At a school track meet two loudspeakers were used to summon athletes to their ev

Question

At a school track meet two loudspeakers were used to summon athletes to their events. One student claimed that she never heard her event called. The speakers had very poor high frequency reproduction capability so that the dominant wavelength in the announcements was about 1 m. Assume that the speakers were placed 5 m apart and that the girl was sitting across the field 50 m from the speakers. How far from the centerline between the speakers and the bleachers does the first minimum in the sound pattern occur? The girl's ears are about 10 cm apart. Do you think it is possible that she actually did not hear her event called? Why?

Explanation / Answer

a)

r1 = sqrt((y-2.5)*(y-2.5)+50*50) = sqrt((y-2.5)*(y-2.5)+2500)

r2 = sqrt((y+2.5)*(y+2.5)+50*50) = sqrt((y+2.5)*(y+2.5)+2500)

path difference = r2 - r1 = sqrt((y+2.5)*(y+2.5)+2500) - sqrt((y-2.5)*(y-2.5)+2500)

lamda = 1 m

for the first destructive interference, path difference must be lamda/2 :

sqrt((y+2.5)*(y+2.5)+2500) - sqrt((y-2.5)*(y-2.5)+2500) = 0.5

==> y = 5.03 m

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