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The statistical distribution of the maxima of a random function By D. E. Cartwright and M. S. Longuet-Higgins National Institute of Oceanography, Wormley (Communicated by G. E. R. Deacon, Received 14 April 1956) This paper studies the statistical distribution of the maximum values of a random function which is the sum of an infinite number of sine waves in random phase. The results are applied to sea waves and to the pitching and rolling motion of a ship.
Statistical distribution of the maxima of a random function 213 i In § 1 we outline briefly Rice’s derivation of the statistical distribution of the naxima £. The discussion shows that the distribution depends, surprisingly, on only (vo parameters: the root-mean-square value of which we denote by m\, and fparameter e which, as we show in § 2, represents the relative width of the frequency pectrum of f(t). When e is small, the distribution of £ tends to a Rayleigh dis tribution, as we should expect, and when e approaches its maximum value 1 the distribution of £ tends to a Gaussian distribution.
214 D. E. Cartwright and M. S. Longuet-Higgins where p(gx, £a, £3) is the joint probability distribution of ( f o f c f . ) (!*«) The mean frequency of maxima in the range £x </< £x d£x is therefore ^ i ) d ^ = f C ^ 0 ^ 8)|^ |d ^ ]d f8, (1-7) J — 00 and the probability distribution of maxima is found by dividing this distribution by the total mean frequency of maxima, which is Nx = P" f° P(£v 0 ,^)1 ^ d£xd£3. (18) J —oo J —oo Now from (1-6) we have £l =/(0 = IX C°S (<7j + en)’ n £2 = /'(*) = “ 5X<rnsin(°V* + eJ> . (1-9) 71 I = AO = -2^0-2 COS (<rn£ + eJ.
218 D. E. Cartwright and M. S. Longuet-Higgins But from Rice (1944, 1945) and equation (1-18) we have (3*5) So equation (3*4) can be written 1 r Y m2 |[ l - ( l - e 2)i]. (3*6) 2 (m0m4)l Hence the proportion of negative maxima increases steadily with the relative width of the spectrum. Conversely, we have e2 = l-(l-2 r)2. (3-7) This relation provides us with a ready means of estimating e by simply counting the numbers of positive and negative maxima in a length of record.
Statistical distribution of the maxima of a random function 219 In particular we have = 1, = l- e 2)*, (4-8) /4 = 2 — e2, /4 = ($»)*(!-«*)*.3., fe see that the mean /i[ is a steadily decreasing function of e, the width of the pectrum. A non-dimensional quantity depending on e is the ratio /42 1 — e2 (4*9) (i77-) 2 —e2* ’he width of the spectrum is given in terms of p by the relation g-4p (4-10) Tl — Ip* Figure 2. Graphs of the mean pi, variance fi2, skewness /?, proportion r of negative maxima, and P(— as functions of e.
Statistical distribution of the maxima of a random function 221 Phe coefficient of skewness is given by 1 — e2 P = i7*-3) (4-14) L l-(|7r-l)(l-e2)J M ,Ve see that the standard deviation steadily increases as e increases. /?, on the ither hand, steadily decreases.