from the parabola’s focus reflected from the
parabola will travel the same distance. Thus, the two
paths shown in Figure 3 are equidistant.
In general, a parabola will have a second-order
equation in x, like this:
y = ax2+ bx + c
Although the parabola in Figure 3 has the
y = x2
The focus for this parabola is at x = 0, y = 0.25.
As a fun trigonometry problem, see if you can prove
that the distance traveled from the focus to the
dashed line is a constant and is independent of
where on the parabolic curve the projected beam
Note that the parabola works the same way in
reverse. A line parallel to the y axis coming toward
the parabola will reflect to the focus.
Turning the Parabola into
a Parabolic Antenna
To make a parabolic antenna, we take the two-dimensional parabola of Figure 3 and rotate it about the y
axis (referred to as the “axis of symmetry”). Figure 4 shows
such a reflector (y axis is pointing out).
Using such a paraboloid as the antenna, the ultrasonic
transducer is placed at the focus, with the transducer in a
backfire configuration (that is, the transducer is pointing in
Figure 3. Both the green and magenta paths are the
same distance from the focus to the dashed
horizontal line at the top.
Figure 5. Ultrasonic transducer firing into the
antenna, with collimated energy out.
Figure 4. Paraboloid formed by rotating a
parabola about the y axis.
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