The Doppler effect is the change in observed frequency caused by motion between a source and an observer. In everyday life, it explains why a siren sounds higher in pitch as it approaches and lower as it moves away. For sound, the effect must be analyzed relative to the medium through which the sound travels, because sound waves move through air (or another material medium) at a finite speed \(v\).
In the non-relativistic sound model, the observed frequency is
\[
f' = f\frac{v \pm v_o}{v \pm v_s}.
\]
Here:
\[
f = \text{emitted frequency},\quad
f' = \text{observed frequency},\quad
v = \text{sound speed in the medium},
\]
\[
v_o = \text{observer speed},\quad
v_s = \text{source speed}.
\]
The signs are not chosen the same way in the numerator and denominator. For the observer, use \(+\) when the observer moves toward the source and \(−\) when moving away. For the source, use \(−\) in the denominator when the source moves toward the observer and \(+\) when it moves away.
The reason is physical. A moving observer changes how quickly wavefronts are encountered. If the observer moves toward the source, wavefronts are met more often, so the observed frequency increases. A moving source changes the spacing of the wavefronts themselves. When the source moves toward the observer, successive wavefronts are emitted from positions closer and closer to the observer, so they are compressed. That means the wavelength in front becomes shorter and the observed pitch increases.
For example, if the observer is stationary and the source moves toward the observer, the formula becomes
\[
f' = f\frac{v}{v-v_s}.
\]
Using the sample values \(f=500\ \text{Hz}\), \(v=343\ \text{m/s}\), and \(v_s=20\ \text{m/s}\),
\[
f' = 500\cdot\frac{343}{343-20}
= 500\cdot\frac{343}{323}
\approx 531\ \text{Hz}.
\]
The observer therefore hears a higher pitch than the emitted one.
A useful extra quantity is the Mach number,
\[
M=\frac{\text{speed}}{v}.
\]
For the source, \(M_s=v_s/v\). When \(M_s\) approaches 1, the denominator \(v-v_s\) becomes very small for an approaching source, and the simple classroom Doppler formula begins to break down. At and above Mach 1, shock-wave effects become important and a sonic boom can occur. That is why this calculator includes a Mach-number check.
This calculator uses the standard non-relativistic sound formula, which assumes a stationary medium and speeds well below the speed of light. At university level, the topic expands to relativistic Doppler shift for light, Doppler radar, moving-media corrections, and shock-front analysis for supersonic motion.
The animation in this tool helps build intuition by showing compressed wavefronts in front of a moving source and stretched wavefronts behind it. The plot then shows how the observed frequency changes as either source speed or observer speed varies.
