Acoustic Doppler echoes and standing-wave boundaries
| English | Français |
|---|---|
| Doppler factor | Doppler factor |
| displacement antinode | displacement antinode |
A decision before an answer
- A driver hears an echo from a stationary wall at a shifted pitch: the outbound and returning sound each acquire a Doppler factor.
- Your goal: Apply source and observer Doppler factors in the medium frame.
Read the relationship
- For sound in a stationary medium, separate source motion from observer motion. A source approaching at speed u_s compresses wavefront spacing and produces received frequency f c/(c−u_s) at a stationary observer. An observer approaching a stationary source at speed u_o meets more wavefronts per second and measures f(c+u_o)/c. Speeds are measured relative to the medium; the acoustic source and observer formulas are not symmetric under exchanging roles. Receding motion reverses the appropriate sign. These expressions assume subsonic motion along the propagation line.
- Derive allowed pipe frequencies from displacement boundary conditions.
A moving siren approaches a stationary wall at speed u in still air with speed c. The driver’s echo frequency divided by emitted frequency is:
The outbound moving-source factor and return moving-observer factor multiply.
Use the defining rule
- For a siren moving toward a stationary reflecting wall with speed u, the wall first receives f_wall=f c/(c−u). Reflection from a stationary wall preserves frequency in the medium frame. The moving driver then approaches the returning wavefronts and receives f_echo=f_wall(c+u)/c=f(c+u)/(c−u). At small u/c the fractional shift is approximately 2u/c, but the exact expression has different numerator and denominator. A moving reflecting surface needs its own Doppler step; do not reuse the stationary-wall result blindly.
- Distinguish frequency, wavelength and boundary changes.
An ideal both-open pipe has fundamental 120 Hz. After closing one end, its first three allowed frequencies are:
The new fundamental is 60 Hz and allowed multiples are odd.
Check the conditions
- At an ideal open pipe end, air displacement is an antinode and pressure variation is a node. At a rigid closed end, displacement is a node and pressure is an antinode. For both ends open, length L contains n half-wavelengths, giving f_n=nc/(2L), n=1,2,… . For one end closed, it contains an odd number of quarter-wavelengths, giving f_n=(2n−1)c/(4L). Real pipes may need end corrections; the ideal GRE model uses the stated length without inventing an adjustment.
- Distinguish frequency, wavelength and boundary changes.
A 500 Hz siren approaches a fixed wall at 10 m/s in air with c=340 m/s. The echo heard by the driver is 500·350/330=530.3 Hz. For a 0.85 m ideal pipe in the same air, the both-open fundamental is 200 Hz, while the one-closed spectrum begins 100, 300, 500 Hz. None of the original 200, 400, 600 Hz lines is retained.
For c=330 m/s and u=30 m/s toward a stationary wall, f_echo/f=____ (decimal).
(330+30)/(330−30)=360/300=1.2.
Apply the task format
- Closing one end of a previously both-open pipe halves its fundamental and leaves only odd multiples of that new fundamental. Its old frequencies were integer multiples of c/(2L), which are even multiples of c/(4L), so none is an allowed frequency of the new ideal one-closed spectrum. This differs from simply deleting even harmonics while retaining the old fundamental. When the medium is unchanged, wave speed is unchanged; frequency and wavelength change together to satisfy the new boundary geometry.
- Distinguish frequency, wavelength and boundary changes.
An echo needs both Doppler factors. Closing a pipe changes the fundamental; odd harmonics are counted from the new fundamental, not the old one.
Which answer fits this case?
Apply source and observer Doppler factors in the medium frame
Closing one end of an ideal both-open pipe retains its original fundamental as the new second harmonic.
The one-closed spectrum has no even multiples of its new fundamental; that original line is absent.
Keep the distinctions
- displacement antinode 位移波腹 — A standing-wave position where the air displacement amplitude is maximal.
- Doppler factor 多普勒因子 — Frequency multiplier produced by a specified source or observer motion.
- Apply source and observer Doppler factors in the medium frame.
- Derive allowed pipe frequencies from displacement boundary conditions.
- Distinguish frequency, wavelength and boundary changes.
Match each term with its precise meaning in this lesson.
Keep the distinctions stated in the teaching example.
Put this lesson’s reasoning or event sequence in order.
The order follows the stated process; check each stage before the next.