Question

Figure shows two isotropic point sources of sound ${\mathbf{S}}_1$ and ${\mathbf{S}}_2$The sources emit waves in phase at wavelength 0.50m; they are separated by$\mathbf{D}\mathbf{=}\mathbf{\text{1.75m}}$ . If we move a sound detector along a large circle centered at the midpoint between the sources, at how many points do waves arrive at the detector(a) Exactly in phase and (b) Exactly out of phase ?

Short answer

1. Total 14 points wave arrive at the detector exactly in phase.
2. Total 14 points wave arrive at the detector exactly out of phase.

Step-by-step solution

Listing the given quantities 

$\mathrm{D}=1.75\text{\hspace{0.17em}cm}$

Sources emit waves at wavelength 0.50 m

Understanding the concept

The problem is based on the concept of destructive interference pattern. In destructive interference the resultant intensity is highly reduced, as the waves are out of phase. For destructive interference, the two waves should have a phase difference equal to $\mathbf{180}\mathbf{°}$ .

(a) To find points where the wave arrive at the detector exactly in phase. 

It can be observed that destructive interference will occur at all the points along the x-axis. This happens because the path difference (for the waves traveling from their respective sources) divided by wavelength gives the (dimensionless) value 3.5, implying a half-wavelength or 180º phase difference i.e. destructive interference between the waves. To distinguish the destructive interference along the +x axis from the destructive interference along the -x - axis, we label one with $+3.5$ and the other $-3.5$. This labeling is useful in that it suggests that the complete enumeration of the quiet directions in the upper-half plane (including the x axis) is: $-3.5,-2.5,-1.5,-0.5,+0.5,+1.5,+2.5,$$+\mathrm{3.5.}$

Similarly, the complete enumeration of the loud directions in the upper-half plane is: $-3,-2,-1,0,+1,+2,+3.$ Considering similar values for the lower-half plane as well, thus there are a total of $7+7=14$“loud” directions.

(b) To find total 14 points wave arrive at the detector exactly out of phase.

The points of constructive interference are: $-3.5,-2.5,-1.5,-0.5,+0.5,+1.5,$$+2.5,+3.5$ along with$-2.5,-1.5,-0.5,+0.5,+1.5,+2.5$ (for the lower-half plane) is 14. There are 14 “quiet” directions.