By: Yueyan Li (#17417149)

Physics 101 202

• 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.

• 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.

• 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.

• Explaination and Derivation

• 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.

But how often do the beats occur? Let's write (4) this way:

ytotal = {2A cos (2π Δf/2)} * cos (2π fav) (4).

The term inside the curly brackets {} can be considered as the slowly varying function that modulates the carrier wave with frequency fav. (It is indeed an example of amplitude modulation or AM.) 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. (One cycle of the green curve is from time (i) to time (v). There are beats at (i), (iii) and (v), and quiet spots at (ii) and (iv).)

So the beat frequency is simply Δf: the number of beats per second equals the difference in frequency between the two interfering waves,