What are room modes in acoustics?

Room modes (also called standing waves or eigenmodes) are resonant frequencies at which a room strongly amplifies sound. They occur when the room dimensions are integer multiples of half-wavelengths: f_n = n�c/(2L) for axial modes in a rectangular room of length L. In a rectangular room there are axial (one axis), tangential (two axes), and oblique (all three axes) modes. At low frequencies, rooms typically have sparsely spaced modes leading to severe coloration of sound � bass notes at certain pitches are dramatically boosted while adjacent pitches are attenuated.

How does the Doppler effect change sound frequency?

The Doppler effect occurs when there is relative motion between a sound source and observer. If the source moves toward the observer at speed v_s in a medium where sound travels at c, the observed frequency is f_obs = f_source � c/(c - v_s). As the source moves away: f_obs = f_source � c/(c + v_s). In this simulation, the moving source travels at ~0.4c (Mach 0.4), so approaching frequency is boosted by factor 1/(1-0.4) = 1.67� and receding frequency is reduced to 1/(1+0.4) = 0.71� � a clearly audible pitch shift straddling the original frequency.

What is the FDTD method used in this simulation?

FDTD (Finite-Difference Time-Domain) solves the acoustic wave equation ?�p/?t� = c�?�p by replacing derivatives with finite differences on a regular grid. Space is discretised into cells of width ?x and time into steps ?t. The stability condition (Courant-Friedrichs-Lewy criterion) requires the Courant number r = c�?t/?x = 1/v2 � 0.707 for 2D grids. This simulation uses r = 0.4 giving R� = 0.16, well within the stability limit. Damping is added via the term a�?p/?t which models absorption by wall materials. Boundary conditions use a sponge (exponential decay) layer at open edges to absorb outgoing waves.

How does noise cancellation work physically?

Noise cancellation (active noise control) works by superimposing an anti-phase pressure wave on the original noise. If two identical sinusoidal sources are separated by half a wavelength and driven 180� out of phase, their pressure fields add destructively along the axis between them (destructive interference). The Noise Cancel preset places two sources separated by ~0.22 � domain width with inverted polarity. Where the waves meet with equal amplitude and opposite phase, pressure cancels to near-zero � visible as dark bands in the pressure field.

Acoustic Wave Simulator

2D FDTD pressure wave simulation � standing waves, room modes, Doppler shift, reverberation & noise cancellation.

Method: 2D FDTD wave equation

Grid: 200 � 120 cells

Courant: r = 0.4 (stable)

Keyboard: 1�6 presets � P pause � R reset � S save

Presets

Parameters

Frequency 280 Hz

Room Size Medium

Absorption 0.30

Speed 3�

Physics & Method

2D FDTD � Finite-Difference Time-Domain

Solves the acoustic wave equation on a 200�120 Cartesian grid. At each timestep, pressure p is updated from the two previous timesteps using finite differences.

Wave equation: ?�p/?t� = c�(?�p/?x� + ?�p/?y�) + source(x,y,t) FDTD discretisation (damped): p_new = [(2 - a)�p - (1 - a)�p_old + R��?�p] / (1 + a/2) R� = (c�?t/?x)� = 0.16 (c=0.4 cells/step, stable: R = 0.707) a = base_damping + absorption_slider � 0.04 Courant-Friedrichs-Lewy (CFL) stability: r = c�?t/?x = 1/v2 � 0.707 (2D) ? use r=0.4 ?

Boundary Conditions

Rigid walls: zero pressure (hard reflection, p = 0 at wall cells)
Open edges: exponential sponge layer (R = 0.72 per step) absorbs outgoing waves, preventing spurious reflections
Open pipe ends: edges are free (pressure antinodes at pipe mouth)
Closed pipe ends: rigid wall columns (pressure nodes at closed ends)

Preset Guide

🎵 Open Pipe

A horizontal pipe with open ends (Dirichlet BC). Standing waves form with pressure nodes at both ends. Resonant lengths: L = n�?/2 (n = 1, 2, 3 ...). Fundamental mode: f1 = c/2L.

🔇 Closed Pipe

One end closed (rigid wall), one open. Pressure antinode at closed end, node at open end. Only odd harmonics: L = (2n-1)�?/4. Fundamental: f1 = c/4L (one octave below equal-length open pipe).

🏛️ Room Modes

Rectangular room with rigid perimeter walls. Axial modes form along each dimension; tangential and oblique modes arise from combinations. Use the Room Size slider to alter the room dimensions and observe how the resonant mode frequencies change.

Axial mode frequencies (rectangular room L � W): f_nx,ny = (c/2) � v((nx/L)� + (ny/W)�) Lowest (1,0) mode: f = c / (2L)

🚗 Doppler Shift

A sinusoidal point source moves at ~0.4� the wave speed (Mach 0.4). Wavefronts compress ahead of the source (higher frequency) and stretch behind it (lower frequency). The wavefront pattern traces characteristic Mach cones at this subsonic speed.

Doppler formula: f_obs = f_s � c / (c � v_s) (- ahead, + behind) At v_s = 0.4c: ahead 1.67�f_s behind 0.71�f_s

⛪ Reverb Chamber

Ultra-low absorption (a � 0.00025). Pressure waves reflect repeatedly � observe energy building up to steady-state and, when source stops, the gradual exponential decay (reverberation time RT60 = time for energy to fall 60 dB).

🔇 Noise Cancellation

Two sources separated by ~0.22 of the domain width driven with 180� phase difference. Their pressure fields interfere destructively, creating quiet zones visible as dark bands between and beyond the sources.

Key Equations

Quantity	Formula
Speed of sound (air, 20 �C)	c = 343 m/s
Wavelength	? = c / f
Open pipe modes	f_n = n�c / 2L, n = 1,2,3�
Closed pipe modes	f_n = (2n-1)�c / 4L, n = 1,2,3�
Room axial mode	f = c / 2L (along dimension L)
Schroeder frequency	f_s � 2000v(T60/V) Hz
RT60 (Sabine)	T60 = 0.161�V / (a�S) seconds
Doppler shift	f_obs = f_s�c / (c � v_s)

Curriculum Links

Level	Topics
GCSE Physics	Waves, reflection, longitudinal/transverse, wave equation
A-Level Physics	Standing waves, harmonics, Doppler effect, sound intensity
AP Physics	Superposition, interference, resonance, wave speed in medium
IB Physics	Standing waves in pipes, Doppler, diffraction and interference
University	Acoustic wave equation, FDTD, room acoustics, psychoacoustics