Question

Two transverse waves in a given medium are represented by the curve as shown in the figure. Find the ratio of their intensities. All the measurements are in $\text{S.I}$ units.

• A. $\frac{10}{11}$
• B. $\frac{24}{5}$
• C. $\frac{25}{36}$
• D. $\frac{25}{18}$

Answer 1

The correct option is C $\frac{25}{36}$

For wave $1$,
$A_1=5\text{m}$ and time period $\left(T_1\right)=T$
For wave 2
$A_2=2\text{m}$ and time period $\left(T_2\right)=\frac{T}{3}$
$\therefore \frac{A_1}{A_2}=\frac{5}{2}$ and $\frac{T_1}{T_2}=3$
or
$\Rightarrow \frac{T_1}{T_2}=\frac{\frac{2\pi }{{\omega }_1}}{\frac{2\pi }{{\omega }_2}}=\frac{{\omega }_2}{{\omega }_1}=3$
Intensity of the wave,
$(I=2{\pi }^2{f}^2\rho v{A}^2=\frac{1}{2}\rho v{A}^2{\omega }^2)$
$\Rightarrow \frac{I_1}{I_2}=\frac{\frac{1}{2}\left(\rho v\right){\omega }_1^2{A}_1^2}{\frac{1}{2}\left(\rho v\right){\omega }_2^2{A}_2^2}={(\frac{{\omega }_1}{{\omega }_2}\times \frac{A_1}{A_2})}^2$
$\Rightarrow \frac{I_1}{I_2}={\left(\frac{1}{3}\right)}^2\times {\left(\frac{5}{2}\right)}^2$
$\therefore \frac{I_1}{I_2}=\frac{25}{36}$