The Doppler Effect and Shock Waves
For sound in still air, source motion changes wave spacing and observer motion changes the rate of meeting waves. Their formulas agree approximately at low speeds. Supersonic motion produces shock waves; an ideal Mach cone relates its half angle to source speed.
What a learner can do afterwards
- Applies the correct Doppler expression for a moving source and for a moving observer
- Explains why the two cases differ and when the difference matters
- Relates the angle of a shock cone to the speed of the source
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Recall v = fλ. A steady siren keeps its emitted frequency while moving. Its motion places sound compressions closer together ahead and farther apart behind. A stationary listener hears higher pitch ahead and lower pitch behind. A moving listener changes the rate of meeting compressions without changing their spacing in the air. The formulas below use still air and motion along the line joining source and listener.
Let f_s be emitted frequency, f_o heard frequency and v sound speed. For a source moving at air-relative speed u_s < v and a stationary listener, approach gives f_o = f_s × v ÷ (v - u_s). Recession gives f_o = f_s × v ÷ (v + u_s). The approach denominator is smaller, giving a higher frequency. These are sound formulas, not formulas for light in a vacuum.
For a stationary source, let u_o be the listener's air-relative speed, with u_o < v. Approach gives f_o = f_s × (v + u_o) ÷ v. Recession gives f_o = f_s × (v - u_o) ÷ v. The wavelength in the air stays v ÷ f_s. The listener meets compressions at relative speed v + u_o or v - u_o, so listener motion changes the numerator.
Compare equal approach speeds u and put r = u ÷ v. Source motion gives f_o = f_s ÷ (1 - r); listener motion gives f_o = f_s × (1 + r). For r much smaller than 1, the source result is also approximately f_s × (1 + r). At larger r, the results differ more. With f_s = 300 Hz, v = 320 m/s and u = 80 m/s, source motion gives 400 Hz and listener motion gives 375 Hz.
A supersonic aircraft moves faster than sound and produces shock waves, sharp changes in air pressure. A listener hears a sonic boom when a shock reaches them. Shocks can continue during supersonic flight; the boom is not limited to the instant the aircraft first exceeds sound speed. Do not extend the subsonic Doppler approach formula to u_s at or above v.
For an ideal Mach cone in a uniform medium, θ is the half angle between the cone surface and its backward axis along the source's path. For an acute angle in a right triangle, sin θ = opposite side ÷ hypotenuse. Then sin θ = v ÷ u_s = 1 ÷ M, where Mach number M = u_s ÷ v. Rearranging gives u_s = v ÷ sin θ. A faster source gives a narrower cone. This weak-disturbance geometry does not give the angle of every finite-strength shock.
Source motion changes sound-wave spacing; listener motion changes encounter rate. Use the matching formula. For an ideal Mach cone, sin θ = v ÷ u_s.
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More to read
Identify what moves
Before choosing a sound Doppler formula, identify the moving object and the medium. Use v for sound speed in the air, f_s for emitted frequency and f_o for heard frequency. Here the air is still, motion is along the line between source and listener, and the moving object's speed is below v.
A moving source changes the positions from which successive compressions begin. For source speed u_s, a stationary listener hears f_o = f_s × v ÷ (v - u_s) during approach and f_o = f_s × v ÷ (v + u_s) during recession.
A moving listener crosses an existing pattern of compressions. With a stationary source and listener speed u_o, approach gives f_o = f_s × (v + u_o) ÷ v. Recession gives f_o = f_s × (v - u_o) ÷ v.
The source case changes the denominator, while the listener case changes the numerator. Equal approach speeds need not produce equal heard frequencies. When speed is a small fraction of v, the formulas give approximately the same shift. Do not use these sound formulas for light in a vacuum.
The shock travels to the listener
A supersonic aircraft moves faster than sound and produces shock waves. Across a shock, air pressure changes sharply. A sonic boom is the sound heard when a shock reaches a listener. Shock production can continue throughout supersonic flight, so the boom is not confined to the moment the aircraft first exceeds sound speed.
An ideal Mach cone describes weak disturbances from steady supersonic motion through a uniform medium. Its half angle θ is measured between the cone surface and the backward axis along the source's path. Let source speed be u_s and sound speed be v. The relation is sin θ = v ÷ u_s.
Mach number is M = u_s ÷ v, so the same relation is sin θ = 1 ÷ M. Increasing M narrows the ideal cone. If an angle measurement gives sin θ = 0.2, the model gives M = 5.
This angle relation describes the ideal Mach cone. Real finite-strength shocks can have different angles. Likewise, a zero denominator in the subsonic Doppler approach formula does not establish an infinite physical pitch; it marks a limit of that model.
Frequency shift is not yet a speed
Sound speed and source speed are different quantities. In air at about 20 degrees Celsius, sound speed is approximately 343 m/s. It varies with conditions, including temperature. Use the sound speed specified in a problem rather than applying 343 m/s everywhere.
To infer motion from a frequency measurement, first choose a model. For a source approaching a stationary listener in still air, f_o = f_s × v ÷ (v - u_s). Rearranging gives u_s = v × (1 - f_s ÷ f_o). The frequency ratio is dimensionless; multiplying by v gives a speed.
Dividing a frequency shift in hertz by a wave speed does not give a velocity. Reflected-wave instruments also need the reflection geometry and direction of motion. Doppler ultrasound and radar cannot be treated as the same one-way sound measurement. Radar uses electromagnetic waves, so a sound-in-air formula is not its general rule.
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