AcademySound
Academy
Doppler Shifts
Level 1 - Physics topic page in Sound.
Principle
Doppler shifts come from motion changing wavefront spacing at the listener.
Notation
\(f_S\)
source frequency
\(\mathrm{Hz}\)
\(f_L\)
frequency heard by listener
\(\mathrm{Hz}\)
\(v\)
sound speed in the medium
\(\mathrm{m\,s^{-1}}\)
\(v_L\)
listener velocity toward the source
\(\mathrm{m\,s^{-1}}\)
\(v_S\)
source velocity toward the listener
\(\mathrm{m\,s^{-1}}\)
Method
For sound, source and listener speeds are measured relative to the medium. Motion toward the other object raises the heard frequency.
Moving listener
\[f_L=f_S\frac{v+v_L}{v}\]
A listener moving into wavefronts meets them more often.
Moving source
\[f_L=f_S\frac{v}{v-v_S}\]
A source moving toward the listener emits crests closer together ahead of it.
Combined motion
\[f_L=f_S\frac{v+v_L}{v-v_S}\]
With this sign convention, \(v_L>0\) means the listener moves toward the source and \(v_S>0\) means the source moves toward the listener.
Rules
These are the compact Doppler relations for motion along the line joining source and listener.
Moving listener
\[f_L=f_S\frac{v+v_L}{v}\]
Moving source
\[f_L=f_S\frac{v}{v-v_S}\]
Combined shift
\[f_L=f_S\frac{v+v_L}{v-v_S}\]
Examples
Question
A \(600\,\mathrm{Hz}\) source moves toward a stationary listener at \(25\,\mathrm{m\,s^{-1}}\). Use \(v=340\,\mathrm{m\,s^{-1}}\).
Answer
Here \(v_L=0\) and \(v_S=25\,\mathrm{m\,s^{-1}}\):
\[f_L=600\frac{340}{340-25}=648\,\mathrm{Hz}\]
Checks
- Sound Doppler speeds are relative to the medium, not just relative to each other.
- Approaching motion raises frequency; separating motion lowers frequency.
- Keep the source-speed term in the denominator.
- The formula above is for subsonic motion along one line.