Updated Animation Specification (≈ 6 seconds)

1. Overview

• Title: Interference of Two Sinusoidal Waves
• Purpose: Visually demonstrate constructive and destructive interference by overlaying two traveling sinusoidal waves of equal amplitude and frequency but with a controllable phase difference.
• Scene: Single Manim InterferenceScene class (the class should be named InterferenceScene).
• Duration: Approximately 6 seconds (well within the 30‑second guideline).

2. Mathematical Elements

• Primary wave equations (in one spatial dimension $x$ and time $t$):
  $$y_1(x, t) = A \sin(kx - \omega t)$$
  $$y_2(x, t) = A \sin(kx - \omega t + \phi)$$
• Resultant wave (superposition):
  y_{\text{total}}(x, t) = 2A \cos\left(\frac{\phi}{2}\r\right) \sin\left(kx - \omega t + \frac{\phi}{2}\r\right)
• Parameters (default values, adjustable during the animation):
  • Amplitude $A = 1$
  • Wavenumber $k = 2\pi / \lambda$ with wavelength $\lambda = 4$ units
  • Angular frequency $\omega = 2\pi f$ with frequency $f = 0.5$ Hz
  • Phase offset $\phi$ will be animated from $0$ (perfect constructive) to $\pi$ (perfect destructive) and back.

3. Visual Elements

Element	Description	Color / Style
Wave 1	Thin sinusoidal curve representing $y_1$.	Light blue, opacity 0.8
Wave 2	Thin sinusoidal curve representing $y_2$.	Orange, opacity 0.8
Resultant Wave	Thicker curve showing $y_{\text{total}}$.	White, line width 4, glow effect
Constructive Zones	Highlighted regions where $\cos(\phi/2) > 0$.	Semi‑transparent green overlay (opacity 0.2)
Destructive Zones	Highlighted regions where $\cos(\phi/2) < 0$.	Semi‑transparent red overlay (opacity 0.2)
Axes	Horizontal axis $x$ from $-8$ to $+8$; vertical axis $y$ from $-2.5$ to $+2.5$.	Gray, thin lines
Phase Indicator	Small rotating arrow at a fixed $x$ location showing the instantaneous phase difference between the two waves.	Yellow arrow, length proportional to $\phi$

4. Animation Sequence & Timing (compressed to ~6 s)

Time (s)	Action
0.0‑0.5	Fade‑in axes and static background.
0.5‑1.0	Draw Wave 1 and Wave 2 simultaneously (Create).
1.0‑1.5	Introduce Resultant Wave (Write).
1.5‑3.0	Animate the phase offset $\phi$ smoothly from $0$ to $\pi$. Component waves shift, resultant amplitude modulates, overlay transitions from green to red, and the phase‑indicator arrow rotates from 0° to 180°.
3.0‑3.2	Hold at $\phi = \pi$ to emphasize the destructive pattern.
3.2‑4.5	Animate $\phi$ back from $\pi$ to $0$, reversing the interference pattern and restoring constructive zones.
4.5‑5.0	Brief pause with all three waves static at $\phi = 0$.
5.0‑5.5	Fade‑out all objects simultaneously, ending the scene.
5.5‑6.0	Optional short buffer (black screen) to reach the target duration.

5. Camera & Perspective

• Camera Position: Orthographic, centered on the origin, showing the full $x$-range $[-8, 8]$.
• Zoom: Fixed; no zooming or panning needed.
• Perspective: Straight‑on 2‑D view; no rotation.

6. Transitions & Easing

• Phase‑offset animation uses smooth (cubic‑ease‑in‑out) interpolation.
• Fade‑ins and fade‑outs use a linear opacity transition over 0.5 s (adjusted to fit the compressed timeline).
• Overlay color changes follow the sign of $\cos(\phi/2)$ and transition continuously with the phase animation.

7. Additional Details

• No Text Labels: All information is conveyed visually.
• Background: Dark navy to make the white resultant wave and colored overlays stand out.
• Lighting: Minimal; a subtle ambient glow on the resultant wave helps it pop.
• Duration Compliance: Total runtime ≈ 6 seconds, satisfying the ≤ 30 second constraint.