The Doppler Effect — Question 4

Question 4

Approach and recession frequencies determine the source speed

A stationary observer records a source moving at constant speed along the line of sight. The approaching tone is $600\,\mathrm{Hz}$ and the receding tone is $500\,\mathrm{Hz}$. Take $c=340\,\mathrm{m/s}$ and a constant emitted frequency; use the collinear approach and recession limits.

1. Infer the source speed and emitted frequency.

2. Compare the emitted frequency with the arithmetic mean of the two measured tones.

Question 4 - Solution

Approach and recession frequencies determine the source speed

(a) Method and calculation

The measured frequencies obey $f_a=f_sc/(c-v)$ and $f_r=f_sc/(c+v)$. Taking their ratio and solving gives$$\frac vc=\frac{f_a-f_r}{f_a+f_r}=\frac1{11},\qquad v=30.909\,\mathrm{m/s}.$$$$f_s=\frac{2f_af_r}{f_a+f_r}=545.455\,\mathrm{Hz}.$$

(b) Interpretation and further calculation

The arithmetic mean is $550\,\mathrm{Hz}$, too high. The emitted frequency is the harmonic mean:$$\frac1{f_s}=\frac12\left(\frac1{f_a}+\frac1{f_r}\right).$$The approaching shift is larger than the receding shift in magnitude because the source speed appears in a denominator. Averaging the measured frequencies directly ignores this asymmetry.

Physical check

The inferred emitted frequency lies between the observed frequencies, and the inferred speed is subsonic.