The Doppler Effect is the change in frequency or wavelength of a wave in relation to an observer who is moving relative to the wave source. Named after the Austrian physicist Christian Doppler, who described the phenomenon in 1842, it is a fundamental property of waves that applies across multiple domains of physics, from acoustics to electromagnetism.

Kinematics of Wavefronts

Consider a stationary source emitting a wave with frequency $f_0$ and period $T = 1/f_0$. The wave propagates through the medium with speed $c$. The wavefronts form concentric spheres (or circles in 2D) centered on the source. The wavelength $\lambda$ is simply the distance the wave travels in one period: $\lambda = cT = c/f_0$.

Now, suppose the source moves with velocity $v_s$ towards a stationary observer. In the time $T$ between the emission of two consecutive wavefronts, the source moves a distance $v_s T$. The distance between these two wavefronts (the observed wavelength $\lambda'$) in the direction of motion is reduced:

$$ \lambda' = \lambda - v_s T = \frac{c}{f_0} - \frac{v_s}{f_0} = \frac{c - v_s}{f_0} $$

The observer perceives a frequency $f' = c/\lambda'$, which gives the equation for a moving source:

$$ f' = \left( \frac{c}{c - v_s} \right) f_0 $$

If the source is moving away, the sign of $v_s$ is reversed, leading to a decreased frequency (redshift).

Moving Observer

Alternatively, if the source is stationary and the observer moves towards it with velocity $v_o$, the wavelength $\lambda$ in the medium remains unchanged. However, the relative speed of the wavefronts with respect to the observer increases to $c + v_o$. The number of wavefronts intercepted per unit time (the observed frequency) increases:

$$ f' = \frac{c + v_o}{\lambda} = \frac{c + v_o}{c/f_0} = \left( \frac{c + v_o}{c} \right) f_0 $$

Combining both effects into a single generalized equation, where relative motion towards each other corresponds to positive velocities in the numerator and negative in the denominator:

$$ f' = \left( \frac{c \pm v_o}{c \mp v_s} \right) f_0 $$

The Relativistic Doppler Effect

For electromagnetic waves in a vacuum, the medium-dependent classical derivation fails, as the speed of light $c$ is constant for all inertial frames. Instead, we must account for relativistic time dilation. For a source moving away at velocity $v$, the observed frequency $f'$ is derived via Lorentz transformations:

$$ f' = f_0 \sqrt{ \frac{c - v}{c + v} } $$

This longitudinal relativistic Doppler effect is essential in astrophysics for measuring the cosmological redshift of distant galaxies and confirming the expansion of the universe.

Supersonic Regimes and Mach Cones

A fascinating threshold is reached when the source velocity $v_s$ exceeds the wave speed $c$ ($v_s > c$). In this supersonic regime, the source outpaces the wavefronts it generates.

The expanding wavefronts construct an envelope—a conical shock wave known as a Mach cone. The geometry of this cone is defined by the Mach angle $\alpha$, representing the half-angle of the cone's vertex. At a time $t$, the source has traveled a distance $v_s t$, while the wavefront emitted at $t=0$ has expanded to a radius $c t$. The sine of the Mach angle is given by the ratio of these distances:

$$ \sin(\alpha) = \frac{c t}{v_s t} = \frac{c}{v_s} = \frac{1}{M} $$

where $M = v_s/c$ is the Mach number. Along the surface of this cone, the constructive interference of the piled-up wavefronts results in a massive surge of energy—experienced as a sonic boom in acoustics or observed as Cherenkov radiation when charged particles exceed the local phase velocity of light in a dielectric medium.