Interference and Sound - Beats

1. By: Yueyan Li (#17417149)
Physics 101 202

2. • When two sound waves of different frequency approach
your ear, the alternating constructive and destructive
interference.
• The interference causes the sound to be alternatively
soft and loud - a phenomenon which is called "beating"
or producing beats.

3. • Beats are caused by the interference of two
waves at the same point in space.
* This plot of the variation of resultant amplitude with time shows the
periodic increase and decrease for two sine waves.
The beat frequency is equal to the absolute value of
the difference in frequency of the two waves.

4. • When you superimpose two
cosine waves of different frequencies,
you get components at the sum and
difference of the two frequencies.
( This can be shown by using a sum
rule from trigonometry.)
• For equal amplitude cosine waves
(Think about the explaination and derivation)
• The first term gives the phenomenon of beats with a beat frequency equal to
the difference between the frequencies mixed.

5. • Explaination and Derivation

6. • Ytotal = {2A cos (2π Δf/2)} * cos (2π faverage) (4).
• The term inside the curly brackets {} can be considered as the slowly varying function that
modulates the carrier wave with frequency faverage.
• This function-the modulation of the amplitude-is the green wave in the diagram. It has frequency
Δf/2, but notice that there is a maximum in the amplitude or a beat when the green curve is
either a maximum or a minimum, so beats occur at twice this frequency.
• So the beat frequency is simply Δf: the number of beats per second equals the difference in
frequency between the two interfering waves.
y fav. (It is indeed an example of amplitude modulation or AM.) This function--the modulatio
e is from time (i) to time (v). There are beats at (i), (iii) and (v), and quiet spots at (ii) and (iv).
waves,