I was watching the rainbow today and started thinking about the effects of the rain falling in different directions.
The idea I had was that normally we model rain drops as small spheres, and this gives a characterizing angle for the perceived rainbow.
Now, if the rain falls with a large horizontal velocity component, will this deform the raindrops, making them more elliptic, thus changing the angle of the perceived rainbow?
Or formulated in other words: Can one determine the speed of the falling rain from the angle of the rainbow?
PS: I know there could also be a minuscule relativistic effect here, due to length contraction, but for this particular question I'm not considering relativistic rain.
Best Answer
There is a theory of rainbows due to elliptical droplets. It was started by Willy Möbius [1] but is perhaps more clearly described in the modern papers [2,3]. Unfortunately the math is messy. While this may change the appearance of the rainbow, the biggest effect is on the angles between the different order rainbows: by measuring them one can presumably work backwards and get a value of the ellipticity.
Except for one problem: raindrops made ellipsoidal because of wind oscillate (they are water after all). They will hence change their ellipticity rapidly; to make matters worse, the degree of maximal ellipticity change scales roughly parabolically with droplet size and droplets can become non-ellipsoidal [4,5,6]. Hence a real rainstorm (with different droplet sizes and vibration modes) will produce a rather messy signal.
So the answer to your question is likely "yes, for rigid uniform droplets" and "no, not for real rain".
Citations:
[1] Möbius, W. (1910). Zur Theorie des Regenbogens und ihrer experimentellen Prüfung. Annalen der Physik, 338(16), 1493-1558.
[2] Lock, J. A., & Können, G. P. (2017). Rainbows by elliptically deformed drops. I. Möbius shift for high-order rainbows. Applied Optics, 56(19), G88-G97. http://www.guntherkonnen.com/articles/322
[3] Können, G. P., & Lock, J. A. (2017). Rainbows by elliptically deformed drops. II. The appearance of supernumeraries of high-order rainbows in rain showers. Applied Optics, 56(19), G98-G103. http://www.guntherkonnen.com/articles/323
[4] Szakáll, M., Mitra, S. K., Diehl, K., & Borrmann, S. (2010). Shapes and oscillations of falling raindrops—A review. Atmospheric research, 97(4), 416-425. http://e-science.sources.ru/sites/default/files/upload_forums_files/y0/sdarticle2.pdf
[5] Thurai, M., Bringi, V. N., Manić, A. B., Šekeljić, N. J., & Notaroš, B. M. (2014). Investigating raindrop shapes, oscillation modes, and implications for radio wave propagation. Radio Science, 49(10), 921-932.
[6] https://www.joanneum.at/uploads/tx_publicationlibrary/img494.pdf