A Study of the Growth of the Wave Spectrum with Fetch

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Abstract
The development of the wave spectrum with fetch in a steady wind has been studied with a line of consecutive wave buoys in the Bothnian Sea in 1976 and 1979. The relationship that was found between dimensionless peak frequency ωm (=ωmU10/g) and dimensionless fetch X̄ (=gX/U102) was close to previous observations. The dimensionless energy ¯σ2 (=g2ω2/U104) was about twice that observed in the JONSWAP experiment. In the saturation range when ¯ω>4 the frequency spectrum was found to have the form S(ω) = αuU10gω−4 where αu=4.5 × 10−3, independent of the dimensionless fetch X̄. The deviation from the Phillips −5 power law could not be explained by the influence of currents or finite depth. Near the peak, the spectra were satisfactorily described by the JONSWAP spectrum; above frequencies twice the peak frequency the difference becomes significant. A qualitative explanation is proposed for the dependence of the spectrum on the wind speed in the saturation range. The semi-theoretical method of Longuet-Higgins (1969) to estimate the Phillips saturation-range constant is applied to estimate αu. The result (4.4–6.4) × 10−1 agrees with the experimental value. The growth of a component of the dimensionless spectrum with the fetch was found to be exponential within the accuracy of the data. The exponential growth parameter agreed with previous observations. A simple model is proposed to predict the growth rate without assuming nonlinear transfer of energy by wave-wave interactions; the results agree well with observations.
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