Doppler Effect with Moving Sound Source · USAPhO 2016 #A1

Physics

Problem

The Doppler effect for a source moving relative to a stationary observer is described by f0=f1−(v/c)cos⁡θf_0 = \frac{f}{1-(v/c)\cos\theta} where ff is the frequency measured by the observer, f0f_0 is the frequency emitted by the source, vv is the speed of the source, cc is the wave speed, and θ\theta is the angle between the source velocity and the line between the source and observer. (Thus θ=0\theta = 0 when the source is moving directly towards the observer and θ=π\theta = \pi when moving directly away.) A sound source of constant frequency travels at a constant velocity past an observer, and the observed frequency is plotted as a function of time: [Figure: Graph showing observed frequency vs time. Y-axis shows frequency from 420-450 Hz, X-axis shows time from 0-14 seconds. The curve starts at about 448 Hz at t=0, decreases smoothly in an asymptotic fashion, reaching about 422 Hz by t=14s.] The experiment happens in room temperature air, so the speed of sound is 340 m/s. a. What is the speed of the source? b. What is the smallest distance between the source and the observer?
Intuitive explanation

The Doppler formula shows that observed frequency depends on the angle θ\theta between the source's velocity and the line-of-sight to the observer. As the source passes by, θ\theta changes continuously—starting near 0° (approaching), passing through 90° (closest point), and ending near 180° (receding)—so the observed frequency traces out a smooth curve. The key insight is that the extreme frequencies (highest and lowest) occur at the turning points, while the minimum frequency shift happens when the source is closest, where θ=90°\theta = 90° and the Doppler factor simplifies. By reading the graph's asymptotic limits and using the geometry of closest approach, you can extract both the source speed and the minimum distance.

Intuitive explanation

The Doppler formula shows that observed frequency depends on the angle θ\theta between the source's velocity and the line-of-sight to the observer. As the source passes by, θ\theta changes continuously—starting near 0° (approaching), passing through 90° (closest point), and ending near 180° (receding)—so the observed frequency traces out a smooth curve. The key insight is that the extreme frequencies (highest and lowest) occur at the turning points, while the minimum frequency shift happens when the source is closest, where θ=90°\theta = 90° and the Doppler factor simplifies. By reading the graph's asymptotic limits and using the geometry of closest approach, you can extract both the source speed and the minimum distance.

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