Question

Two waves $y_1=8\sin (2x-3t)$ and $y_2=6\sin (2x-3t-\frac{\pi }{2})$ are superimposed. Then, which of the following is/are correct.
$(y_1$ and $y_2$ are in $\text{cm})$

• A. Phase difference between $y_1$ and $y_2$ is $\frac{\pi }{2}$.
• B. Velocity of $y_1$ and $y_2$ is $1.5{\text{cm s}}^{-1}$
• C. Amplitude of resultant wave is $20\text{cm}$.
• D. Amplitude of resultant wave is $12\text{cm}$.

Answer 1

Answer: (A), (B)

Velocity of $y_1$ and $y_2$ is $1.5{\text{cm s}}^{-1}$

Given that,
$y_1=8\sin (2x-3t)$

$y_2=6\sin (2x-3t-\frac{\pi }{2})$

Comparing the equations of $y_1$ and $y_2$, we get

Phase difference $\varphi =\frac{\pi }{2}$

The velocity of the wave is given by,

$v=\frac{\omega }{k}$

Comparing the given equation with standard equation

$y=\sin (kx-\omega t+\varphi )$

$\Rightarrow k=2,\omega =3$

$\Rightarrow v=\frac{w}{k}=\frac{3}{2}=1.5{\text{cm s}}^{-1}$

The amplitude of the resultant wave is given by,

$A_R=\sqrt{(A_1{)}^2+(A_2{)}^2+2A_1{A}_2\cos \varphi }$

Given that, $A_1=8,A_2=6$

And, as the phase differences, $\varphi =\frac{\pi }{2}$

$A_R=\sqrt{8^2+6^2}=10\text{cm}$

Hence, option $\left(A\right)\text{and}\left(C\right)$ are correct.

Why this Question? This question help you in revision of superimposition of wave and other parameter of wave equation like $\omega ,k,\varphi$ etc.