KAHIBARO
Table of Contents
Changing Frequency Due to Motion
The Doppler effect is the change in observed frequency of a wave when there is relative motion between the source of the wave and the observer. For sound, this means the pitch you hear changes if the source moves, if you move, or if both move.
A common example is a passing siren. As the ambulance approaches, the siren sounds higher in pitch. As it moves away, the siren sounds lower. The actual sound produced by the siren does not need to change. What changes is how often the sound wavefronts reach the listener.
Basic Idea
Sound travels through a medium such as air with speed $v$. If the source moves toward the observer, the wavefronts in front of the source become closer together. A shorter wavelength means a higher observed frequency. If the source moves away, the wavefronts spread out, the wavelength becomes longer, and the observed frequency becomes lower.
If the observer moves, the wavelength in the air does not change, but the observer meets the wavefronts more quickly or more slowly. Moving toward the source increases the rate of meeting wavefronts, so the observed frequency increases. Moving away decreases it.
Doppler Formula for Sound
For sound in a medium, the general formula is
$$
f' = f \frac{v \pm v_o}{v \mp v_s}
$$
where $f'$ is the observed frequency, $f$ is the emitted frequency, $v$ is the speed of sound in the medium, $v_o$ is the speed of the observer relative to the medium, and $v_s$ is the speed of the source relative to the medium.
The signs must be chosen carefully. Use the sign that makes the frequency increase when source and observer move toward each other, and decrease when they move apart.
Important rule:
$$
f' = f \frac{v \pm v_o}{v \mp v_s}
$$
Choose signs so that motion toward increases $f'$ and motion away decreases $f'$.
Sign Convention
A useful way to remember the formula is given in the table below.
| Situation | Effect on observed frequency | Formula choice |
|---|---|---|
| Observer moves toward source | Increases | $v + v_o$ in numerator |
| Observer moves away from source | Decreases | $v - v_o$ in numerator |
| Source moves toward observer | Increases | $v - v_s$ in denominator |
| Source moves away from observer | Decreases | $v + v_s$ in denominator |
The observer speed affects the numerator. The source speed affects the denominator.
Source Motion and Wavelength
When the source moves, it changes the spacing of the sound waves in the medium. If the source emits sound of frequency $f$, then in time $T = 1/f$ it moves a distance $v_s T$.
For a source moving toward the observer, the wavelength in front becomes
$$
\lambda' = \frac{v - v_s}{f}
$$
so the observed frequency is
$$
f' = \frac{v}{\lambda'} = \frac{vf}{v - v_s}
$$
For a source moving away,
$$
\lambda' = \frac{v + v_s}{f}
$$
and therefore
$$
f' = \frac{vf}{v + v_s}
$$
This shows clearly why source motion changes wavelength.
Observer Motion
If the source is at rest and the observer moves toward it, the wavelength stays
$$
\lambda = \frac{v}{f}
$$
but the observer encounters wavefronts at speed $v + v_o$. Then
$$
f' = \frac{v + v_o}{\lambda} = f\frac{v + v_o}{v}
$$
If the observer moves away,
$$
f' = f\frac{v - v_o}{v}
$$
So observer motion changes how quickly wavefronts are received, not the wavelength in the air.
Visualizing the Effect
The waves are compressed in front of the source and stretched behind it.
Example
Suppose a car horn emits sound at $f = 500 \,\text{Hz}$. The speed of sound is $v = 340 \,\text{m/s}$. The car moves toward a stationary observer at $v_s = 20 \,\text{m/s}$.
Using the source moving toward observer formula,
$$
f' = f\frac{v}{v - v_s}
$$
$$
f' = 500 \cdot \frac{340}{340 - 20}
= 500 \cdot \frac{340}{320}
= 531.25 \,\text{Hz}
$$
So the observer hears about $531 \,\text{Hz}$.
If the same car moves away,
$$
f' = 500 \cdot \frac{340}{340 + 20}
= 472.2 \,\text{Hz}
$$
The pitch is then lower than the emitted frequency.
When Both Source and Observer Move
If both are moving, use the full formula. For example, let the observer move toward the source at $v_o = 10 \,\text{m/s}$ while the source also moves toward the observer at $v_s = 15 \,\text{m/s}$.
Then
$$
f' = f \frac{v + v_o}{v - v_s}
$$
The numerator becomes larger and the denominator becomes smaller, so both motions increase the observed frequency.
For sound, source motion and observer motion are not symmetric.
The observer changes how quickly wavefronts are received.
The source changes the wavelength in the medium.
Special Cases
If there is no relative motion, then $v_o = 0$ and $v_s = 0$, so
$$
f' = f
$$
If only the observer moves,
$$
f' = f\frac{v \pm v_o}{v}
$$
If only the source moves,
$$
f' = f\frac{v}{v \mp v_s}
$$
These simpler forms are often enough for basic problems.
Limitation for Sound
The Doppler effect for sound depends on the medium, because sound travels through air, water, or another material. Speeds in the formula are measured relative to that medium. Wind can affect the situation because it changes motion relative to the air.
The formula given here is used for ordinary speeds much smaller than the speed of sound. More advanced cases, such as shock waves and sonic booms, belong to other topics.
What the Listener Actually Hears
As a source passes a listener, the sound changes from a higher frequency before passing to a lower frequency after passing. This sudden shift is often very noticeable. The effect is strongest when the motion is directly along the line between source and observer. If the motion is sideways, the change is smaller because only the component of motion along the line of sight matters.
Key Results
Main Doppler formulas for sound:
$$
f' = f \frac{v \pm v_o}{v \mp v_s}
$$
Observer toward source:
$$
f' = f\frac{v + v_o}{v}
$$
Source toward observer:
$$
f' = f\frac{v}{v - v_s}
$$
Motion toward increases observed frequency.
Motion away decreases observed frequency.
The Doppler effect explains why moving sound sources change pitch for a listener. It is one of the clearest examples of how motion affects wave observation.