Let me share a little experiment I did last night. Listen to the following audio first. This plays a sine wave of 500 Hz for 2 seconds followed by another sine wave of 500.1 Hz for 2 more seconds. You can hear a tick after each second. The 500.1 Hz sine wave starts at the second tick.

Could you notice any difference in the pitch of the two tones? The difference in the frequencies is so small that you wouldn't be able to notice it by hearing.

Now, let us see what happens when we superimpose both the waves and listen to it.

You would notice that the loudness decreases rapidly at about 5 seconds and then returns to full volume by 10 seconds, decreases rapidly again at 10 seconds and returns to full volume by 15 seconds and so on. In other words, the loudness decreases every 10 seconds.

Let us see why this happens from the graph of the waves. The first wave can be represented with the function f(t) = 0.4 sin (2π · 500 · t) and the second one with g(t) = 0.4 sin (2π · 500.1 · t)

At t = 0 seconds, both the waves look very similar. The crests and troughs of both the waves appear almost simultaneously. On superposition, the amplitude almost doubles. The graph of the superimposed waves is shown below. Note that the amplitude of each wave in the figure above is 0.4 while that of superimposed waves is almost 0.8 as can be seen below. In fact, it is exactly 0.8 at t = 0 seconds and then gradually decreases in the region 0 < t < 5.

The time period of the second wave is slightly shorter than the first one as the frequency of the second wave is slightly greater than the first one. So, the two waves are not identical. In fact, the second wave looks like a very slightly compressed form of the first wave. At about t = 5 seconds, the difference is obvious.

The crests and troughs of both the waves do not appear simultaneously at around t = 5 seconds. In fact, the crest of one wave and the trough of the other wave appear simultaneously. This cancels out each other on superposition and thus the resultant wave has almost no amplitude at this time.

The amplitude is exactly 0 at t = 5 seconds and it increases gradually till t = 10 seconds. At t = 10 seconds, the amplitude becomes 0.8 again and then starts decreasing till t = 15 seconds when it becomes 0 again. This cycle continues. This can be seen in the graph of the superimposed waves given below.

Let us understand this mathematically. Let us represent both the sine
waves with the functions: y_{1} = A sin 2πft and
y_{2} = A sin 2π(f + Δf)t, where A is the amplitude of
the waves, and f and f + Δf are the frequency of the two waves.

So, the superimposed waves can be represented as

y_{1} + y_{2}

= A sin 2πft + A sin 2π(f + Δf)t

= (2A cos 2π · ^{Δf}⁄_{2} ·
t) sin 2π(f + ^{Δf}⁄_{2})t

To understand this function intuitively, imagine it as a sine wave with
a frequency of f + ^{Δf}⁄_{2} and an
amplitude of
2A cos 2π · ^{Δf}⁄_{2} · t.
In our case, f + ^{Δf}⁄_{2} = 500.5 Hz. So,
the superimposed waves look like a sine wave with a frequency of 500.5
Hz but with its amplitude varying with time at a frequency of
^{Δf}⁄_{2}. In other words, the amplitude
oscillates between 0 and 2A every ^{1}⁄_{Δf}
seconds. Since Δf = 0.1 Hz and A = 0.4 in our case, we see that
the amplitude of the 500.5 Hz sine wave oscillates between 0 and 0.8
every 10 seconds.

## 1 comment

## Jarav said:

This phenomenon is called beats but this is usually illustrated for integral frequency differences between the two superposed waves. This is the first time I am listening to the superposition of two waves with frequencies that differ by less than 1.

Here are two interactive applets that I made that illustrate this, one in Flash and one in HTML5 (using Webaudio, works only in Chrome):