Question

Two coherent sources of sound, $S_1$ and $S_2$, produce sound waves of the same wavelength, $\lambda =1m$, in phase. $S_1$and $S_2$ are placed $1.5m$ apart (see fig). A listener, located at $L$, directly in front of $S_2$finds that the intensity is at a minimum when he is $2m$ away from $S_2$. The listener moves away from $S_1$, keeping his distance from $S_2$ fixed. The adjacent maximum of intensity is observed when the listener is at a distance $d$ from $S_1$. Then, $d$ is :

• A. $12m$
• B. $2m$
• C. $3m$
• D. $5m$

Answer 1

The correct option is C

$3m$

Step 1: Given;

1. $S_{2}L=2m$
2. $S_1{S}_2=1.5m$
3. same wavelength, $\lambda =1m$

Step 2: Join the points $S_1$ and $L$.

Step 3: Find the value of $S_{1}L$and $S_{2}L$.

From the figure, $S_{2}L=2m$

$$\begin{gathered} S_{1}L=\sqrt{{\left(S_{2}L\right)}^2+{\left(S_1{S}_2\right)}^2} \\ {S}_{1}L=\sqrt{{\left(2\right)}^2+{(1.15)}^2} \\ =2.5m \end{gathered}$$

Step 4: Find the initial path difference,$∆x_1$ minima is observed.

$∆x_1=S_{1}L-S_{2}L=2.5-2=0.5m$

Step 5: Find the path difference at new position, $∆x_2$ where maxima is observed.

$∆x_2=d-2$

Step 6: Find d.

The path difference between adjacent maxima and minima is $\frac{\lambda }{2}$.

$$\begin{gathered} ∆x_2-∆x_1=\frac{\lambda }{2} \\ \Rightarrow d-2-0.5=\frac{1}{2}(\because \lambda =1m) \\ \Rightarrow d=3m \end{gathered}$$

Thus, option C is the correct answer.