Question

Two point sources of sound are kept at a separation of $10cm$. They vibrate in phase to produce waves of wavelength $5cm$. What would be the phase difference between the two waves arriving at a point $20cm$ from the source.

1. on the line joining the sources and
2. on the perpendicular bisector of the line joining the sources?

Answer 1

Step 1: Given data

Wavelength, $\lambda =5cm$

Distance between the slits$=10cm$

Step 2: To find

We have to find the phase difference between the two waves arriving at a point $20cm$ from one source.

Step 3: Formula used

Path difference,

$∆x=x_1-x_2$

Phase difference,

$∆\varphi =2\mathrm{\pi }\times \frac{∆\mathrm{x}}{\mathrm{\lambda }}$

where $\lambda$ is the wavelength

Step 4: The phase difference between the two waves arriving at a point $20cm$ from one source on a line joining the sources.

$$\begin{gathered} x_1=20+10=30cm \\ {x}_2=20cm \end{gathered}$$

Path difference,

$∆x=x_1-x_2=30-20=10cm$

Phase difference,

$$\begin{gathered} ∆\varphi =2\mathrm{\pi }\times \frac{∆x}{\lambda } \\ \Rightarrow ∆\varphi =2\mathrm{\pi }\times \frac{10}{5} \\ \Rightarrow ∆\varphi =2\mathrm{\pi }\times 2 \end{gathered}$$

Since it is a multiple of $2\mathrm{\pi }$, the phase difference will be $0$

Step 5: The phase difference between the two waves arriving at a point 20 cm from on the perpendicular bisector of the line joining the sources.

From, the figure, it is clear that $y_1=y_2$

Therefore, path difference,

$∆y=y_1-y_2=0$

Phase difference,

$$\begin{gathered} ∆\varphi =2\mathrm{\pi }\times \frac{∆y}{\lambda } \\ \Rightarrow ∆\varphi =2\mathrm{\pi }\times \frac{0}{\lambda } \\ \Rightarrow ∆\varphi =0 \end{gathered}$$

Here the phase difference is $0$.

In both cases, a and b, the phase difference is $0$.