Practical_Antenna_Handbook_0071639586

714 A p p e n d i x A : U s e f u l M a t h
+15
x (feet)
0
5 10 15 20
time
(seconds)
–15
Figure A.3.3 Horizontal position of Ferris wheel chair versus elapsed time.
smoothly drawn curve, suddenly you’ll have a duplicate of Fig. A.3.1—a sine wave, in
other words!
How can this be? You know, from having watched your friends take many rides,
that once the ride has started, the Ferris wheel rotates at a constant rate. Yet your shadow
has obviously moved along the ground beneath the Ferris wheel at a varying rate.
What’s going on here?
The answer is this: The shadow on the ground was an exact replica of how the
horizontal position of your chair varied with time. You, of course, were moving vertically
as well as horizontally, so what you saw when your eyes followed the car your
friends were in was a result of their total motion, but what your friend saw when he
watched the shadow of your car intently was just the horizontal component of your
total motion.
Now, it turns out that if you had ridden the Ferris wheel at night and we had shone
a very bright light at the Ferris wheel from the side and marked a utility pole or other
post on the opposite side of the wheel with the same white stripes as before, with y = 0
marked at the midheight of your travel, your friend could collect the same kind of data
about your vertical motion. If we use the same time intervals and the same exact starting
time as before, that curve will look like Figure A.3.4. Note that while it, too, is a sine
wave, it “starts” at a different time than the first one. Specifically, when your horizontal
position was halfway between the left and right extremes of travel (the first or leftmost
data point in Fig. A.3.3), your vertical position was at the very bottom or the very top of
its range.
For reasons we don’t need to discuss right now, we call the curve of Fig. A.3.1 or Fig.
A.3.3 a sine curve or a sine function, and we call the curve of Fig. A.3.4 a cosine curve or a
cosine function. (This particular cosine function is negative—that is, since we assumed
the Ferris wheel rotation started when your chair went past the loading gate, the cosine
function is starting at y = -15.) But both are sine waves or sinusoidal functions. When we
travel a complete circle, or one complete rotation, we say we have gone through 360
degrees (often written 360°), just as there are 360 degrees on a compass. Thus, each of
our sine waves repeats every 360 degrees, and the sinusoidal wave of your vertical position
lags the sinusoidal wave of your horizontal position by a quarter rotation, or 90
degrees. We say these two waves are in quadrature with respect to each other.