4bDoppler Effect and Resonance
Apply the Doppler formula to moving sources and observers, and explore resonance in open and closed pipes.
The Doppler effect powers radar speed guns, weather forecasting, and medical ultrasound. Resonance explains everything from musical instruments to bridge collapses — understanding both phenomena is critical for physics and engineering.
Lesson Overview
The Doppler effect describes the change in observed frequency when a source or observer is in motion relative to the medium. Resonance occurs when a system is driven at its natural frequency, producing standing waves with large amplitude. In this lesson you will apply the Doppler formula, analyze open and closed pipe harmonics, and connect these phenomena to real-world applications.
Key Concepts
Doppler Effect
f′ = f(v ± v_obs) / (v ∓ v_src); observed frequency shifts with relative motion
Approaching vs Receding
Approaching → higher f′; receding → lower f′
Doppler Applications
Radar speed guns, weather Doppler radar, medical ultrasound
Resonance
Large-amplitude oscillation when driving frequency = natural frequency
Open Pipe Harmonics
fₙ = nv / 2L (n = 1, 2, 3, …); both ends are antinodes
Closed Pipe Harmonics
fₙ = nv / 4L (n = 1, 3, 5, …); only odd harmonics present
A fire truck emitting a 800 Hz siren approaches you at 30 m/s. What frequency do you hear? (v = 343 m/s, observer stationary)
The same fire truck (800 Hz, 30 m/s) is now moving away from you. What frequency do you hear?
An open organ pipe is 0.85 m long. What is its fundamental frequency? (v = 343 m/s)
A closed organ pipe is 0.50 m long. Find the frequencies of the first three harmonics. (v = 343 m/s)
A bat flying at 12 m/s emits 40 000 Hz toward a stationary wall. What frequency does the echo return at? (v = 343 m/s)
A train whistle emits 600 Hz. An observer moves toward the stationary train at 20 m/s. What frequency does the observer hear? (v = 343 m/s)
Hint: Use f′ = f(v + v_obs) / v when the observer moves toward a stationary source.
An open pipe resonates at a fundamental frequency of 440 Hz. What is the length of the pipe? (v = 343 m/s)
Hint: Rearrange f₁ = v / 2L to solve for L.
Why does a closed pipe produce only odd harmonics while an open pipe produces all harmonics?
Hint: Think about the boundary conditions: a closed end must be a node; an open end must be an antinode.
A police radar gun emits microwaves at 24 GHz and detects a return frequency of 24.002 GHz. Is the car approaching or receding? How do you know?
Hint: Compare the observed frequency to the emitted frequency — higher means approaching.
A 1.2 m open pipe and a 1.2 m closed pipe are compared. Which has the lower fundamental frequency, and by what factor?
Hint: f_open = v/2L; f_closed = v/4L. Find the ratio.
Key Vocabulary
Doppler Effect
The change in observed frequency of a wave due to relative motion between the source and the observer.
Example: A siren sounds higher-pitched as an ambulance approaches and lower-pitched as it drives away.
Resonance
The phenomenon in which a system oscillates with maximum amplitude when driven at its natural (resonant) frequency.
Example: A wine glass shatters when a singer hits the exact frequency that matches the glass's natural frequency.
Harmonic
A frequency that is an integer multiple of the fundamental frequency of a vibrating system.
Example: If the fundamental of a guitar string is 110 Hz, the second harmonic is 220 Hz and the third is 330 Hz.
Standing Wave
A wave pattern formed by the superposition of two identical waves traveling in opposite directions, producing fixed nodes and antinodes.
Example: Plucking a guitar string creates standing waves between the two fixed ends (nodes).
Interactive Practice — 5 Questions
A source emitting 500 Hz moves away from a stationary observer at 34.3 m/s (v = 343 m/s). What frequency does the observer hear?
Which harmonics are present in a closed-end pipe?
Resonance occurs when the driving frequency equals:
An open pipe of length L has fundamental frequency f₁. If the length is halved, the new fundamental frequency is:
Medical ultrasound uses the Doppler effect to:
Independent Practice
A car horn emits 400 Hz. The car moves toward a stationary observer at 25 m/s (v = 343 m/s). Calculate the observed frequency.
An open pipe has a fundamental frequency of 256 Hz. List the frequencies of the first four harmonics.
Explain in your own words why the Doppler effect causes a pitch shift. Use the concept of wavefronts in your explanation.
A closed pipe is 0.75 m long (v = 343 m/s). Find the fundamental frequency and the next two allowed harmonics.
★ A weather radar station emits 3 GHz microwaves. A storm cell moving toward the station at 40 m/s reflects the signal. Calculate the Doppler-shifted frequency of the return signal. Explain how meteorologists use this to track storm movement.
ChallengeCommon Mistakes
Using the wrong sign in the Doppler formula — adding v_obs when the observer moves away.
Use f′ = f(v + v_obs)/(v − v_src): + v_obs when observer approaches source; − v_obs when receding. − v_src when source approaches; + v_src when receding.
Thinking closed pipes support all harmonics like open pipes.
Closed pipes (one closed end) support only odd harmonics because the closed end must be a displacement node.
Math Tips
Doppler sign rule: the signs in the formula always make f′ > f when source and observer approach each other, and f′ < f when they move apart. Use this as a quick sanity check after calculating.