Two small speakers are separated by a distance of 6 cm, as shown. The speakers a

Question

Two small speakers are separated by a distance of 6 cm, as shown. The speakers are driven in phase with a sine wave signal of frequency 10 kHz. A small microphone is placed a distance 1.2 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 of the interference pattern from the speakers as measured from the axis? The speed of sound in air is 343 m/s

m (from axis)   

and the first maximum

m (from axis)   

Explanation / Answer

For destructive interfence the condition is d*sin(theta) = m*lamda/2

for first minimum m = 1

d*sin(theta) = lamda/2

sin(theta) = lamda/(2*d) = (v/f)/(2*d) = (343/10^4)/(2*0.06) = 0.285
theta = asin(0.285) = 16.55 degrees

required distnce for first minimum is 1.2*tan(16.55) =0.356 m = 35.6 cm

For constructive interference
d*sin(theta) = m*lamda

m= 1 and lamda = (343/10^4) = 0.0343

sin(theta) = 0.0343/d = 0.0343/0.06 = 0.571

theta = asin(0.571) = 34.8 degrees

required distnace for first maximum is 1.2*tan(34.8) = 0.834 m
83.4 cm

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