The Doppler Effect — Question 2

Question 2

Signed velocities keep the two-endpoint Doppler formula consistent

Sound travels rightward in still air at $c=340\,\mathrm{m/s}$. A source to the left emits $640\,\mathrm{Hz}$ while moving right at $u_s=20\,\mathrm{m/s}$. The observer moves left at $10\,\mathrm{m/s}$, so its signed velocity is $u_o=-10\,\mathrm{m/s}$.

1. Find the wavelength ahead of the source and the observed frequency.

2. Derive the signed one-dimensional formula for arbitrary subsonic $u_s,u_o$ while sound propagates rightward.

Question 2 - Solution

Signed velocities keep the two-endpoint Doppler formula consistent

(a) Method and calculation

In one source period, a crest advances $c/f_s$ while the source advances $u_s/f_s$. Thus$$\lambda_+=\frac{c-u_s}{f_s}=\frac{320}{640}=0.500\,\mathrm m.$$The observer meets the rightward fronts at relative speed $c-u_o=350\,\mathrm{m/s}$, giving $f_o=350/0.500=700\,\mathrm{Hz}$.

(b) Interpretation and further calculation

Combining the spacing and encounter rate yields$$f_o=f_s\frac{c-u_o}{c-u_s}.$$The velocities are signed relative to one fixed axis in the medium frame. Recession of the observer to the right lowers the numerator; approach of the source to the right lowers the denominator and raises the frequency. These signs follow from geometry rather than memorized approach symbols.

Physical check

The observer and source effects reinforce here. The emitted frequency remains $640\,\mathrm{Hz}$ in the source clock convention of classical acoustics.