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Summary

The basics of wave propagation in three dimensions, including the wave nature of sound, are considered. When waves from a common source reach a common end point, but by different paths, wave interference can result. Wave amplitudes that add or subtract result in constructive and destructive interference, respectively. Diffraction results when the components of a continuum of sources, initially from the same source, arrive at an end point following different paths. Diffraction becomes very significant when the wavelength becomes comparable to, or larger than, the objects in its path. Diffusion of sound energy in a room will occur if the reflections look as if they are randomized. Increasing the amount of sound diffusion in a room is often desirable, so special acoustic treatments have been devised for this purpose that are known as diffusers. Some of those treatments are based on variable-depth walls, where the depths are determined by a numerical series, including pseudo-random numbers and some results from numbers theory.

Let $n$ be a positive integer and $a$ an integer prime to $n$. Multiplication by $a$ induces a permutation over $\mathbb{Z}/n\mathbb{Z}=\{\overline{0},\overline{1},\ldots ,\overline{n-1}\}$. Lerch’s theorem gives the sign of this permutation. We explore some applications of Lerch’s result to permutation problems involving quadratic residues modulo $p$ and confirm some conjectures posed by Sun [‘Quadratic residues and related permutations and identities’, Preprint, 2018, arXiv:1809.07766]. We also study permutations involving arbitrary $k$th power residues modulo $p$ and primitive roots modulo a power of $p$.

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16 - Interference, Diffraction, and Diffusion

Summary

APPLICATIONS OF LERCH’S THEOREM TO PERMUTATIONS OF QUADRATIC RESIDUES

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16 - Interference, Diffraction, and Diffusion

Summary

APPLICATIONS OF LERCH’S THEOREM TO PERMUTATIONS OF QUADRATIC RESIDUES