Doppler Effect — Formula for Sound & Light

The Doppler effect is the change in observed frequency when a source and an observer move relative to each other along the line of sight. If they move toward each other, the observed frequency goes up. If they move apart, it goes down.

For sound, this explains why an ambulance siren sounds higher as it approaches and lower after it passes. For light, the same idea appears as blueshift and redshift, but the formula is different because light is not treated with the sound-in-air model.

What Actually Changes

The quantity that shifts is the observed frequency. In many sound problems, the wave speed is set by the medium, while relative motion changes how often wave crests reach the observer.

That is why it helps to separate three ideas:

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

If you mix those up, Doppler problems become much harder than they need to be.

Doppler Effect Formula For Sound

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

$$f_{\mathrm{obs}} = f \frac{v + v_o}{v - v_s}$$

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. If source and observer move away from each other, the signs change and the observed frequency becomes smaller.

This is a sound-in-a-medium formula, not a universal formula for every wave. It also assumes motion along the line joining source and observer. If the source speed reaches the speed of sound or higher, this simple formula is no longer the right model.

Doppler Effect Formula For Light

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 from the observer is

$$f_{\mathrm{obs}} = f \sqrt{\frac{1 - \beta}{1 + \beta}}$$

where

$$\beta = \frac{v}{c}$$

and $c$ is the speed of light in vacuum.

If the source moves toward the observer along the same line, the factor reverses:

$$f_{\mathrm{obs}} = f \sqrt{\frac{1 + \beta}{1 - \beta}}$$

So motion away gives a lower observed frequency, often called redshift, and motion toward gives a higher observed frequency, often called blueshift.

Worked Example: Ambulance Siren

Suppose 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 in air as $v = 343\ \mathrm{m/s}$, and let the observer be at rest, so $v_o = 0$.

Use the sound formula for motion toward the observer:

$$f_{\mathrm{obs}} = 700 \cdot \frac{343 + 0}{343 - 30}$$ $$f_{\mathrm{obs}} = 700 \cdot \frac{343}{313} \approx 767\ \mathrm{Hz}$$

So the observer hears about $767\ \mathrm{Hz}$ as the ambulance approaches.

After the ambulance passes, it is moving away. In the same sign convention,

$$f_{\mathrm{obs}} = 700 \cdot \frac{343}{343 + 30}$$ $$f_{\mathrm{obs}} = 700 \cdot \frac{343}{373} \approx 644\ \mathrm{Hz}$$

Now the siren sounds lower. This one example shows the whole pattern: motion toward raises the observed frequency, and motion away lowers it.

Common Mistakes

Using the sound formula for light

The sound formula assumes a medium such as air. Light does not need a material medium in the same way, so its Doppler shift must be treated relativistically.

Forgetting that sign conventions vary

Different books may place plus and minus signs in different positions. The safe check is physical: motion toward should increase the observed frequency, and motion away should decrease it.

Ignoring the role of the medium for sound

For sound, speeds in the formula are measured relative to the medium. If you use only the source-observer relative speed and ignore the air or other medium, you can get the wrong result.

Mixing up emitted frequency and observed frequency

The source emits one frequency. The Doppler effect describes what a particular observer receives when there is relative motion.

Thinking higher pitch means the wave speed changed

In the usual sound model, the medium still sets the wave speed. The shift mainly shows up as a change in observed frequency and wavelength.

Where The Doppler Effect Is Used

You see the Doppler effect in everyday sound, but it also matters in radar speed measurements, medical ultrasound, and astronomy.

In astronomy, the observed shift in light helps scientists infer motion along the line of sight. That does not tell you everything about the object's motion by itself, but it is a powerful clue.

Try A Similar Case

Keep the siren at $700\ \mathrm{Hz}$ and change the ambulance speed to $20\ \mathrm{m/s}$ or $40\ \mathrm{m/s}$. Compute the observed frequency before and after the pass, then check whether the size of the shift changes the way you expect.

If you want one useful next step after that, compare this idea with the wave equation. That makes it easier to keep frequency, wavelength, and wave speed separate.