驻波与频闪效应教学动画

Overview

A concise three‑scene animation illustrating standing‑wave formation on a fixed string, the stroboscopic (frequency‑shutter) effect that makes the wave appear stationary, and how doubling the driving frequency doubles the number of antinodes. The key take‑away is the relationship between wave frequency, wavelength, and visual perception under stroboscopic illumination.

Code fixes applied:

• All cosine terms now use the angular frequency $\omega = 96\pi$ (48 Hz).
• In Phase 3 the lower‑row wave uses $\omega = 192\pi$ (96 Hz).
• The standing‑wave equation text has been shifted upward (≈ $y=1.8$) to avoid overlapping the string.

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Phases

#	Phase Name	Duration	Description
1	Intro – Standing Wave	~6 s	Dark background fades in, title “驻波与频闪效应” appears briefly, then the fixed string (black blocks at both ends) is drawn horizontally. The standing‑wave equation $y(x,t)=A\cos(96\pi t)\sin(kx)$ appears in yellow at $y=1.8$ (moved up), with parameters $\omega = 96\pi\ \text{rad/s}$, $k = \frac{4\pi}{L}$ (four half‑wavelengths). The string animates with a sinusoidal vertical displacement using np.cos(96*np.pi*t_val); nodes (red dots) at $x=0,\frac{L}{4},\frac{L}{2},\frac{3L}{4},L$ remain stationary. A dashed line across the string marks the wavelength $\lambda = \frac{L}{2}$.
2	Stroboscopic Verification (Option C)	~8 s	Screen splits vertically. Left pane shows the same standing wave in blue, continuously vibrating (still using cos(96πt)). Right pane shows a white strobe flash (brief white flash) occurring every $\frac{1}{96}$ s. Overlaid text (yellow) states $f_{\text{vib}}=48\text{ Hz},\ f_{\text{strobe}}=96\text{ Hz}=2f_{\text{vib}}$. The strobe synchronisation makes the wave appear at only two positions (visual “still” effect).
3	Frequency Doubling (Option D)	~10 s	Screen splits horizontally into two rows. Upper row: original rotation speed $n=2880\ \text{r/min}$ producing 4 antinodes; wave uses cos(96πt) (48 Hz) – warm‑orange color. Lower row: doubled speed $n=5760\ \text{r/min}$ producing 8 antinodes; wave uses cos(192πt) (96 Hz) – cool‑blue color (twice the vibration speed). Both rows animate with appropriate speed (lower row twice as fast). A yellow formula comparison appears: $v=\lambda f$ constant, so $f'\to \lambda' = \lambda/2$ and node count doubles.
4	Outro	~4 s	Fade out all elements, leaving only the title and a brief caption “波速不变，频率加倍导致波长减半，节点数加倍” in yellow.

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Layout

┌───────────────────────────────────────────────────────┐
│                     TOP AREA (Title)                 │
├───────────────────────┬───────────────────────────────┤
│ LEFT AREA (Scene 1)    │ RIGHT AREA (Scene 1)            │
│ – Fixed string,       │ – Equation & parameters        │
│   nodes, wavelength   │   (yellow, y≈1.8)               │
├───────────────────────┼───────────────────────────────┤
│ LEFT AREA (Scene 2)    │ RIGHT AREA (Scene 2)            │
│ – Blue vibrating wave │ – White strobe flashes, text   │
│   (standing wave)      │   (yellow)                      │
├───────────────────────┼───────────────────────────────┤
│ UPPER ROW (Scene 3)    │ LOWER ROW (Scene 3)             │
│ – Warm‑orange 4‑node   │ – Cool‑blue 8‑node               │
│   standing wave (48 Hz)│   standing wave (96 Hz)          │
├───────────────────────┴───────────────────────────────┤
│ BOTTOM AREA (captions, equations, notes)            │
└───────────────────────────────────────────────────────┘

Area Descriptions

Area	Content	Notes
Top	Title “驻波与频闪效应” (large, fades in)	Appears in phase 1, fades out in outro
Left (Scene 1)	Fixed string with black end blocks, red node markers, blue wave animation	Primary visual; wavelength dashed line shown
Right (Scene 1)	Standing‑wave equation $y(x,t)=A\cos(96\pi t)\sin(kx)$ and parameter values	Yellow text, centered at $y≈1.8$ to avoid overlap
Left (Scene 2)	Blue vibrating standing wave (same as Scene 1)	Continuous motion, uses cos(96πt)
Right (Scene 2)	White strobe flash indicator, yellow frequency labels	Flash lasts ~0.02 s, repeats at 96 Hz
Upper Row (Scene 3)	4‑node standing wave, warm‑orange color	Frequency 48 Hz → cos(96πt)
Lower Row (Scene 3)	8‑node standing wave, cool‑blue color	Frequency 96 Hz → cos(192πt)
Bottom	Comparative formula $v=\lambda f$ and explanatory caption	Yellow, small font, appears in phase 3

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Notes

• Background color: deep dark #1a1a2e. All other elements use high contrast (yellow, red, white, orange, blue) as specified.
• Node emphasis: pulsing red circles to highlight that they remain stationary.
• Strobe flash: brief full‑screen white overlay (opacity ≈ 0.6) that appears exactly at the 1/96 s intervals; timing is synchronized with the wave so that only two apparent positions are visible.
• Animation speed: tied to the physical frequency. cos(96πt) yields a true 48 Hz vibration (48 cycles per second). cos(192πt) yields 96 Hz (twice as fast).
• Code fix: get_wave_path now uses np.cos(96*np.pi*t_val) for the base wave and np.cos(192*np.pi*t_val) for the doubled‑frequency wave in Phase 3. This ensures the displayed vibration rates match the intended frequencies.
• Equation placement: the standing‑wave equation text has been moved upward (≈ $y=1.8$) to keep it visually separate from the string, eliminating overlap.
• No textual narration is required; all information is conveyed visually via equations, labels, and color coding.
• The entire animation is contained within a single Manim Scene class.