Question

The beat frequency observed when two sound waves $ y=0.5\sin (410t) $ and $ y=0.5\sin (454t) $ travel in the same direction is
A. 5
B. 3
C. 7
D. 2

Answer 1

Hint: Two waves with different frequencies when added together will beat with a frequency equal to the difference in the two original frequencies. We will find the value of beat frequency using the equation.

Formula used:
 $ \omega =2\Pi f $
Where, $ \omega $ is the angular frequency of the sound wave and $ f $ is the number of oscillations per second.

Complete step-by-step answer:
When two sound waves, with different frequencies interfere then beats are created and the beat frequency is equal to the absolute value of the difference in the two original frequencies. For example if two sound waves with frequencies $ {{f}_{1}} $ and $ {{f}_{2}} $, when added together will pulse with a frequency of $ {{f}_{1}}-{{f}_{2}} $.
Given two sound waves $ y=0.5\sin (410t) $ and $ y=0.5\sin (454t) $
Calculating frequency of first wave,
Here, $ {{\omega }_{1}}=410 $
By using, $ \omega =2\Pi f $
We get,
 $ \begin{align}
  & 410=2\Pi {{f}_{1}} \\
 & {{f}_{1}}=\dfrac{410}{2\Pi }=\dfrac{205\times 7}{22} \\
\end{align} $
Similarly, $ {{\omega }_{2}}=454 $
 $ \begin{align}
  & 454=2\Pi {{f}_{2}} \\
 & {{f}_{2}}=\dfrac{454}{2\Pi }=\dfrac{227\times 7}{22} \\
\end{align} $
For beat frequency $ f={{f}_{2}}-{{f}_{1}} $
 $ f=\dfrac{227\times 7}{22}-\dfrac{205\times 7}{22}=\dfrac{7}{22}\left( 227-205 \right)=7 $
 $ f=7 $
Beat frequency is equal to 7
Hence the correct option is C.

Note: Students should not get confused between the terms $ \omega $ and $ f $ . $ \omega$ is the angular frequency of a wave, that is the number of radians per second or we can say the rate of change of the function argument expressed in the units of radians per second, while $ f $ is the number or turns or the oscillations performed by the wave per second. Remember that each turn is $ 2\pi $ radians.
Also, keep in mind while calculating beat frequency; always take the absolute value of difference in the frequencies.