Two in-phase speakers 2.0 m apart in a plane are emitting 1800 Hz sound waves into a room where the speed of sound is 340 m/s. Is the point 4.0 m in front of one of the speakers, perpendicular to the plane of the speakers, a point of maximumn constructive interference, maximum destructive interference, or something in between?

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uiteensitnh · Accepted Answer

First of all, we must get the image right: the two loudspeakers are facing the same way, perpendicular to the line connecting them. Therefore, the point which we are interested in is 4 m away from one of the loudspeakers, and the distance from the other one can be found by the Pythagorean theorem to be      d          2        =            4              2              +          2              2              =  2      5  Clearly, the path difference is  △  d  =  2      5    −  4  =  2  (      5    −  2  )If we write this path difference as a function of the wavelength, it will be given as  △  d  =  a  λ .where lambda is a positive constant. If a is an integer, the interference will be perfectly constructive; if it is given as the modulus 2 of an odd number (i.e. 1.5, 2.5, .159.5,..) then the interference is perfectly destructive. If it is none of these, then we have something inbetween. Let us find a to find the state of the point in question.  a  =            △      d        λ    =            △      d              v              /            f        =            △      d      f        v    =            2      (      1800              5            −      2      )        340    =  2.5Therefore, at the point in question we will have a perfect destructive interference.Result:Perfect destructive.

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