The Doppler Effect
When a source of waves moves relative to an observer, the observed frequency changes. This phenomenon, known as the Doppler Effect, explains why a passing siren drops in pitch and why distant galaxies are redshifted. Explore how the speed of the source affects the wavefronts in a continuous medium.
The Mathematics
For a moving source and a stationary observer in a medium where waves propagate at speed $c$, the observed frequency $f'$ is given by:
Where $f$ is the emitted frequency, $c$ is the speed of waves in the medium, and $v_s$ is the speed of the source. The minus sign is used when the source is approaching the observer, and the plus sign when receding.
Mach Cone
When the source exceeds the wave speed ($v_s > c$), it outruns its own waves. The waves pile up to form a conical shock front, known as the Mach cone. The half-angle $\theta$ of this cone is defined by:
Where $M$ is the Mach number.