Question

Two waves in the same medium are represented by $y-t$ curves in the figure. Find the ratio of their average intensities.

• A. $\frac{25}{16}$
• B. $\frac{16}{25}$
• C. $\frac{4}{5}$
• D. $\frac{5}{4}$

Answer 1

The correct option is A $\frac{25}{16}$

Let $f_1$ be the frequency of $\text{wave}1$ and $f_2$ be the frequency of $\text{wave}2$.
From the figure,
$\frac{A_1}{A_2}=\frac{5}{2}$ and, $f_1=\frac{f_2}{2}$
$\Rightarrow \frac{f_1}{f_2}=\frac{1}{2}$
Let $I$ be the intensity of the wave.
$I={\omega }^2{A}^2\rho v=2{\pi }^2{f}^2{A}^2\rho v$
$\Rightarrow I\propto f^2{A}^2$ (for same medium)
So, $\frac{I_1}{I_2}=\frac{f_1^2{A}_1^2}{f_2^2{A}_2^2}$
$\Rightarrow \frac{I_1}{I_2}={\left(\frac{f_1}{f_2}\right)}^2{\left(\frac{A_1}{A_2}\right)}^2={\left(\frac{1}{2}\right)}^2\times {\left(\frac{5}{2}\right)}^2$ $=\frac{25}{4\times 4}=\frac{25}{16}$
Hence, option $\left(a\right)$ is the correct answer.