Question

Two progressive waves are represented by the following equations
$y_{1}=10 \sin 2 \pi(10 t-0.1 x)$
$y_{2}=20 \sin 2 \pi(20 t-0.2 x)$
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 $1$
$10 \sin 2 \pi(10 t-0.1 x)$
$A \sin \left(\frac{2 \pi t}{T}-\frac{2 \pi x}{\lambda}\right)$
$A=10$
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 $v_{2}=\frac{1}{0.2} \times(20)$
$\therefore \frac{2 \pi^{2} \rho A_{1}^{2} f_{1}^{2} v_{1}}{2 \pi^{2} \rho A_{2}^{2} f_{2}^{2} v_{2}}=\frac{(10)^{2}(10)^{2}(100)}{(20)^{2}(20)^{2}(100)}$
$\Rightarrow \frac{I_{1}}{I_{2}}=\frac{1}{16}$