1. “Beats” – a simple (and easy to hear)
example of an effect resulting from
the interference of sound waves.
Suppose that there are two sources
emitting sound waves of frequency
f1 and f2:

2. Suppose that both waves reach a “receiver”
located at certain point x .
Due to the interference of the two, the sound
signal “heard” by the receiver will be:

3. However, we have a total freedom of choosing
the coordinate system – we can choose such
in which the location of our receiver is x = 0.
Then:
All we need now is a trigonometric identity
you must remember from high school:

4. So:

5. The sum of two sine functions with slightly different periods:
Time (arbitrary units)

6. Remember, the the frequency of the beats is twice
the frequency of the modulation function!

7. Doppler Effect for sound waves
Let’s start with a simpe model. The transmitter
(loudspeaker) emits a sound wave, and at some
distance from it there is a receiver (e.g., a human
ear, in the picture below symbolized by the
question-mark-like shape).

8. We can think of the receiving process that
DE for sound waves, cont.
We can think of the receiving process that
each time a sound wave “crest”reaches the
receiver, it produces a PING! Many such
“pings” heard in a single second would give
us the impression of a continuous sound:
(the real mechanism of hearing is surely more complicated, but the basic
physics in our model is OK -- actually, for analyzing the basic principles
of DE, instead of considering a continuous wave, one can think of a sound
signal in the form of a sequence of short sound signals:
ping!-ping!-ping!-ping-ping!

9. Well, here is the same animation as in the
preceding slide, but slowed down – note
that each PING! occurs exactly at the mo-
ment when a “crest” (i.e., a maximum)
reaches the “ear”:

10. Now, consider two observers, of which one is stationary, and the
other moves towards the sound source with constant speed:
Note that the moving observer registers more Pings! per time unit than
the stationary one – it means that she/he registers a higher frequency.
Question: how would the frequency change if the second observer
moved away from the sound source?
Yes, of course – then the frequency
would be lower,

11. The same as before, but slowed down a bit: note that in both
cases the PING! Appears precisely at the moment a consecutive
maximum reaches the ear: