Two small speakers are separated by a distance of 4 cm, as shown. The speakers a

Question

Two small speakers are separated by a distance of 4 cm, as shown. The speakers are driven in phase with a sine wave signal of frequency 14 kHz. A small microphone is placed a distance 1.4 m away from the speakers on the axis running through the middle of the two speakers, and the microphone is then moved perpendicular to the axis. Where does the microphone record the first minimum and maximum of the interference pattern from the speakers as measured from the axis? The speed of sound in air is 343 m/s. Do not use the small angle approximation for this problem.

Explanation / Answer

When the microphone records first maximum there constructive interference takes place

condition for constructive and destructive interference is d sin theta= m* lambda, d Sin theta= (m+1/2) lambda, (m=0,1,2,3,............)

given   d = 4 cm, f = 14 kHz, L= 1.4 m, v = 343 m/s

       lambda = v/f = 343/14*10^3 = 0.0245 m

d sin theta = m*lambda ==> theta = Sin-1(lambda/d)

               = sin-1(0.0245/4)= 0.35
for minimum                d sin(theta) = (1/2 *lambda)
                       theta = sin-1(lambda/(2*d))= sin-1(0.0245/8)

                  = 0.175
   now y max = L tan theta max
          = 1.4*tan(0.35) = 85.5 cm

        y min = L*tan(theta) = 1.4*tan(0.175)= 42.76 cm

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