# Acoustic Cavity Modes Visualizer

Interactive 3D normal-mode analysis of a rectangular room — see the pressure nodes and antinodes of each acoustic eigenmode, and how room shape and listener position shape the frequency response.

> Canonical page: https://elysiatools.com/en/visualizations/acoustic-cavity-modes

- **Category:** Physics

## Overview

Interactive 3D normal-mode analysis of a rectangular room — distinct from the existing helmholtz-resonator (a single-cavity Helmholtz resonator, lumped-element), standing-wave (1D string/tube), and sound-attenuation/resonance-phenomenon cases; this is the only case treating the full 3D eigenmode structure of an enclosure, the foundation of architectural and studio acoustics. The model implements the Morse–Ingard rigid-wall eigenfunctions: pressure p_{n,m,l}(x,y,z) = cos(nπx/Lx)·cos(mπy/Ly)·cos(lπz/Lz) with eigenfrequencies f = (c/2)·√((n/Lx)²+(m/Ly)²+(l/Lz)²), classified by how many indices are nonzero into axial (one — bouncing between one pair of walls, strongest), tangential (two — skimming four walls), and oblique (three — involving all six walls, weakest but densest at high frequency) modes. It detects modal degeneracy (different index triples sharing one frequency, the hallmark of bad room shapes like cubes), computes the Schroeder frequency f_s = 2000·√(RT60/V) marking the modal-to-diffuse transition, and evaluates the exact and Weyl-asymptotic modal density N(f) ≈ (4π/3)V(f/c)³ + (π/4)S(f/c)² + (1/8)Le(f/c). The steady-state response at a listener is the modal sum of Lorentzians Σ_k ψ_k(r₀)ψ_k(r_s)/√[(f²−f_k²)²+(f·Δf)²] with half-width Δf ≈ 2.2/RT60. Three visualization panels: (1) A Three.js 3D room rendered as a wireframe box with a 14×12×8 voxel grid sampling the pressure eigenfunction — red cubes for positive-pressure antinodes, blue for negative, transparent near nodal surfaces (p≈0) — with an animate toggle oscillating the field in time cos(ωt), hand-rolled spherical-camera orbit controls (drag to rotate, scroll to zoom, pinch on touch), and draggable listener (green) and source (gold) markers raycast onto the listener's z-plane. (2) Frequency-response plot of |P(f)| at the listener position with the Schroeder frequency (gold dashed) and current-mode frequency (red dashed) marked, reshaping live as the listener moves between nodes (peaks vanish) and antinodes (peaks swell). (3) Mode-list bar chart color-coded by type (axial red, tangential gold, oblique blue), with degenerate modes visibly clustered, and the current mode highlighted with its (n,m,l) label. Adjustable parameters: room dimensions Lx/Ly/Lz (1–20 m each), mode indices n/m/l (0–6), reverberation time RT60 (0.1–3 s, driving mode Q), and a pressure-animation toggle. Five room presets: Studio (golden-ish 7:5:3), Cube (bad — triple degeneracy), Golden ratio (1.618:1:0.618), Shoebox (15:8:6), and Concert hall (20:15:10, low Schroeder). Real-time diagnostics: mode frequency, type (axial/tangential/oblique), wavelength, room volume, Schroeder frequency, quality factor Q, count of modes below the current one, and modal-vs-diffuse region classification. Educational content covers the wave equation with rigid-wall boundary conditions, the cosine-product eigenfunctions and their nodal/antinodal surfaces, the axial/tangential/oblique classification, the Schroeder frequency as the modal-to-diffuse transition, modal degeneracy and room-proportion design (Bolt, Sepmeyer, golden ratio, Bonello criterion), and applications to recording studios (bass traps, non-parallel walls), concert halls (shoebox acoustics), home audio (subwoofer/listening-position trade-offs), and architectural finite-element simulation. Multi-language support (zh, en, es, fr, de, ru, pt).

## Related content

- [Zero-Order Reaction - Interactive Visualization](https://elysiatools.com/en/visualizations/zero-order-reaction): Interactive visualization of zero-order reaction kinetics and concentration changes over time
- [First-Order Reaction - Interactive Visualization](https://elysiatools.com/en/visualizations/first-order-reaction): Interactive visualization of first-order reaction kinetics and exponential concentration decay
- [Second-Order Reaction - Interactive Visualization](https://elysiatools.com/en/visualizations/second-order-reaction): Interactive visualization of second-order reaction kinetics and bimolecular collision dynamics
- [Arrhenius Equation](https://elysiatools.com/en/visualizations/arrhenius-equation): Interactive visualization of temperature effect on reaction rate - Explore activation energy, pre-exponential factor, and rate constant relationship
- [Reversible Reaction](https://elysiatools.com/en/visualizations/reversible-reaction): Interactive visualization of A ⇌ B reversible reaction kinetics - Explore forward and reverse reaction rates, equilibrium constant, and concentration changes over time
- [Consecutive Reaction](https://elysiatools.com/en/visualizations/consecutive-reaction): Interactive visualization of A → B → C consecutive reaction kinetics - Explore intermediate concentration peaks, rate-determining steps, and complete evolution of all species
- [Chain Reaction](https://elysiatools.com/en/visualizations/chain-reaction): Interactive visualization of free radical chain reaction polymerization - Explore initiation, propagation, termination steps, chain growth animation, and molecular weight distribution
- [Le Chatelier's Principle](https://elysiatools.com/en/visualizations/le-chateliers-principle): Interactive visualization of Le Chatelier's Principle - Explore how changes in concentration, pressure, and temperature affect chemical equilibrium with animated molecular dynamics