The Doppler effect is a common phenomenon in everyday life, although we often overlook it due to familiarity. A well-known example is the sound of a siren on an approaching ambulance. The siren emits sound waves with a fixed frequency and wavelength, and these waves propagate through air at the speed of sound. If the ambulance moves toward an observer, the pitch (frequency) of the siren appears higher. After it passes and moves away, the pitch appears lower.
In this example, the source (ambulance) is moving, while the observer is stationary. In another case — such as a stationary bell at a railroad crossing — a moving observer (e.g., passengers on a train) hears a higher pitch when approaching the bell and a lower pitch after passing it.
Let us analyze the case of a moving source (like an ambulance). The speaker emits waves at wavelength $\lambda$, but since it’s moving toward the observer, the wavefronts are compressed:
$$
\lambda' = \lambda - \Delta \lambda
$$
The shift $\Delta \lambda$ depends on the ratio between the source’s speed $v$ and the wave speed $c$:
$$
\lambda' = \lambda \left(1 - \frac{v}{c}\right)
$$
Using $f = \frac{c}{\lambda}$, the frequency observed becomes:
$$
f' = \frac{c}{\lambda'} = \frac{c}{\lambda (1 - \frac{v}{c})} = \frac{f}{1 - \frac{v}{c}}
$$
If the source is moving away, use $+v$ instead of $-v$. This gives the general formula for a moving source:
$$
\boxed{f'_\mathrm{source} = \frac{f}{1 \mp \frac{v}{c}}}
$$
If the observer moves toward a stationary source, the wavelength doesn’t change — but the observer encounters more wavefronts per second:
$$
f' = \frac{c + v}{\lambda} = f \left(1 + \frac{v}{c}\right)
$$
If the observer moves away, use $-v$. The general formula for a moving observer becomes:
$$
\boxed{f'_\mathrm{observer} = f \left(1 \pm \frac{v}{c}\right)}
$$
If both source and observer are moving (e.g. a speeding car approaching a siren), combine the two results:
$$
\boxed{f'_\mathrm{combined} = f \cdot \frac{c + v_\mathrm{E}}{c - v_\mathrm{Q}}}
$$
Where:
- $v_\mathrm{E}$: speed of the observer (positive if toward source),
- $v_\mathrm{Q}$: speed of the source (positive if toward observer).
When a source moves at the speed of sound ($v = c$), the denominator in the Doppler formula goes to zero, and the perceived frequency approaches infinity — a singularity. Physically, this corresponds to a buildup of wavefronts into a sharp shock wave, perceived as a sonic boom.
Supersonic aircraft cause such booms when breaking the sound barrier. Behind the aircraft, a conical wavefront forms — the Mach cone. The half-angle $\theta$ of this cone can be derived geometrically:
$$
\boxed{\sin \theta = \frac{c}{v}}
$$
The greater the aircraft’s speed, the narrower the cone.
A similar effect occurs with charged particles moving faster than the speed of light in a medium (not in vacuum). These emit a Cherenkov cone of light, visible in nuclear reactors filled with water.