surface Slope and Equilibrium Wind-Wave Spectra

The measured mean-square slope of the sea surface is shown to vary logarithmically with the wind velocity, and the rates of variation change with different boundary-layer regimes of the wind. Based on mean-square slope data, the form of a newly proposed equilibrium wind-wave spectrum for the capillary range is verified, the spectral coefficients for both gravity and capillary ranges are determined, and the cutoff wavenumber for the slope data is shown to be much smaller than the neutrally stable wavenumber obtained from laminar model. Finally, an approximate correlation is presented between the mean-square slope and the roughness height of the sea surface, implying that the growth of small waves is mainly due to the direct action of the wind.

Nonstationary surface roughness measurement by light scattering

A noncontacting method of measuring the mean-square slope of realistic, non-stationary surface roughness is suggested. The method is based on the relationship between the mean scattered power and the two-dimensional slope probability density function, which is valid in the high-frequency limit.

Slope and Curvature Distributions of Wind-Disturbed Water Surface

The microstructure of the wind-disturbed water surface, characterized by surface-slope and surface-curvature distributions, is studied in a laboratory tank with a newly developed optical instrument. Some features of these distributions under various wind and wave conditions are systematically presented. It is also shown that wind waves arise at about the time when the airflow boundary layer becomes turbulent. At lower wind velocities, the formation of parasitic waves causes a skewed slope distribution; at high wind velocities, the wave breaking causes a peaked slope distribution. With a turbulent wind, similar variation of the mean-square slope with the shear velocity is shown in the present laboratory results and in older oceanic data, suggesting that the laboratory facility may be able to simulate the microstructure of the air–sea interface.

Probability distributions of water‐wave slopes under conditions of short fetch

Slope probability distributions were measured for a fetch of about 15 inches, using a small wind tunnel containing a water boundary. For this very short fetch, the meansquare slope was found to be higher than previous longer‐fetch data for wind velocities greater than about 10 kn. At 18 kn the short‐fetch mean‐square slope was 70 pct higher. Evidence of peakedness of the slope distributions was found in the short‐fetch data. The mean value of the distributions was measured to reach a maximum in the downwind direction and was about 1.5° in the region of 18 kn.