# Mathematical relation between wavelength of wave and area of surface for reflection

I was reading about the reflection of sound waves, and the condition provided is that the reflecting surface's area should be greater than the wave's wavelength. But there isn't a single mention of the minimum area required for reflection for a particular wavelength.
Is there any relation to this? If present, please provide the mathematical relation and some references to learn these topics; I am new to this field. Also, while reading this answer-
Why size of the reflecting surface must be greater than the wavelength of sound wave?, URL (version: 2018-03-09): https://physics.stackexchange.com/q/391121 There was a mention of subwavelength surfaces. How do we decide the surface area for reflection of waves whose wavelength is very large( even if we know min. area, how to take tolerances, etc. )? For example-Finding minimum area for reflection for a sound wave of 20Hz.

• If the area of the reflecting surface is smaller than the wavelength, then for some very small angles of reflection, close to grading, a part of the wavelength (close to the edge) will not be reflected, it will be diffracted at the edge and all reflection theory will not hold for this position. I am not sure if there's another phenomenon taking place in such a case though. Additionally, the link doesn't seem to work due to the last backslash. Remove it and it will be OK. Commented May 29, 2023 at 13:52
• There is no such area. For objects that are much smaller than the wavelength the effective scattering cross section is given by Rayleigh scattering, for objects that are much larger it's classical reflection. In the range between these two extremes we have to solve the wave equation explicitly to get the correct scattering function, which is admittedly hard to do for the general case. Commented May 29, 2023 at 14:26

In wave scattering, everything reflects and everything diffracts, the question is in what relative amount. In this sense forward scattering is diffraction, backward scattering is reflection. If the scattering object is much larger, say 10-fold, in all its geometric dimensions than the wavelength, then we can talk about primarily backward scattering and with little or insignificant diffraction. Contrariwise, if the wavelength is about of the order or less than the size of the object then scattering is mostly forward, i.e., diffracted and not reflected. This is true for acoustic as well as for radio or water waves. At $$20Hz$$ audio the wavelength in air is ~17m and will diffract around most man-made objects with very little be reflected. In water, where the sound velocity is more like 1,500m/sec the wavelength is 75m, only the bottom and the top is reflecting, everything else is diffracting.