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Question 16.18: DOPPLER EFFECT V: A DOUBLE DOPPLER SHIFT The police car is m...

DOPPLER EFFECT V: A DOUBLE DOPPLER SHIFT

The police car is moving toward a warehouse at 30 m/s. What frequency does the driver hear reflected from the warehouse?

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IDENTIFY: This situation has two Doppler shifts (Fig. 16.34). In the first shift, the warehouse is the stationary “listener.” The frequency of sound reaching the warehouse, which we call f_W, is greater than 300 Hz because the source is approaching. In the second shift, the warehouse acts as a source of sound with frequency f_W, and the listener is the driver of the police car; she hears a frequency greater than f_W because she is approaching the source.

 

SET UP: To determine f_W, we use Eq. (16.29) with f_L replaced by f_W. For this part of the problem, v_L = v_W= 0 (the warehouse is at rest) and v_S = -30 m/s (the siren is moving in the negative direction from source to listener). To determine the frequency heard by the driver (our target variable), we again use Eq. (16.29) but now with f_S replaced by f_W. For this second part of the problem, v_S = 0 because the stationary warehouse is the source and the velocity of the listener (the driver) is v_L = +30 m/s. (The listener’s velocity is positive because it is in the direction from listener to source.)

f_{\mathrm{L}}=\frac{v+v_{\mathrm{L}}}{v+v_{\mathrm{S}}} f_{\mathrm{S}}                    (16.29)

EXECUTE:

The frequency reaching the warehouse is

f_{\mathrm{W}}=\frac{v}{v+v_{\mathrm{S}}} f_{\mathrm{S}}=\frac{340 \mathrm{~m} / \mathrm{s}}{340 \mathrm{~m} / \mathrm{s}+(-30 \mathrm{~m} / \mathrm{s})}(300 \mathrm{~Hz})=329 \mathrm{~Hz}

Then the frequency heard by the driver is

f_{\mathrm{L}}=\frac{v+v_{\mathrm{L}}}{v} f_{\mathrm{W}}=\frac{340 \mathrm{~m} / \mathrm{s}+30 \mathrm{~m} / \mathrm{s}}{340 \mathrm{~m} / \mathrm{s}}(329 \mathrm{~Hz})=358 \mathrm{~Hz}

 

EVALUATE: Because there are two Doppler shifts, the reflected sound heard by the driver has an even higher frequency than the sound heard by a stationary listener in the warehouse.

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