“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:

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:

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:

So:

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

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

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).

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!

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”:

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,

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: