波的传播与干涉分析

Overview

This animation visualizes two synchronized wave sources, the propagation of their waves via 60 discrete sample particles distributed along the 0–20 m axis, the arrival at observation particles A and B, the calculated distance between them, and the interference pattern that emerges from superposition. The key takeaway is how wave speed, travel time, and particle positions determine the interference outcome: particles at odd-half-wavelength path-difference positions undergo destructive interference (zero amplitude), while particles at even-half-wavelength path-difference positions undergo constructive interference (doubled amplitude). The ABCD option verification section is removed entirely.

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Phases

#	Phase Name	Duration	Description
1	Intro	~1 s	Show the static layout: two black source points $S_1$ and $S_2$ at $x=0$ m and $x=20$ m, yellow particle $A$ at $x=7$ m and green particle $B$ at $x=12$ m between them. Display 60 small grey dots evenly spaced along the axis (spacing ≈ 0.33 m). Display the vibration equation $y = 0.01\sin(\pi t)$ m near the sources.
2	Wave Emission & Propagation	~3 s	At $t=0$ both sources begin vertical oscillation. Each of the 60 sample particles begins vibrating only when the wavefront from its nearest source reaches it. The red rightward wave from $S_1$ activates particle $i$ (at position $x_i$) at time $t = x_i / v$; the blue leftward wave from $S_2$ activates particle $j$ at time $t = (20 - x_j)/v$. Before activation a particle remains stationary at zero. After activation by a single wave the particle oscillates according to the relevant wave equation. The wavefront "sweep" is thus visualized as a cascade of particles waking up one by one from left to right (red) and right to left (blue), each small dot bobbing vertically in real time.
3	Arrival at A	~2 s	Red wavefront reaches $A$ at $t = x_A/v = 7/2 = 3.5$ s. Highlight $A$'s dot, then $A$ begins vertical vibration as $y_A = 0.01\sin(\pi(t-3.5))$ m. Show the formula $x_A = v\,t_1 = 2\times3.5 = 7$ m in yellow beside $A$ for 3 s.
4	Arrival at B	~2 s	Blue wavefront reaches $B$ at $t = (20-x_B)/v = 8/2 = 4$ s. Highlight $B$'s dot, then $B$ begins vertical vibration as $y_B = 0.01\sin(\pi(t-4))$ m. Show the formula $x_B = 20 - v\,t_2 = 20 - 2\times4 = 12$ m in green beside $B$ for 3 s.
5	Superposition & Interference Steady State	~4 s	Once both wavefronts have swept the entire axis (physical time $t=10$ s, i.e. 5 s after $t=0$ in animation time), every sample particle superposes both wave displacements strictly according to the superposition principle: $y_i(t) = 0.01\sin(\pi t - \pi x_i/2) + 0.01\sin(\pi t - \pi(20-x_i)/2)$. The path difference for particle at $x_i$ is $\Delta_i = |x_i - (20 - x_i)| = |2x_i - 20|$ m. With $\lambda = 4$ m: Destructive interference (amplitude = 0) at $x = 1, 3, 5, 7, 9, 11, 13, 15, 17, 19$ m — path difference is an odd multiple of $\lambda/2 = 2$ m. These particles remain completely stationary and their dots are rendered in grey. Constructive interference (amplitude = 0.02 m) at $x = 0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20$ m — path difference is an even multiple of $\lambda/2$ (i.e. an integer multiple of $\lambda$). These particles oscillate with doubled amplitude and their dots are rendered in bright white/purple. The 60-particle array displays the full standing-wave interference pattern as a living, oscillating ensemble. Annotate representative destructive nodes with "振动减弱 (A=0)" in grey and constructive antinodes with "振动加强 (A=2A₀)" in bright green.
6	Distance Between A & B	~1.5 s	Draw a yellow line segment connecting $A$ ($x=7$ m) and $B$ ($x=12$ m) along the axis. Label its length "5 m". Display the calculation $|x_B - x_A| = |12 - 7| = 5$ m in red next to the segment for 3 s. Note that $A$ at $x=7$ m is a destructive node (path difference = 6 m = 1.5λ) and $B$ at $x=12$ m is a constructive antinode (path difference = 4 m = λ).
7	Vibration Plots & Outro	~2 s	On the right panel, display two small time-series plots spanning $t = 0$ to $t = 12$ s: Red sinusoid for $A$ ($x=7$ m, destructive node): oscillates with single-wave amplitude 0.01 m from $t=3.5$ s until the second wavefront arrives at $t=(20-7)/2=6.5$ s, then amplitude collapses to zero for $t \geq 6.5$ s — the plot flatlines at zero. Green sinusoid for $B$ ($x=12$ m, constructive antinode): oscillates with single-wave amplitude 0.01 m from $t=4$ s until the second wavefront arrives at $t=7/2=3.5$ s (already passed), so from $t=6$ s onward amplitude grows to 0.02 m — the plot shows doubled amplitude. Both plots are clearly labeled with their interference type. Fade out all elements.

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Layout

┌─────────────────────────────────────────────┐
│                 TOP AREA                    │
│   (Title: "波的传播与干涉分析")               │
├───────────────────────┬─────────────────────┤
│        LEFT AREA       │      RIGHT AREA      │
│  (Main wave scene,    │  (Interference      │
│   sources S1 & S2,   │   summary labels,   │
│   60 sample particle │   distance label,   │
│   dots oscillating   │   vibration plots   │
│   vertically along   │   for A and B       │
│   the axis, yellow   │   showing zero      │
│   particle A, green  │   amplitude at A    │
│   particle B,        │   and doubled       │
│   distance line,     │   amplitude at B    │
│   interference       │   after full        │
│   annotations)       │   superposition)    │
└───────────────────────┴─────────────────────┘

Area Descriptions

Area	Content	Notes
Top	Title "波的传播与干涉分析"	Fade‑in at start of Intro
Left	Black sources $S_1, S_2$; 60 small sample particle dots evenly spaced from $x=0$ to $x=20$ m, each oscillating vertically according to its activation time and the superposition of both wave equations once both fronts have passed; yellow particle $A$ at $x=7$ m (destructive node, eventually stationary); green particle $B$ at $x=12$ m (constructive antinode, eventually doubled amplitude); yellow distance line; interference annotations ("振动加强 (A=2A₀)" in bright green at constructive positions, "振动减弱 (A=0)" in grey at destructive positions)	Primary visual focus throughout phases. Each dot's vertical displacement is computed every frame using the exact superposition formula. Destructive-node dots are rendered grey and stationary after full superposition; constructive-antinode dots are rendered bright white/purple with visibly larger oscillation. No continuous sinusoidal curve is drawn — the ensemble of dots is the wave visualization. Dot color transitions: grey (inactive) → red tint (only $S_1$ wave) or blue tint (only $S_2$ wave) → grey/stationary (destructive) or bright white/purple with large excursion (constructive) after both waves arrive.
Right	Interference summary: list of destructive positions ($x = 1,3,5,...,19$ m, "路程差为半波长奇数倍，振幅为0") and constructive positions ($x = 0,2,4,...,20$ m, "路程差为半波长偶数倍，振幅为2A₀"); vibration time-series plots for $A$ (red, flatlines to zero after $t=6.5$ s) and $B$ (green, amplitude doubles after full superposition)	Appears from Phase 5 onward; vibration plots appear in Phase 7

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Notes

• Vibration equation for both sources at their respective origins: $y = 0.01\sin(\pi t)$ m, giving amplitude $A_0 = 0.01$ m, angular frequency $\omega = \pi$ rad/s, and period $T = 2$ s.
• Wave speed: $v = 2$ m/s; wavelength: $\lambda = vT = 4$ m; half-wavelength: $\lambda/2 = 2$ m.
• 60 sample particles are placed at positions $x_i = i \times (20/59)$ m for $i = 0, 1, \ldots, 59$ (spacing ≈ 0.339 m).
• Displacement rules for each sample particle at position $x_i$ (strictly enforcing superposition principle):
  • $t < x_i/v$ AND $t < (20-x_i)/v$: $y_i = 0$ (neither front arrived).
  • $x_i/v \leq t$ AND $t < (20-x_i)/v$: $y_i = 0.01\sin(\pi t - \pi x_i/2)$ (only $S_1$ wave).
  • $(20-x_i)/v \leq t$ AND $t < x_i/v$: $y_i = 0.01\sin(\pi t - \pi(20-x_i)/2)$ (only $S_2$ wave).
  • Both fronts arrived ($t \geq x_i/v$ AND $t \geq (20-x_i)/v$): $y_i = 0.01\sin(\pi t - \pi x_i/2) + 0.01\sin(\pi t - \pi(20-x_i)/2)$.
• Interference analysis (steady state, both waves present) — rigorous derivation:
  • Path difference: $\Delta_i = |x_i - (20 - x_i)| = |2x_i - 20|$ m.
  • Destructive interference (amplitude = 0): $\Delta_i = (2n-1)\cdot\lambda/2$ for integer $n$, i.e. $\Delta_i = 2, 6, 10, 14, 18$ m → $x_i = 1, 3, 5, 7, 9, 11, 13, 15, 17, 19$ m. These particles are completely stationary in steady state.
  • Constructive interference (amplitude = $2A_0 = 0.02$ m): $\Delta_i = 2n\cdot\lambda/2 = n\lambda$ for integer $n\geq0$, i.e. $\Delta_i = 0, 4, 8, 12, 16, 20$ m → $x_i = 0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20$ m.
  • Particle A at $x=7$ m: $\Delta = |14-20|=6$ m $= 1.5\lambda$ → destructive, amplitude = 0. After $t=6.5$ s (when $S_2$ wave arrives), $A$ becomes stationary.
  • Particle B at $x=12$ m: $\Delta = |24-20|=4$ m $= \lambda$ → constructive, amplitude = 0.02 m. After $t=6$ s (when $S_1$ wave arrives at $t=6$ s and $S_2$ wave arrived at $t=4$ s), $B$ oscillates with doubled amplitude.
• Vibration plots must faithfully reflect the physics: the plot for $A$ must show the amplitude collapsing to zero after both waves are present; the plot for $B$ must show the amplitude doubling. These plots are the primary pedagogical verification of the superposition principle.
• The ABCD multiple-choice option verification section is completely removed from the animation.
• Particle dots $A$ and $B$ are rendered larger and more prominently than the 60 sample dots.
• The "cascade wake-up" effect in Phase 2 is the primary visual representation of wave propagation — no vertical line wavefront indicator is used.
• All motions use smooth sinusoidal interpolations; no abrupt jumps.
• The bottom time axis is removed entirely from the layout. Time information is conveyed through on‑screen formula annotations and the vibration plots in the Right area only.
• Colors: source points black, $A$ yellow, $B$ green, sample particles grey (inactive) → red tint ($S_1$ only) or blue tint ($S_2$ only) → grey/stationary (destructive node) or bright white/purple (constructive antinode), distance line yellow.
• Text annotations appear only when needed; they fade out after their specified display time.
• The entire scene fits within a single Manim Scene class.
• Total runtime ≈ 16 s, well under the 30 s limit.
• Pedagogical priority: the animation must be physically rigorous and suitable for classroom teaching. Every visual element must correctly represent the underlying wave physics, especially the interference outcomes at each particle position.