Make Electronics

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Experiment 29: Filtering Frequencies

theory

Waveforms

If you blow across the top of a bottle, the mellow sound
that you hear is caused by the air vibrating inside the bottle,
and if you could see the pressure waves, they would have a
distinctive profile.

If you could slow down time and draw a graph of the alter-
nating voltage in any power outlet in your house, it would
have the same profile.

If you could measure the speed of a pendulum swinging
slowly to and fro in a vacuum, and draw a graph of the speed
relative to time, once again it would have the same profile.

That profile is a sine wave, so called because you can derive
it from basic trigonometry. In a right-angled triangle, the
sine of an angle is found by dividing the length of the side
opposite the angle by the length of the hypoteneuse (the
sloping side of the triangle).

To make this simpler, imagine a ball on a string rotating
around a center point, as shown in Figure 5-48. Ignore the
force of gravity, the resistance of air, and other annoying
variables. Just measure the vertical height of the ball and di-
vide it by the length of the string, at regular instants of time,
as the ball moves around the circular path at a constant
speed. Plot the result as a graph, and there’s your sine wave,
shown in Figure 5-49. Note that when the ball circles below
its horizontal starting line, we consider its distance negative,
so the sine wave becomes negative, too.

Why should this particular curve turn up in so many places
and so many ways in nature? There are reasons for this
rooted in physics, but I’ll leave you to dig into that topic if it
interests you. Getting back to the subject of audio reproduc-
tion, what matters is this:

• Any sound can be broken down into a mixture of sine
  waves of varying frequency and amplitude.

Or, conversely:

• If you put together the right mix of audio sine waves,
  you can create any sound at all.

Suppose that there are two sounds playing simultaneously.
Figure 5-50 shows one sound as a red curve, and the other
as pale blue. When the two sounds travel either as pressure
waves through air or as alternating electric currents through
a wire, the amplitudes of the waves are added together to
make the more complex curve, which is shown in black.
Now try to imagine dozens or even hundreds of different
frequencies being added together, and you have an idea of
the complex waveform of a piece of music.

b

b

a

a

Start

Figure 5-48. If a weight on the end of a string (length b, in

the diagram) follows a circular path at a steady speed, the

distance of the weight from a horizontal center line (length a,

in the diagram) can be plotted as a graph relative to time. The

graph will be a sine wave, so called because in basic trigonom-

etry, the ratio of a/b is the sine of the angle between line b and

the horizontal baseline, measured at the center of rotation.

Sinewaves occur naturally in the world around us, especially in

audio reproduction and alternating current.

Time

Ratio of a/b

Figure 5-49. This is what a “pure” sinewave looks like.

Figure 5-50. When two sinewaves are generated at the same

time (for instance, by two musicians, each playing a flute),

the combined sound creates a compound curve. The blue

sinewave is twice the frequency of the red sinewave. The

compound curve (black line) is the sum of the distances of the

sinewaves from the baseline of the graph.