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This is in answer to a question in conference. I didn't want to
fill up my 5MB server space with it so I put it here. It's in three
parts: An explanation of how sine waves are derived, a thing about
radians for those who haven't met them before, and finally the practical
application to this course down at the bottom. I'm not brilliant
at explaining things, but I've taken time over this. If it's still
confusing, let me know and I'll fix it or just trash the thing.
Whatevah!
 1.
What the hell are sine curves, and why?
Sines are derived by looking at right-angled triangles. In the
usual case, the angle is less than 90 degrees. The sine of the angle
is what you get if you divide the length of the side opposite the
angle by the length of the hypotenuse (which is the longest side
of the triangle).
In this picture, where the angle is smallish, if the side opposite
it (labelled 'o' for opposite!) is 3cm, and the hypotenuse is 5,
then the sine of the angle (which I've labeled with the Greek letter
theta) would be 3 divided by 5, which is 0.6. If you want to make
life even easier, take a circle with radius 1cm. The radius forms
the hypotenuse, so now all the calculations have the sine equal
to the length of the side 'o' divided by one...which is a nice easy
calculation!
If you try to imagine a situation where the angle shrunk to nothing,
then the side 'o' would shrink to nothing, so the sine would have
to be zero because nothing divided by anything is still nothing!
This wouldn't be a 'proper' triangle, of course, because one of
its sides would have disappeared but in theory, it marks the start
of the sine curve:
sin0=0.
That's why a standard sine curve starts from the point where the
two axes cross.
 Similarly,
if the angle was 90 degrees, the two lines would merge together,
pointing straight up, so you still wouldn't have a triangle. However,
the sine would be 1 because in theory you'd have an infinitely thin
triangle with 'o' and 'h' both equal to 1.
sin90=1 divided by 1=1.
 OK,
it's not a very good picture is it? That's how you get from 0 to
1 anyway. When the angle gets over 90 degrees, you have to picture
a triangle being made on the other side...
Between 90 and 180 degrees, the sine goes back down from one to
zero in a mirror image of the curve from 0 to 90.
And from 180 onwards...
...the line opposite the angle is negative, so the sine is negative
too, and moves from 0 to -1 and back in just the same way.
2. So why is a cycle 2Pi in those graphs on
the disk?
Scientific types don't like measuring in degrees. Luckily, just
as you can measure temperatures in Fahrenheit or centigrade, so
you can chuck away degrees and use radians instead.
 There
are (conveniently enough), 6.28 (i.e., 2 Pi) radians in a circle,
instead of 360 degrees. When you see 2pi written on an axis, you
can read it as 360 degrees. Why do they do this? Well, it makes
life a whole lot easier (for them, not us!) because so many scientific
and engineering equations use pi...anyway, that's another story.
So the standard, no-frills sine wave moves from zero to one, back
to zero, down to minus one and then to zero again as the angle moves
from zero through pi/2 (90degrees), pi (180), 3pi/2 (270) and on
to 2pi (360 degrees, the full circle.)
3. What does this have to do with radio waves?
Radio waves have the same shape as sine waves. So do a lot of
other things. It's quite a common kind of motion - see animation
'shm' on the CD-ROM. Waves and simple harmonic motions are based
on the idea that speed varies with 'displacement' (i.e., how far
above or below the central position you are). Well, displacement
is the same as the length of the side 'o', which in turn is directly
related to the sine. So when something is going up and down like
a spring or a piston or a wave, sines are really handy for telling
us about their behaviour at any point in the cycle. However, waves
and SHMs are much more variable than the standard sine wave I've
drawn above. For example, they don't stop at a maximum of one and
a minimum of minus one, That's why, in the equations you need to
add in parameters specific to the wave in question, like its period
(T) and its amplitude (A)
Dividing 2 Pi by T and multiplying by t will get you an exact location
on the curve. e.g., a wave which repeats every 2 sec, after 9 sec
will have gone through 4.5 cycles and reached 9 Pi, and the equation
above should come out right. You also need to add in an extra value
if the wave doesn't start at rest (i.e., of the graph doesn't start
at zero) - again, see the SHM animation, which gives the finished
equation.
If you need a better explanation, try Catcode's
'sines and cosines do the wave', and other bits.
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