Question

There are three sources of sound of equal intensity with frequencies $400,401$ and $402vibse{c}^{-1}.$ The number of beats heard per second is:

• A. $0$
• B. $1$
• C. $2$
• D. $3$

Answer 1

Answer: (C)

For source of frequency, $400Hz$ , $y_1=asin2\pi \left(f-1\right)t$
For source of frequency, $401Hz$ , $y_2=asin2\pi ft$
For source of frequency, $402Hz$ , $y_3=asin2\pi \left(f+1\right)t$
Resultant of these waves,
$Y=y_1+y_2+y_3\Rightarrow Y=asin2\pi \left(f\right)t+\left[asin2\pi \left(f-1\right)t+asin2\pi \left(f+1\right)t\right]\Rightarrow Y=a\left(1+cos2\pi t\right)sin2\pi ft$
For maximum amplitude,
$cos2\pi t=1\Rightarrow 2\pi t=2\pi m\Rightarrow t=m\left[m=0,1,2,\mathrm{3....}\right]$
For minimum amplitude,
$cos2\pi t=-\frac{1}{2}\Rightarrow 2\pi t=2\pi m+\frac{2\pi }{3}\Rightarrow t=m+\frac{1}{3}\left[m=0,1,2,\mathrm{3....}\right]t=\frac{1}{3},\frac{4}{3},\frac{7}{3},.......$
Time between successive minima, $t_2-t_1=\frac{4}{3}-\frac{1}{3}=1$
Hence, number of beats per second is $1$ .