In Fig. 17-25, two point sources $\mathbf{S}\mathbf{1}$and$\mathbf{S}\mathbf{2}$, which are in phase, emitidentical sound waves of wavelength$\mathbf{2}\mathbf{.}\mathbf{0}\mathbf{}\mathit{m}$. In terms of wavelengths, what is the phase differencebetween the waves arriving atpoint Pif (a)$\mathbf{L}\mathbf{1}\mathbf{}\mathbf{=}\mathbf{38}\mathbf{}\mathit{m}\mathbf{}$and$\mathbf{L}\mathbf{2}\mathbf{}\mathbf{=}\mathbf{}\mathbf{34}\mathbf{}\mathit{m}$, and (b)$\mathbf{L}\mathbf{1}\mathbf{}\mathbf{=}\mathbf{39}\mathbf{}\mathit{m}$and$\mathbf{L}\mathbf{2}\mathbf{}\mathbf{=}\mathbf{}\mathbf{36}\mathbf{}\mathit{m}$? (c) Assuming that the source separation is much smaller than$\mathbf{L}\mathbf{1}$and$\mathbf{L}\mathbf{2}$, what type of interference occurs atin situations (a) and (b)?

1. The wavelength of the two in phase waves =$\lambda =2.0m$
2. In situation (a), the distance of point P from the source are$L_1=38m,L_2=34m$
3. In situation (b), the distance of point P from the source are$L_1=39m,L_2=36m$

The interference between two waves with identical wavelengths occurs when they cross at a point. Constructive interference is observed at that point if the phase difference is an integral multiple of 2π and destructive interference is observed if the phase difference is an odd integral multiple of π. The phase difference is decided by the path difference between the two waves.

Formulae are as follows:

$f=\frac{∆L}{\lambda }2\mathrm{\pi }$

Condition for constructive interference$f=m\left(2\pi \right)\dots \dots \dots \dots form=0,1,2,\dots .$

Condition for destructive interference $f=\left(2m+1\right)\pi \dots \dots .form=0,1,2,\dots .$

Where, L is distance from source,$\lambda$is wavelength.

Determine the phase difference between the waves as,

$\Delta L=L_1-L_2=38-34=4.0m$

Using the formula,

$f=\frac{∆L}{\lambda }2\mathrm{\pi }$

$\therefore f=\frac{4.0}{2.0}\times 2\pi$

$\therefore f=4\pi$

Hence,the phase difference between the points arriving at P if$L_1=38m$ and $L_2=34m$is$4\mathrm{\pi }$

Determine the phase difference between the waves as ,

$\Delta L=L_1-L_2=39-36=3.0m$

Using the formula,

$f=\frac{∆L}{\lambda }2\mathrm{\pi }$

$\therefore f=\frac{3.0}{2.0}\times 2\pi$

$\therefore f=3\pi$

Hence, the phase difference between the points arriving at P if $L_1=39m$and $L_2=36m$is$3\mathrm{\pi }$

In part (a), it is seen that the phase difference is an integer multiple of $2\mathrm{\pi }$hence, the interference will be constructive.

Hence, type of interference at P in situation (a) is constructive interference

In part (b), the phase difference is an odd integer multiple of. Hence, the interference will be destructive.

Hence, type of interference at P in situation (b) is destructive interference

Therefore, the two waves having identical wavelengths interfere when they cross a point at the same instant. The type of interference is decided by the phase difference between the waves at that point. Hence, we calculate the phase difference to determine the type.