Relativity class | Syllabus
| Readings |
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.
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
Based on this rudimentary analysis, it
seems likely that this mirror is circular, as claimed by this
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
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