Question

The number of beats heard per second if there are three sources of sound of frequencies $(n-1),n$ and $(n+1)$ of equal intensities sounded together is:

• A. $2$
• B. $1$
• C. $4$
• D. $3$

Answer 1

The correct option is B $1$

The frequencies of the sources are n-1, n and n+1. Their resul tan t displacement is given by

$y=asin2\pi (n-1)t+asin2\pi nt+asin2\pi (n+1)t$

$y=2asin2\pi nt-cos2\pi t+asin2\pi nt$

$y=a(1+2cos2\pi nt)sin2\pi nt$

$y=Asin2\pi nt$

where $A=a(1+2cos2\pi t)$

It is maximum when $2\pi t=+1$ Or

$2\pi t=2m$, where m = 0,1,2 ... so t = 0,1,2...

so the time interval between two consecutive maxima is 1 second.

In the similar way amplitude is minimum when

$1+2cos2\pi t=0$

$cos2\pi t=-\frac{1}{2}$

$2\pi t=2m\pi +\frac{2\pi }{3}$ where m = 0,1,2 ...

so mimima appears after an interval of ne second which indicate that the frequency of beat is also one hence one beat is heard per second.

so option (B) is correct.