Question

There are three sources of the sound of equal intensities and frequencies $400,401$ and $402$ vibrations per second. The maximum number of beats/sec is:

• A. $0$
• B. $1$
• C. $3$
• D. $2$

Answer 1

The correct option is B $1$

Equation of the three sources can be written as,

$y_1=asin(2\times 400\pi t)=asin\left(800\pi t\right)$

$y_2=asin(2\times 401\pi t)=asin\left(802\pi t\right)$

$y_3=asin(2\times 402\pi t)=asin\left(804\pi t\right)$

$\therefore y=y_1+y_2+y_3$

$y=asin\left(800\pi t\right)$+$asin\left(802\pi t\right)$$+asin\left(804\pi t\right)$

$y=asin\left(802\pi t\right)[1+cos2\pi t]$

$\therefore$ Amplitude $A=a[1+cos2\pi t]$

$A$ is maximum when $cos\pi t=+1$

$\therefore 2\pi t=2k\pi$ ....where $(k=0,1,2,3,4,....)$

or $t=k$

$\therefore$ at $t=0,1,2,\mathrm{3....}$ we hear maximum sound.

$\therefore$the time interval$=1sec$

and the beats frequency $=\frac{1}{1}=1$$/sec$