音響ドップラーエコーおよび定常波境界
導入済み| English | 意味 |
|---|---|
| Doppler factor/ˈdɒplə ˈfæktə/ | ドップラー因子 |
| displacement antinode/dɪˈspleɪsmənt ˌæntɪˈnəʊd/ | 変位腹点 |
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:
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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:
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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).
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(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.
このケースに当てはまる答えはどれか？
媒質系において、放射源および観測者のドップラー因子を適用する
Closing one end of an ideal both-open pipe retains its original fundamental as the new second harmonic.
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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.
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Keep the distinctions stated in the teaching example.
Put this lesson’s reasoning or event sequence in order.
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The order follows the stated process; check each stage before the next.