Question

When two sound waves of slightly different ftequencies $f_1$ and $f_2$ are sounded together, then the time interval between successive maxima is

• A. $\frac1{f_1 +f_2}$
• B. $\frac1{f_1 -f_2}$
• C. $\frac1{f_1} +\frac1{f_2}$
• D. $\frac1{f_1 f_2}$
• E. $\frac1{f_1} -\frac1{f_2}$

Answer 1

The Correct Option is B

Beats and Time Interval Between Successive Maxima 

When two sound waves of slightly different frequencies \(f_1\) and \(f_2\) are sounded together, we hear a phenomenon called "beats". The time interval between successive maxima (loudest sounds) is related to the beat frequency.

Step 1: Define Beat Frequency

\(f_{beat} = |f_1 - f_2|\)

The beat frequency, \(f_{beat}\), is the absolute difference between the two frequencies. It represents how many times per second the sound gets louder and softer (the number of "beats" we hear).

Step 2: Relate Beat Frequency to the Time Interval

\(T_{beat} = \frac{1}{f_{beat}}\)

The period of the beat, \(T_{beat}\), represents the time interval between successive maxima (loudest sounds). The period is the inverse of the frequency. In other words, it is the time it takes for one complete beat cycle to occur (loud -> soft -> loud).

Step 3: Substitute the Expression for Beat Frequency

\(T_{beat} = \frac{1}{|f_1 - f_2|}\)

Substituting the expression for the beat frequency into the period equation gives us the time interval between successive maxima.

Conclusion

The time interval between successive maxima is: \(T_{beat} = \frac{1}{|f_1 - f_2|}\).

Answer 2

When two waves of slightly different frequencies interfere, they produce beats. The beat frequency is the difference in frequencies: \[ f_{\text{beat}} = |f_1 - f_2| \] The time interval between successive maxima (i.e., between beats) is the reciprocal of the beat frequency: \[ T = \frac{1}{f_{\text{beat}}} = \frac{1}{|f_1 - f_2|} \]