Loudspeakers A and B are vibrating in phase and are playing the
same tone, which has a frequency of 420 Hz. They are set up as in
the figure, and point C is located as shown there. However, the
distance between the speakers and the distance between speaker B
and point C have the same value d. The speed of sound is
343 m/s. What is the smallest value of d, such that
constructive interference occurs at point C?
The segments BA and BC are at right angles.
If that's the case, then constructive interference first occurs
where the difference in lengths CA minus CB equals one
wavelength.
Let l = 1 wavelength ( l as in lambda)
f = frequency = 420 Hz (given)
d = distance between speakers, and from B to C
v = speed of sound = 343 m/s (given)
Constructive interference first occurs when:
CA - CB = l
But l = V/f
CB = AB = d
CA = d x sqrt(2) (where sqrt(2) = square root of 2)
Substituting into the above equation:
CA - CB = l
d x sqrt(2) - d = V/f
d = V / (f x ( sqrt(2)-1 ) )
d = 343 /(420 x 0.414213)
d = 1.97 m
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