# Why size of the reflecting surface must be greater than the wavelength of sound wave?

My book says that "the only requirement for the reflection of sound wave is that the size of the reflecting surface must be bigger than the wavelength of sound wave". Why is that so? Shouldn't the amplitude be the factor?

## 3 Answers

To get a specular reflection (angle of incidence equals angle of reflection), the surface must be large in comparison with the wavelength. A small object will scatter all around.

And a row of trees or a a wall with corrugation can act as a diffraction grating.

You are correct, there is a nonzero amplitude reflexion from a subwavelength surface. Its reflexion will simply look like a point source in the farfield, and it is an inefficient scatterer. That is, for greater than wavelength breadth reflectors, the reflected power from an incident plane wave is roughly proportional to the area of the reflecting surface. For smaller than wavelength scatterers, the decrease in reflected power with decreasing wavelength is much faster than linear.

The corresponding phenomenon with light / electromagnetic waves is called Mie scattering, and, for very small reflectors, Rayleigh scattering. In the Rayleigh régime, the decrease is with the square of the cross sectional area, not linearly.

There are analogous phenomena for scalar acoustic waves, which, like electromagnetic waves, fulfill the Helmholtz equation. The final equations are slightly different because the electromagnetic boundary conditions force slightly different expressions, but the broad principles will be the same.

Below is a plot I made some years ago to check that some software I was working with would reproduce Figure 13-14 in Born and Wolf, "Principles of Optics". It is a plot of the effective cross-section of a scatterer as a function of its physical size. The vertical axis is twice the ratio of power scattered from a homogeneous sphere of water to the power one calculates assuming ray theory. The horizontal axis is the size factor $\sigma = \frac{2\,\pi\,a}{\lambda}$.

The curve is saying that for size factors of 2 or greater, the scattered power can be calculated by ray theory, by thinking of light as little pellets that bounce off a macroscopic object blocking them. That is, the scattering cross section roughly equals the cross sectional area of the target. For much lower size factors, the scattering cross section is much smaller than the "physical" target cross sectional area, and indeed as the object gets very small, the scattering is proportional to the inverse fourth power of the wavelength - much smaller than the linear decrease you would expect from ray theory.

Sound is a longitudinal wave which travels through the medium (it can’t travel without a medium), by creating a wave of compression and rarefaction alternatively and being a wave it has a fundamental property called diffraction which means when a wave passes through an obstacle it bends towards its edges as you might have observed in case of light passing through a hole or a slit it spreads outwards. You can verify it by using a Laser torch, point it on a wall you will see a small red dot now put a needle or a pen in between the beam of light so that only a part of light can hit the wall, you will see the red line instead of dot, thats diffraction. The diffraction is maximum if the obstacle is smaller and its maximum if the obstacle is smaller then the wavelength of the wave in such case most of the wave is diffracted from the edge and that’s why the amount of wave reflected is negligible. That answers your first question

Now, Scattering is the phenomenon in which the wave get deflected from its path multiple times and in random directions. Its more like the wave gets diffused in the medium. Reflection is one of the causes of scattering, refraction and diffraction being other causes. Tyndall Effect on foggy morning is a perfect example of scattering.

The size of the particles is an important factor in scattering. If the size is more then the wavelength of wave then the scattering is caused by multiple geometric reflections, which is justified by the first paragraph. Tyndall effect, muddy water are examples of this type of scattering. If the size of the particles is smaller then the wavelength of the wave, for example size of the air molecules is in order of 10^(-12) Picometers whereas the wavelength of visible light is of the order of 10^(-7) (100 nanometers), in such cases the photon (the light particles) hits the gas molecules like oxygen and nitrogen which are very abundant in our atmosphere, the electron of the molecule jumps in the higher energy levels by absorbing the photon and then re-emitting a photon of equal or smaller energy. But the absorption happens only when a single photon has sufficient energy to make the electron jump, two photons can’t contribute to make a single jump. Which is why in our example the blue light which has higher frequency (or smaller wavelength) which means higher energy get scattered by the atmospheric gases almost 9.5 times more then the red light which has lover frequency which means less energy, and thats why we see the sky blue, This is called Raman Scattering.

I didn’t plan for such a long answer but just couldn’t finish in any less. There is a lot more to scattering which you can check in the books or on the Internet