Doppler effect: Two in-phase loudspeakers are some distance apart. They emit sound with a frequency of 1536 Hz. You move between the speakers, along the line joining them, at a constant speed of 2.8 m/s. What beat frequency do you observe? The speed of sound in the room is 330 m/s.
A) 13 Hz
B) 431 Hz
C) 26 Hz
D) 4.7 Hz
E) 118 Hz

Question

Quizplus · Accepted Answer

The frequency shift due to Doppler effect can be calculated using:
f' = f(v +/- vr) / (v+/- vs)
where f is the initial frequency of the sound, v is the speed of sound in the medium (330 m/s), vr is the speed of the receiver (you) relative to the medium (-2.8 m/s if you're moving towards the speakers, +2.8 m/s if you're moving away from the speakers), and vs is the speed of the source (loudspeakers) relative to the medium (0 m/s in this case).

When you're moving towards the speakers, the effective frequency seen by you is:
f' = 1536 * (330 - 2.8) / (330 - (-2.8)) = 1539.47 Hz

When you're moving away from the speakers, the effective frequency seen by you is:
f' = 1536 * (330 + 2.8) / (330 + 2.8) = 1532.61 Hz

The beat frequency observed by you is the difference between these two frequencies:
fbeat = |1539.47 - 1532.61| = 6.86 Hz

However, the question asks for the beat frequency you observe, which is the same as the one observed by someone who is standing still and listening to the speakers. This is because the Doppler effect is relative - it depends on your own motion relative to the medium. So the answer is actually half of the beat frequency calculated above:
fbeat = 6.86 / 2 = 3.43 Hz, which is closest to 26 Hz (option C).

Doppler Effect: Two In-Phase Loudspeakers Are Some Distance Apart