Beard and Chuang model

The Beard and Chuang model is a well known and leading theoretical force balance model used to derive the rotational cross-sections of raindrops in their equilibrium state by employing Chebyshev polynomials in series.

Beard and Chuang model of raindrop

The radius-vector of the raindrop's surface in vertical angular direction is equal to

,

where shape coefficients are defined for the raindrops with different equivolumetric diameter as in following table

d(mm) n = 0 1 2 3 4 5 6 7 8 9 10
2.0 -131 -120 -376 -96 -4 15 5 0 -2 0 1
2.5 -201 -172 -567 -137 3 29 8 -2 -4 0 1
3.0 -282 -230 -779 -175 21 46 11 -6 -7 0 3
3.5 -369 -285 -998 -207 48 68 13 -13 -10 0 5
4.0 -458 -335 -1211 -227 83 89 12 -21 -13 1 8
4.5 -549 -377 -1421 -240 126 110 9 -31 -16 4 11
5.0 -644 -416 -1629 -246 176 131 2 -44 -18 9 14
5.5 -742 -454 -1837 -244 234 150 -7 -58 -19 15 19
6.0 -840 -480 -2034 -237 297 166 -21 -72 -19 24 23

Applications

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The description of raindrop shape has some rather practical uses. Understanding rain is particularly important with regard to the propagation of electromagnetic signals. A portion of atmosphere that has rain in it, or a rain cell, has the characteristic of attenuating and de-polarizing EM signals that pass through it. The attenuation of such a signal is approximately proportional to the square of the frequency of the signal, and the de-polarization is proportional to the shape distribution of raindrops in the rain cell.

References

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  • K. V. Beard and C. Chuang. A new model for the equilibrium shape of raindrops. Journal of the Atmospheric Sciences, 44:1509-1524, 1987.