The Doppler effect describes how the observed frequency of a wave changes when the source, observer, or both are moving. This cheat sheet focuses on sound waves, where motion toward each other raises the observed pitch and motion apart lowers it. Students need these formulas to solve common problems involving sirens, vehicles, speakers, and moving listeners.
It also connects frequency differences to beats, which are heard when two similar tones interfere.
The main Doppler formula for sound is , where is the speed of sound, is observer speed, and is source speed. The signs are chosen by asking whether the motion makes the observed frequency increase or decrease. Beat frequency is found with , which gives the number of loudness pulses per second.
Interference is strongest when waves meet in phase and weakest when they meet out of phase.
Key Facts
- For sound, the Doppler effect formula is , where is observed frequency and is emitted frequency.
- Use a larger observed frequency when the source and observer move toward each other, so .
- Use a smaller observed frequency when the source and observer move away from each other, so .
- For a moving observer and stationary source, use , with when the observer moves toward the source.
- For a moving source and stationary observer, use , with in the denominator when the source moves toward the observer.
- Beat frequency is the absolute difference between two nearby frequencies: .
- Constructive interference occurs when waves are in phase, and the resulting amplitude is larger than either individual wave.
- Destructive interference occurs when waves are out of phase, and the resulting amplitude is reduced or canceled.
Vocabulary
- Doppler effect
- The apparent change in observed frequency caused by relative motion between a wave source and an observer.
- Observed frequency
- The frequency detected by an observer, written as , which may differ from the emitted frequency.
- Source speed
- The speed of the object producing the sound wave, written as in Doppler effect formulas.
- Observer speed
- The speed of the listener or detector, written as in Doppler effect formulas.
- Beat frequency
- The rate at which loud and soft pulses are heard when two similar frequencies interfere, given by .
- Interference
- The combining of two or more waves to produce a larger, smaller, or canceled resultant wave.
Common Mistakes to Avoid
- Using the wrong sign in is wrong because the sign must match whether the observed frequency increases or decreases.
- Forgetting the absolute value in is wrong because beat frequency cannot be negative.
- Mixing up source motion and observer motion is wrong because a moving source changes the denominator, while a moving observer changes the numerator.
- Using the speed of the moving car instead of the speed of sound for is wrong because represents the wave speed in the medium.
- Assuming beats happen for very different frequencies is wrong because beats are most noticeable when and are close together.
Practice Questions
- 1 A stationary observer hears a siren from an ambulance moving toward them at . If the speed of sound is , find using .
- 2 A listener moves toward a stationary speaker at while the speaker emits . If , find the observed frequency using .
- 3 Two tuning forks have frequencies and . Find the beat frequency using .
- 4 Explain why the pitch of a siren sounds higher as it approaches and lower after it passes, even though the siren emits the same frequency.
Understanding Doppler Effect & Beat Frequencies
A moving sound source changes the spacing of the wavefronts it sends into the air. In front of the source, each new wavefront begins from a position closer to the previous one. The waves are packed into a shorter wavelength.
Behind the source, the wavefronts are spread farther apart. Sound speed through still air stays nearly the same, so a shorter wavelength reaches an observer more often each second.
This is why the change is not caused by the source making a different tone. Its emitted frequency can remain constant while different listeners receive different frequencies.
A moving observer creates a different situation. The source still produces wavefronts with the same spacing in the air. The observer simply meets those wavefronts at a different rate.
Someone moving into the waves encounters more crests per second. Someone moving with the waves encounters fewer. This distinction matters in calculations.
Source motion changes wavelength in the medium. Observer motion changes the rate at which existing wavefronts are encountered. A useful habit is to draw the source, observer, direction of travel, and wavefronts before selecting any signs.
Then predict whether the answer must be above or below the emitted frequency. That prediction catches many sign errors.
The speed of sound depends on the medium, especially temperature. In warmer air, sound generally travels faster than in colder air. Textbook problems often give a fixed sound speed and assume still air.
Real wind complicates the situation because it carries the air itself. The important relative motions are measured compared with the air, not only compared with the ground. For ordinary road speeds, the effect is easy to hear from an ambulance or train horn.
It becomes much larger when an object moves at a sizeable fraction of the wave speed. Aircraft can create a shock wave when they move as fast as sound or faster. The familiar change in pitch is then replaced by a sharp sonic boom as many wavefronts arrive together.
Beats reveal interference over time. Two tones with close frequencies repeatedly drift into alignment, then drift apart. When their pressure changes reinforce, the sound is louder.
When they oppose, it is quieter. The listener hears a repeating swell in loudness rather than two clearly separate notes. Musicians use this effect when tuning instruments.
They adjust one string until the pulses slow down and finally disappear, showing that the two frequencies match closely. Beat calculations require care with units. Frequency is measured in hertz, meaning cycles per second, so the beat result is pulses per second.
Constructive interference does not create energy from nowhere. It redistributes energy as waves overlap, producing regions or moments of greater amplitude along with reduced amplitude elsewhere.