Question

Prove that the frequency of beats is equal to the difference between the frequencies of the two sound notes giving rise to beats.

Hint:

Beats result from interference; frequency difference determines beat rate, useful for tuning instruments.

Answer 1

Beats occur when two sound waves of slightly different frequencies interfere. Let the frequencies be \( f_1 \) and \( f_2 \) (with \( f_1>f_2 \)). The combined wave is: \[ y = A \sin (2\pi f_1 t) + A \sin (2\pi f_2 t) = 2A \cos \left( 2\pi \frac{f_1 - f_2}{2} t \right) \sin \left( 2\pi \frac{f_1 + f_2}{2} t \right). \] The amplitude modulates with frequency \( \frac{f_1 - f_2}{2} \), but the perceived beat frequency is the number of amplitude maxima per second, which is \( f_1 - f_2 \).
Proof: The time between consecutive beats is the period of the cosine envelope, but since each cycle of cosine gives two beats (max and min), the beat frequency is \( f_1 - f_2 \).
Answer: Beat frequency = \( |f_1 - f_2| \).