Question

Two sound sources produce progressive waves given by $y_1=12cos\left(100\pi t\right)$ and $y_2=4cos\left(102\pi t\right)$ near the ear of an observer. When sounded together, the observer will hear :

• A. 2 beats per second with an intensity ratio of maximum to minimum nearly 4 : 1
• B. 1 beat per second with an intensity ratio of maximum to minimum nearly $\sqrt{2}$ : 1
• C. 2 beats per second with an intensity ratio of maximum to minimum nearly 9 : 1
• D. 1 beat per second with an intensity ratio of maximum to minimum nearly 4 : 1

Answer 1

The correct option is D 1 beat per second with an intensity ratio of maximum to minimum nearly 4 : 1

Given,

$y_1=12cos\left(100\pi t\right)$

$y_2=4cos\left(102\pi t\right)$

$a_1=12$, $a_2=4$, ${\omega }_1=100\pi$, ${\omega }_2=102\pi$

We know that, ${\omega }_1=2\pi f_1=100\pi$

$f_1=50Hz$

${\omega }_2=2\pi f_2=102\pi$

$f_2=51Hz$

Beat, $f_2-f_1=51-50=1beat$

Ratio, $\frac{I_{max}}{I_{min}}=\frac{(a_1+a_2{)}^2}{(a_1-a_2{)}^2}$

$\frac{I_{max}}{I_{min}}=\frac{(12+4{)}^2}{(12-4{)}^2}=\frac{256}{64}=\frac{4}{1}$

$I_{max}:I_{min}=4:1$

The correct option is D.