By William F. Tanner (auth.)
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Highly irregular or strongly embayed particles are less amenable to Fourier analysis. Because of the potential that Fourier analysis holds for accurate measurement of particle shape, it is described below in greater detail than were the preceding techniques for shape measurement. Fourier analysis General principles The fundamentals of Fourier analysis of particle shape were introduced to geologists particularly in the papers of Schwarcz and Shane (1969) and Ehrlich and Weinberg (1970). Applications and amplifications of the method have appeared in numerous subsequent papers.
2, p. 319. Copyright 1987 by Van Nostrand Reinhold. ) (higher order harmonics). When investigating the shapes of large numbers of grains in numerous samples, the huge amount of data generated in 20 or more harmonics of each grain in each sample makes cumbersome the analysis of each harmonic separately. It becomes desirable to determine which one or which few harmonics may yield the most useful information about shape for the purposes of a particular study, so that all harmonics need not be studied in equal detail.
Thus the harmonic y ¼ 2 sin ð3x þ p=3Þ has the form shown in Fig. 14D. 13 Method of “unrolling” a grain to produce a periodic wave. A. Grain outline showing radii measured from grain center to points on the grain perimeter. B. Unrolled grain outline as constructed from radii measurements. Note that the form of the unrolled grain is a crude sine wave. A mathematical theorem owing to Fourier shows that any periodic function with frequency ω can be represented as a superposition of harmonic vibrations of frequencies ω, 2ω, 3ω … with various amplitudes (harmonics 1, 2, 3…).
Beaches and Coastal Geology by William F. Tanner (auth.)