The Doppler Effect — Question 6

Question 6

Wind changes the medium-frame velocities used for Doppler shift

Air flows rightward at $w=20\,\mathrm{m/s}$ relative to the ground. Sound speed relative to air is $c=340\,\mathrm{m/s}$. A source moving right at ground speed $30\,\mathrm{m/s}$ emits $440\,\mathrm{Hz}$ toward a ground-stationary observer ahead.

1. Calculate the observed frequency using velocities relative to the air.

2. Recover the answer directly from ground-frame crest spacing. If the source also becomes ground-stationary, does wind alone shift the frequency?

Question 6 - Solution

Wind changes the medium-frame velocities used for Doppler shift

(a) Method and calculation

The source and observer velocities relative to air are $u_s=30-20=10$ and $u_o=0-20=-20\,\mathrm{m/s}$. Therefore$$f_o=440\frac{340-(-20)}{340-10}=480\,\mathrm{Hz}.$$

(b) Interpretation and further calculation

The rightward ground speed of sound is $c+w=360\,\mathrm{m/s}$. The crest spacing is$$\lambda=\frac{360-30}{440}=0.750\,\mathrm m,\quad f_o=360/0.750=480\,\mathrm{Hz}.$$If both endpoints are stationary on the ground, their air-frame velocities both equal $-w$; the Doppler factors cancel. Uniform wind changes travel time and wavelength but not the frequency between stationary endpoints.

Physical check

Using ground source speed with still-air speed in only one part of the formula mixes frames and gives an inconsistent result.