The World Cup Roar: Acoustic-to-Light Physics Engine

Context

You are an expert Physics Engine Developer specializing in wave dynamics and energy conversion. Your task is to design a theoretical and mathematical framework for a system that captures the acoustic energy of a World Cup stadium (the 'Roar') and converts it into a real-time visual light display.

Input Parameters

1. Acoustic Input (S): Define the sampling rate (kHz) and decibel threshold (dB) for crowd noise.
2. Frequency Mapping (F): Categorize sound into 'Base Roar' (low freq), 'Whistles/Vuvuzelas' (high freq), and 'Chants' (mid-range).
3. Stadium Geometry (G): Provide calculations for sound wave reflection and focal points within a parabolic or bowl-shaped arena.

Execution Instructions

• Step 1: The Acoustic-to-Photon Formula: Create a custom mathematical function where $L(t)$ represents light intensity derived from $A(t)$ (amplitude) and $f(t)$ (frequency).
• Step 2: Chromatic Mapping: Assign specific light wavelengths (colors) to frequency bands. For example, assign low-frequency rumbles to deep reds and high-pitched cheers to vibrant cyans or whites.
• Step 3: Particle Physics Simulation: Describe how 'Acoustic Particles' interact with a virtual 'Light Mesh' draped over the stadium. Use fluid dynamics principles to show how sound waves 'push' light ripples through the mesh.
• Step 4: Energy Decay & Reverb: Implement a damping factor to ensure the light visual fades realistically as the sound dissipates.

Output Format

Provide the response in three parts:

1. The Mathematical Core: The primary equations governing the simulation.
2. Pseudo-code Logic: A logic block compatible with C++ or Python for real-time processing.
3. Visual Description: A prose description of what a 'Goal!' moment looks like in this physics engine.