Further Investigation of the
ACOUSTIC PROPERTIES OF PARABOLIC REFLECTORS
Randolph Scott Little
29 September 2007
In 1963 this author reported on “An Investigation of the Acoustic Properties of Parabolic Reflectors.” He developed a complex mathematical model for the acoustic pressure at the focal point of such a reflector due to on-axis incident acoustic excitation. Quantitative solution of the resulting equation was thought to be beyond ordinary means at the time and left as an exercise for the emerging computer generation. That unsolved equation has, for over 40 years, simmered in this author’s mind, waiting for an inspiration or a chance to apply modern computer technology. The Eureka! Moment recently occurred when Steve Pelikan, a birder and math professor at the University of Cincinnati, offered to have a grad student take a stab at a numerical solution, and posed a few questions to make sure we were each thinking alike.
Steve wrote: “I’ve been trying to read your article on Acoustic Properties of Parabolic Reflectors (that I down loaded from your home page, thanks) and am pretty confused about the section “Parabolic Geometry” since when I try to do the computations you outline there, I get a different result. Here I’ve sketched my approach to the problem and hope you can help me understand why I’m getting different results from yours. I understand if you don’t want to revisit something you worked on in 1963, but any suggestions or references to help me figure this out I would be appreciated!”
Steve continued: “My problem is that the defining property of the parabola ensures that all paths entering the parabola at some fixed height, reflecting off the curve, and converging on the focus all have the same length.”
That, of course, is absolutely right. So far we were thinking alike.
Then came the breakthrough statement: “So, I find that your expression D = Di+ Dr= |x − x0| +{(xf− x)2+y2}0.5is a constant independent of y.”
Had I only recognized that obvious truth in 1963, I could have taken that unwieldy term ahead of the integral, thereby leaving an integral that would be readily determinant by ordinary means. In fact, it would have yielded the somewhat trivial result that the sound pressure at the focal point is proportional to the square of the reflector radius. But the more interesting, even astonishing, result is that the sound pressure at the focal point would be independent of frequency according to that model. This we know not to be consistent with either intuition or experience. Therefore, back to the drawing board.
My hunch is that the “right” solution lies in extending Sten Wahlstrom’s analysis of the sound pressure at the precise focal point by considering the mean sound pressure over the finite area representing the microphone diaphragm. How best to do this remains to be discovered. Suggestions are welcome.