Doppler Effect — Formula for Sound & Light

When an ambulance passes you, the siren sounds higher as it approaches and lower once it goes by. That shift in the frequency you actually hear is the Doppler effect: the change in observed frequency when a source and an observer move relative to each other along the line of sight. Toward each other raises the observed frequency; apart lowers it. The same idea appears for light as blueshift and redshift, though light uses a different formula.

The Formulas and Their Symbols

For sound in a stationary medium, one common $1$D form is

where:

• $f$ is the emitted frequency
• $f_{\mathrm{obs}}$ is the observed frequency
• $v$ is the speed of sound in the medium
• $v_o$ is the observer's speed toward the source
• $v_s$ is the source's speed toward the observer

This version uses a specific sign convention: speeds counted toward each other make the observed frequency larger, and the signs flip when source and observer move apart. It assumes motion along the line joining them, and it stops being the right model once the source speed reaches the speed of sound.

For light, the classical sound formula does not apply. For motion along the line of sight, the relativistic Doppler formula for a source moving away is

and for a source moving toward the observer the factor reverses:

Here $c$ is the speed of light in vacuum. Motion away gives a lower observed frequency (redshift); motion toward gives a higher one (blueshift).

Why the Formula Looks This Way

The quantity that shifts is the observed frequency, not the wave speed. In many sound problems the medium fixes the wave speed, and relative motion only changes how often wave crests reach the observer. That is the physical reason the source speed sits in the denominator: as the source chases its own crests, it packs them closer together, so more arrive each second.

That also explains why three ideas must stay separate:

• frequency: how many cycles arrive each second
• wavelength: spacing between wave crests
• wave speed: how fast the disturbance moves through the medium

Mix those up and Doppler problems get much harder than they need to be. For light the medium argument fails, which is exactly why the relativistic factor replaces the simple ratio.

Worked Example: Ambulance Siren

An ambulance emits a siren at $f = 700\ \mathrm{Hz}$ and moves toward a stationary observer at $v_s = 30\ \mathrm{m/s}$. Take the speed of sound as $v = 343\ \mathrm{m/s}$, with the observer at rest, so $v_o = 0$.

Approaching, use the toward-the-observer form:

After it passes, it is moving away, so the sign flips:

The whole pattern is in those two numbers: toward raises the pitch to about $767\ \mathrm{Hz}$, away lowers it to about $644\ \mathrm{Hz}$.

Practice It Yourself

Keep the siren at $700\ \mathrm{Hz}$ and change the ambulance speed to $20\ \mathrm{m/s}$, then to $40\ \mathrm{m/s}$. Compute the approaching and receding frequencies for each.

Check your work against the trend: a faster source should widen the gap between the high and low pitches. At $20\ \mathrm{m/s}$ you should get roughly $743\ \mathrm{Hz}$ and $661\ \mathrm{Hz}$; at $40\ \mathrm{m/s}$, roughly $792\ \mathrm{Hz}$ and $627\ \mathrm{Hz}$. If your numbers move the wrong way, recheck which sign goes with approach.

Calculation Traps to Avoid

• Using the sound formula for light. The sound formula assumes a material medium. Light's shift must be treated relativistically.
• Trusting a memorized sign layout. Books place plus and minus differently. The safe check is physical: toward should raise the frequency, away should lower it.
• Dropping the medium for sound. Speeds in the sound formula are measured relative to the medium, not just as a source-observer relative speed.
• Confusing emitted and observed frequency. The source emits one frequency; the Doppler effect is what a particular observer receives.
• Assuming the wave speed changed. In the usual model the medium still sets the wave speed; the shift shows up in observed frequency and wavelength.

Where It Matters

Beyond sirens, the Doppler effect drives radar speed measurement, medical ultrasound, and astronomy, where the observed shift in light reveals motion along the line of sight. To keep frequency, wavelength, and wave speed cleanly separated as the math gets busier, it helps to pair this with the wave equation.