Question

Two progressive waves are represented by the following equations
$y_1=10\sin 2\pi (10t-0.1x)$
$y_2=20\sin 2\pi (20t-0.2x)$
Find the ratio of their intensities.

• A. $1/2$
• B. $1/4$
• C. $1/8$
• D. $1/16$

Answer 1

Answer: (D)

Intensity of a progressive wave,
$I=2{\pi }^2\rho f^2{A}^{2}v$
where $\rho =$ density of medium
$f=$ frequency of wave
$v=$ wave velocity
$A=$ amplitude
For wave $1$
$10\sin 2\pi (10t-0.1x)$
$A\sin \left(\frac{2\pi t}{T}-\frac{2\pi x}{\lambda }\right)$
$A=10$
For frequency $\omega =\frac{2\pi }{T}=20\pi$
$\Rightarrow f=10$
Wave velocity $=\lambda \times f=\frac{1}{(0.1)}\times (10)$
Similarly, for wave $2$
$A_1=20$
Frequency ${\omega }_2=\frac{2\pi }{T_2}=40\pi$
$\Rightarrow f=20$
Wave velocity $v_2=\frac{1}{0.2}\times (20)$
$\therefore \frac{2{\pi }^2\rho A_1^2{f}_1^2{v}_1}{2{\pi }^2\rho A_2^2{f}_2^2{v}_2}=\frac{(10{)}^2(10{)}^2(100)}{(20{)}^2(20{)}^2(100)}$
$\Rightarrow \frac{I_1}{I_2}=\frac{1}{16}$