Question

The displacement equations of sound waves produced by two sources are given by: \( y_1 = 5 \sin(400t) \) and \( y_2 = 8 \sin(408t) \), where \( t \) is time in seconds. If the waves are produced simultaneously, the number of beats produced per minute is:

• A. \( 4 \)
• B. \( 8 \)
• C. \( 120 \)
• D. \( 240 \) \bigskip

Hint:

The beat frequency is given by the absolute difference of the two wave frequencies: \[ f_{\text{beat}} = | f_2 - f_1 | \] Substituting the given values, \[ f_{\text{beat}} = | 408 - 400 | = 8 \text{ Hz} \] Since beats per second is 8 Hz, the number of beats per minute is: \[ 8 \times 60 = 240 \] Understanding the concept of beats is crucial in musical tuning and wave interference studies.

Answer 1

The Correct Option is D

The number of beats is given by the difference in frequencies of the two sound waves. The frequency of the first wave is \( f_1 = 400 \, \text{Hz} \), and the frequency of the second wave is \( f_2 = 408 \, \text{Hz} \). The difference in frequencies is: \[ \Delta f = f_2 - f_1 = 408 - 400 = 8 \, \text{Hz} \] The number of beats per second is \( \Delta f = 8 \, \text{Hz} \). The number of beats per minute is: \[ 8 \times 60 = 240 \, \text{beats per minute} \] Thus, the number of beats per minute is \( 240 \).