Algorithms for Fresnel diffraction at rectangular and circular apertures

Abstract

This paper summarizes the theory of Fresnel diffraction by plane rectangular and circular apertures with a view toward numerical computations. Approximations found in the earlier literature, and now obsolete, have been eliminated and replaced by algorithms suitable for use on a personal computer.

Synthesis of audio spectra using a diffraction model

“…We will consider cases when approaches zero, unity and large values. 10 For large values of , Eq. ͑2a͒, reduces to Eq.…”

“…The parameter , on the other hand, depends on the model setup values as well as the speed of sound. 10 Fuks has shown that the speed of sound changes significantly during the period of production of sound in wind instruments due to the changing ratio of oxygen and carbon dioxide in the air. 15,16 Hence, the parameter can be assumed to evolve in time, which results in a dynamic spectrum.…”

“…for a fixed frequency of the originating sound wave. [10][11][12] In the proposed model, the intensity and phase of the diffracted sound wave at P are obtained for varying frequencies of the originating sound wave in terms of two new model parameters and , which can be expressed in terms of the Lommel variables, u and v as = v / u and = v / f.…”

“…1. 10 Let X and I denote, respectively, the amplitude and intensity of the sound wave at a point P which is at a specific distance from the center of the aperture as shown in Fig. 1.…”

“…1. The values of X and I can be determined by using Lommel functions, 10 which are basically series expansions in Bessel functions.…”