Question

The displacement vs time graph for two waves $A$ and $B$ which travel along the same string are shown in the figure. Their intensity ratio $I_A/I_B$ is

• A. $\frac{9}{4}$
• B. $1$
• C. $\frac{81}{16}$
• D. $\frac{3}{2}$

Answer 1

The correct option is B $1$

Intensity of sound wave is given by:

$I=\frac{1}{2}\rho c{\omega }^2{x}^2$

Where,

$I$ is intensity of sound wave

$\rho$ is density of medium

$v$ is the speed of sound

$\omega$ is the angular frequency

$x$ is the amplitude of sound

From the given graph:

$x_A=3cm$

$x_B=2cm$

${\omega }_A=\frac{2\pi }{6}=\frac{\pi }{3}$

${\omega }_B=\frac{2\pi }{4}=\frac{\pi }{2}$

Taking ratios of intensities of A and B as

$\frac{I_A}{I_B}=\frac{\frac{1}{2}\rho c{\left(\frac{\pi }{3}\right)}^2{\left(3\right)}^2}{\frac{1}{2}\rho c{\left(\frac{\pi }{2}\right)}^2{\left(2\right)}^2}$

$\frac{I_A}{I_B}=\frac{1}{1}$

So the ratio of intensities is $1:1$