The Doppler effect describes how the observed frequency or wavelength of a wave changes when the source and observer move relative to each other. This cheat sheet helps students compare sound waves, which need a medium, with light waves, which do not. It is useful for solving problems about sirens, moving vehicles, stars, galaxies, radar, and astronomy spectra.
Clear sign conventions are especially important because many wrong answers come from choosing the wrong direction of motion.
For sound, the observed frequency depends on the speeds of the source, observer, and wave in the medium. For light, the shift is usually described using redshift and blueshift, and at high speeds it requires the relativistic Doppler formula. Approaching motion increases observed frequency and decreases wavelength, while receding motion decreases observed frequency and increases wavelength.
The most important formulas connect , , , and redshift .
Key Facts
- For any wave, speed, frequency, and wavelength are related by .
- For sound in still air, the Doppler formula is when the observer moves toward the source and the source moves toward the observer.
- If a sound source and observer move closer together, the observed frequency increases, so .
- If a sound source and observer move farther apart, the observed frequency decreases, so .
- For light at low relative speeds, the approximate Doppler shift is , where is positive for recession.
- Redshift is defined by .
- A positive redshift means the source is moving away, while a negative redshift means the source is moving toward the observer.
- For relativistic light Doppler shift along the line of sight, for a receding source, where .
Vocabulary
- Doppler effect
- The Doppler effect is the change in observed frequency or wavelength caused by relative motion between a wave source and an observer.
- Observed frequency
- Observed frequency is the frequency measured by an observer, often written as or .
- Redshift
- Redshift is an increase in observed wavelength, described by .
- Blueshift
- Blueshift is a decrease in observed wavelength that occurs when a light source moves toward an observer.
- Wave speed
- Wave speed is the speed at which a wave travels through a medium or through space, related by .
- Relative velocity
- Relative velocity is the velocity of one object as measured from another object's frame of reference.
Common Mistakes to Avoid
- Using the sound Doppler formula for light is wrong because sound depends on motion through a medium, while light in vacuum always travels at for all observers.
- Reversing the sign convention gives the wrong shift because approaching motion should make larger and smaller.
- Confusing frequency shift with wavelength shift is wrong because frequency and wavelength change in opposite directions when wave speed is fixed by .
- Forgetting to convert units can make the calculation inconsistent because speeds such as and cannot be used together without conversion.
- Using the low-speed light approximation at very high speeds is wrong because is only accurate when .
Practice Questions
- 1 A police siren emits while moving toward a stationary observer at . If the speed of sound is , find the observed frequency using .
- 2 A stationary sound source emits . An observer moves away from the source at while sound travels at . Find using .
- 3 A galaxy has an emitted spectral line at and an observed line at . Calculate the redshift using .
- 4 Explain why an approaching ambulance has a higher-pitched siren as it comes toward you but a lower-pitched siren after it passes you.
Understanding Doppler Effect for Light & Sound Reference
Sound Doppler problems make more sense when you separate two physical effects. A moving source changes the spacing of the wavefronts it produces in the air. In front of a moving source, wavefronts are crowded together.
Behind it, they are spread out. This changes the wavelength in the medium. A moving observer does not rearrange wavefronts already in the air.
Instead, the observer meets them more often or less often. That is why the source speed appears in a different part of the sound equation from the observer speed. The speed of sound is measured relative to the air, so wind and moving air can matter in careful experiments.
A reliable way to choose signs is to describe the motion before using numbers. State whether the observer moves toward or away from the arriving wavefronts. State whether the source moves toward or away from the observer.
Motion that makes wavefront encounters happen more often must raise the observed frequency. Motion that makes the source place wavefronts farther apart must lower it. Draw the source, observer, and several wavefronts.
This simple sketch prevents many sign errors. It is especially useful when both objects move, or when a vehicle passes by and the direction changes after the closest point.
Light shifts are measured very differently from sound shifts because light has no material medium that defines a preferred rest frame. Only the relative motion along the line joining source and observer produces the ordinary radial Doppler shift. Motion across the sky does not create the same simple redshift or blueshift.
At speeds much smaller than the speed of light, the wavelength change is approximately proportional to radial speed. For fast spacecraft, particles, or distant astronomical objects, relativity matters. The relativistic result includes time dilation.
A source moving away has its wave cycles received farther apart in time, while its clock is observed to run slowly. Both effects contribute to the measured frequency shift.
Astronomers identify shifts by comparing known spectral lines with lines measured from a star or galaxy. Each element absorbs or emits light at particular wavelengths, creating a pattern like a barcode. If the whole pattern moves toward longer wavelengths, the object has a redshift.
If it moves toward shorter wavelengths, it has a blueshift. This method can reveal orbital motion in binary stars, rotation in galaxies, and the expansion of the universe. Students should keep wavelength and frequency trends separate.
For light in vacuum, a longer wavelength means a lower frequency, not a higher one. They should also label the original emitted value and the observed value clearly. Reversing those labels can reverse the meaning of a calculated shift.