Question

Two waves represented by equation,
$y_1=2\sin (\pi t-0.1x)$ and $y_2=2\sin (\pi t-0.1x+\varphi )$ are superimposed. If the resultant wave has a maximum amplitude equal to $4\text{cm}$, the value of $\varphi$ is :

• A. $\pi$
• B. $2\pi$
• C. $\frac{\pi }{2}$
• D. $\frac{3\pi }{2}$

Answer 1

The correct option is B $2\pi$

The phase difference between the given waves is,
$\Delta \varphi =(\pi t-0.1x+\varphi )-(\pi t-0.1x)$
$\Delta \varphi =\varphi$
Here, we can see that the maximum amplitude of resultant wave:
$A_{max}=A_1+A_2$
The maximum amplitude of resultant wave suggests that the waves have interfered constructively.
$\Rightarrow \Delta \varphi =2n\pi$
The phase difference between waves is an integral multiple of $2\pi$
For $n=1$,
$\Rightarrow \Delta \varphi =\varphi =2\pi$