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Reflection of Spherical or Plane Waves

When a spreading spherical wave reaches a wall or other obstacle, it is either reflected or scattered. A wavefront is reflected when it impinges on a surface which is flat over at least a few wavelengths in each direction.3.1 Reflected wavefronts can be easily mapped using ray tracing, i.e., the reflected ray leaves at an angle to the surface equal to the angle of incidence (``law of reflection''). Wavefront reflection is also called specular reflection, especially when considering light waves.

A wave is scattered when it encounters a surface which has variations on the scale of the spatial wavelength. A scattering reflection is also called a diffuse reflection. As a special case, objects smaller than a wavelength yield a diffuse reflection which approaches a spherical wave as the object approaches zero volume. More generally, each point of a scatterer can be seen as emitting a new spherically spreading wavefront in response to the incoming wave--a decomposition known as Huygen's principle, as mentioned in the previous section. The same process happens in reflection, but the hemispheres emitted by each point of the flat reflecting surface combine to form a more organized wavefront which is the same as the incident wave but traveling in a new direction.

The distinction between specular and diffuse reflections is dependent on frequency. Since sound travels approximately 1 foot per millisecond, a cube 1 foot on each side will ``specularly reflect'' directed ``beams'' of sound energy above $ 1$ KHz, and will ``diffuse'' or scatter sound energy below $ 1$ KHz. A good concert hall, for example, will have plenty of diffusion. As a general rule, reverberation should be diffuse in order to avoid ``standing waves'' (isolated energetic modes). In other words, in reverberation, we wish to spread the sound energy uniformly in both time and space, and we do not want any specific spatial or temporal patterns in the reverberation.

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``Physical Audio Signal Processing'', by Julius O. Smith III, W3K Publishing, 2010, ISBN 978-0-9745607-2-4.
Copyright © 2015-05-22 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University