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# Sound Mirrors

### Parabolas or circles?

The large sound mirror near Dungeness, pictured in the satellite photo below (Fig. 1), is designed to focus sound from distance sources. In other words, it is designed to focus a plane wave. While a parabolic shape produces a perfect focus for a wave incident on the axis, it suffers from severe aberrations off-axis. Since this mirror is not steerable, it is reasonable to guess that the mirror has a circular cross section since this shape is more forgiving off-axis.

Fig. 1. Satellite photograph of the 200 foot sound mirror at Denge with grid and reference points.

To check this hypothesis, five points were measured on the satellite photo and plotted in Fig. 2 (arbitrary units). The sum of the mean square residuals for the parabola is 1.7, while it is only 0.06 for the circle. The radii of curvature at the center of the two curves differ from each other by about 15%. Note that the x and y axis scales are not the same, distorting the appearance of both curves.

Fig. 2. Reference points plotted, least-squares fit to a circle and parabola.

Based on this rudimentary analysis, it seems likely that this mirror is circular, as claimed by this site.

### Uncertainties

The satellite photograph was not taken from directly overhead. This introduces a distortion in the shape that would tend to make it appear less like a circle and more like an ellipse. The overhead angle could be estimated by the lengths of the edges, but this is beyond the scope of this study. It is left as an exercise to the reader to show that a parabola remains a parabola under this distortion. Thus it is not possible for a parabola appear to be a circle.

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