As you stand at rest along the side of a road, a car is moving directly away fro
ID: 1431474 • Letter: A
Question
As you stand at rest along the side of a road, a car is moving directly away from you at a steady speed Vc, sounding its horn with a constant power P and frequency f. At a certain moment, the relative intensity of the sound you hear from the horn is b. Find expressions (using only known values) for each of the following:
a. Find the rate of change of the sound’s absolute intensity, I, at that moment.
b. Find the beat frequency you would hear by listening to both the horn of the receding car and its echo, which is returning to you after reflecting off the front of a truck moving toward you at speed v. Model the horn as a spherically symmetric sound source. Ignore any interference other than the beat frequency.
Assume the following known values: VC, P, f, b, Io , Vsound
Explanation / Answer
1) intensity = P / 4 pi r0^2 = b
initial distance, r0 = sqrt ( P / 4pi b)
distance at time t: r = r0 + Vc*t = sqrt ( P / 4pi b) + Vc*t
New Intensity, I = P / 4 pi r^2
= P / ( sqrt ( P / 4pi b) + Vc*t )^2
dI / dt = - 2 [ P Vc/ ( sqrt ( P / 4pi b) + Vc*t )^3
dI/dt = - P Vc / ( sqrt ( P / 4pi b) + Vc*t )^3
b) Applying doppler effect,
hearing horn frequency, f1 = Vsound f / (Vsound + Vc )
truck receiving freuency = reflecting frequency , f' = (Vsound + v)f / (Vsoud - Vc)
echo frequency, f2 = (Vsound ) f' / (Vsound + v)
= (Vsound )(Vsound + v)f / [(Vsoud - Vc) (Vsound + v) ]
beat freuency = f2 - f1
= 2 Vsound (Vsound Vc + Vc v ) / [ (Vsoud - Vc) (Vsound + v)(Vsound + Vc )]
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