On the distribution of the heights of sea waves: Some effects of nonlinearity and finite band width

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Abstract
It is shown that some recent data on the crest‐to‐trough heights of sea waves are fitted just as well as by the one‐parameter Rayleigh distribution as by the two‐parameter Weibull distribution, provided that the rms amplitude ā is taken as 0.925(2m0)1/2, where m0 is the lowest moment of the frequency spectrum. Reasons why the ratio ā/(2m0)1/2 should differ from unity are discussed. It is shown that the effect of finite wave steepness would be to increase the ratio by a factor [1 + (ak)2] approximately, contrary to observation. The effect of finite band width is estimated from a model assuming low background noise superposed linearly on a delta function spectrum. For narrow band widths one obtains the formula ā2/2m0 = 1 − 0.734ν2, where ν is the rms spread of the noise about the mean frequency. Values of ν2 corresponding to Pierson‐Moskowicz spectra give results in close agreement with observation.

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