Figure  shows two  isotopic point sources of  sound, 
$$S_1$$ and $$S_2$$. The sources emit wave in phase at wavelength 0.50 m; they are separated by D = 1.75 m. 
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?



(a) The problem is asking at how many angles will there be "loud" resultant waves and at how many will there be "quiet" ones? 
We note that at all points (at large distance from the origin) along the x axis there will be quiet ones : one way to see this is to note that the path-length difference (for the waves traveling from their respective sources) divided by wavelength gives the (dimensionless) value 3.5, implying a half-wavelength $$\left (180^{0}  \right )$$ phase difference (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, +3.5. 
Similarly, the complete enumeration of the loud directions in the upper half plane is -3, -2, -1, 0, +1, +2, +3. Counting also the "other" -3, -2, -1, 0, +1, +2, +3 values for the lower-half plane, then we conclude there are a total of 7 + 7 = 14 "loud" directions.
(b) The discussion about the "quiet" directions was started in part(a). The number of values in the list: -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.


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