We present the results of a computational study of the waves that leave a point source in a circular enclosure. We work in the geometric optics approximation. A video results from the instantaneous source, and a single picture from the effectively steady source. We thus find Green’s function for the scalar wave equation and for steady waves, respectively. We illustrate and explain some unexpected qualitative features of the waves, including the dependence of the focusing properties of the enclosure on the position of the point source. We compare our work with previous work on the natural focusing of light. The MATHEMATICA code that produces the video discussed in the paper is available on EPAPS.
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See EPAPS Document No. E-AJPIAS-73-010501 for the simulation software and accompanying manual.
A direct link to this document may be found in the online article’s HTML reference section. The document may also be reached via the EPAPS homepage (http://www.aip.org/pubservs/epaps.html) or from ftp.aip.org in the directory /epaps/. See the EPAPS homepage for more information.
7.
A basic inaccuracy of the movie frames, and consequently of our steady state solution, must be pointed out. Our use of Huygens’ wave fronts is not strictly correct. In two, as compared to one or three dimensions, there is a diffusive effect that causes a wake behind the otherwise sharp wave front at all times. For a more detailed discussion see Ref. 2, pp. 686–688, 843. I wish to thank Professor Alexios Polychronakos of CCNY for pointing out this important effect.
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We do not pretend to explain the dynamics of the production of sound in cracking. We only suggest that a rapid (singular) motion of the material near the tip of the whip is plausibly involved.
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2005
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