17-7 The Doppler Effect#
Prompts
What is the Doppler effect? Does motion toward the other object increase or decrease the detected frequency?
Write the general Doppler formula \(f' = f \frac{v \pm v_D}{v \mp v_S}\). What do \(v\), \(v_D\), and \(v_S\) mean? How do you choose the \(\pm\) signs?
A car horn (source) moves toward you (stationary detector). Does the wavelength in front of the car increase or decrease? What happens to \(f'\)?
You (detector) run toward a stationary siren. Does \(f'\) increase or decrease? Write the formula for this case.
Speeds in the Doppler formula are measured relative to what? Why does the medium matter for sound?
Lecture Notes#
Overview#
The Doppler effect is the change in the detected frequency \(f'\) when the source or detector moves relative to the medium (e.g., air).
Motion toward the other object → \(f' > f\) (higher pitch).
Motion away → \(f' < f\) (lower pitch).
Speeds are measured relative to the medium, not the ground.
General Doppler formula#
\(v\): speed of sound in the medium.
\(v_D\): speed of the detector relative to the medium.
\(v_S\): speed of the source relative to the medium.
Sign rule: Choose signs so that motion toward increases \(f'\) and motion away decreases \(f'\).
Who moves |
Direction |
Effect on \(f'\) |
Sign |
|---|---|---|---|
Detector |
Toward source |
\(f' > f\) |
\(+v_D\) in numerator |
Detector |
Away from source |
\(f' < f\) |
\(-v_D\) in numerator |
Source |
Toward detector |
\(f' > f\) |
\(-v_S\) in denominator |
Source |
Away from detector |
\(f' < f\) |
\(+v_S\) in denominator |
Detector moving, source stationary#
Toward source: \(f' = f(v + v_D)/v\) → \(f' > f\).
Away from source: \(f' = f(v - v_D)/v\) → \(f' < f\).
Physical picture: A detector moving toward the source intercepts wavefronts more often → higher frequency.
Source moving, detector stationary#
Toward detector: \(f' = f\,v/(v - v_S)\) → \(f' > f\) (wavefronts compressed in front).
Away from detector: \(f' = f\,v/(v + v_S)\) → \(f' < f\) (wavefronts stretched behind).
Physical picture: A source moving toward the detector bunches wavefronts in front → shorter \(\lambda'\) → higher \(f'\).
Symmetry?
Detector moving and source moving give different formulas for the same relative speed. The asymmetry arises because the medium is the reference frame—the source alters the wavelength in the medium; the detector does not.
Both source and detector moving#
Use the general formula and apply the sign rule to each:
Detector toward source → \(+v_D\).
Detector away → \(-v_D\).
Source toward detector → \(-v_S\).
Source away → \(+v_S\).
Poll: Doppler shift
A train whistle (source) moves away from you (stationary detector). The detected frequency \(f'\) is:
(A) Greater than the emitted frequency \(f\)
(B) Less than \(f\)
(C) Equal to \(f\)
Poll: Doppler with both moving (sign convention)
Speaker moves 200 m/s toward detector; detector moves 100 m/s away from speaker. Two students use different sign choices. Method 1: \(f' = f\frac{v - v_D}{v - v_S}\). Method 2: \(f' = f\frac{v + v_D}{v + v_S}\). Which is correct? (\(v = 343\) m/s)
(A) Both methods are right
(B) Both are wrong
(C) Only method 1 is right
(D) Only method 2 is right
Example: Reflection from wall#
Example: Doppler with reflection from wall
Speaker and detector move together at 100 m/s toward a wall. Emitted frequency 2 kHz. What frequency does the detector hear from the reflected wave? (\(v = 343\) m/s)
Steps: (1) Frequency at wall (wall is “detector” of the speaker). (2) Wall re-emits at that frequency. (3) Frequency at detector (detector receives from the wall as “source”).
Solution: (1) \(f_{\text{wall}} = f \frac{v}{v - v_S} = 2000 \frac{343}{243} \approx 2823\) Hz. (2) Wall emits 2823 Hz. (3) \(f' = f_{\text{wall}} \frac{v + v_D}{v} = 2823 \frac{443}{343} \approx 3645\) Hz.
\(f' = f\,\frac{v \pm v_D}{v \mp v_S}\)—general Doppler formula for sound.
Toward → higher \(f'\); away → lower \(f'\).
Speeds relative to medium (air).
Assumes motion along the line joining source and detector; \(v_S < v\) (see section 17-8 for supersonic).