The Complete Acoustics Cheatsheet: Essential Principles, Formulas, and Applications

Introduction: What is Acoustics?

Acoustics is the science of sound, including its production, transmission, and effects. It encompasses how sound waves propagate through different media, how they interact with surfaces and objects, and how they are perceived by humans. Understanding acoustics is crucial for architects, engineers, musicians, audio professionals, and anyone working with sound in any capacity.

Fundamental Principles of Sound

Sound Wave Properties

Sound Wave Mathematics

• Wave Equation: ∂²p/∂t² = c² ∇²p
• Simple Harmonic Motion: y(t) = A sin(2πft + φ)
• Sound Intensity (I): I = p²/(ρc) (W/m²)
• Sound Power (W): W = I × Area (watts)

Speed of Sound in Various Media

Temperature affects the speed of sound in air: c = 331.3 + 0.606T (where T is in °C)

Sound Measurements and Units

Decibel Scales and Relationships

Decibel Addition Rules

• Two equal sources: Total dB = Single source dB + 3
• 10 dB difference: Total dB ≈ Higher dB + 0.4
• 3 dB difference: Total dB ≈ Higher dB + 1.8
• For multiple sources: dB_total = 10 log₁₀(10^(dB₁/10) + 10^(dB₂/10) + …)

Common Sound Levels Reference Chart

Inverse Square Law

Sound intensity decreases with the square of the distance from the source:

• I₂ = I₁ × (d₁/d₂)²
• For SPL: dB₂ = dB₁ – 20 log₁₀(d₂/d₁)

Acoustic Phenomena

Reflection, Absorption, and Transmission

• Reflection coefficient (r): r = (Z₂ – Z₁)/(Z₂ + Z₁)
  • Z = acoustic impedance
• Absorption coefficient (α): Fraction of sound energy absorbed by a surface (0 to 1)
• Transmission Loss (TL): TL = 10 log₁₀(1/τ), where τ is transmission coefficient

Diffraction and Interference

• Diffraction: Bending of sound waves around obstacles
  • Significant when wavelength is comparable to or larger than obstacle size
• Constructive Interference: Waves align in phase, amplitudes add
• Destructive Interference: Waves align out of phase, amplitudes subtract
• Critical frequency: f_c = c²/(1.8c_L h)
  • Where c_L is longitudinal wave speed in material and h is thickness

Standing Waves and Room Modes

• Axial modes: (n_x, 0, 0), (0, n_y, 0), (0, 0, n_z)
• Tangential modes: (n_x, n_y, 0), (n_x, 0, n_z), (0, n_y, n_z)
• Oblique modes: (n_x, n_y, n_z) where all n values are non-zero

Room mode calculator formula: f = (c/2) × √[(n_x/L_x)² + (n_y/L_y)² + (n_z/L_z)²]

Architectural Acoustics

Room Acoustics Parameters

Sabine Reverberation Formula

RT60 = 0.161 × V / (A + 4mV)

Where:

• V = room volume (m³)
• A = total room absorption (m² sabins)
• m = air absorption coefficient (typically 0.004)

Absorption Coefficients (α) by Frequency

Optimal Room Dimensions

Preferred dimensional ratios to minimize modal problems:

• 1:1.14:1.39 (L×W×H) – European Broadcasting Union
• 1:1.4:1.9 (L×W×H) – Sepmeyer
• 1:1.6:2.33 (L×W×H) – Bolt area
• 1.54:1.25:1 or 2:3:5 – Golden ratio derived

Sound Isolation

Transmission Loss and STC Rating

Sound Transmission Class (STC): Single-number rating of a partition’s ability to block sound

• STC 25-30: Normal speech can be understood
• STC 35-40: Loud speech audible but not intelligible
• STC 40-45: Loud speech barely audible
• STC 45-50: Loud speech not audible
• STC 50-60: Very good isolation, music systems barely heard
• STC 60+: Excellent isolation, most sounds inaudible

Mass Law (approximate): TL = 20 log₁₀(fm) – 47

Where:

• f = frequency (Hz)
• m = mass per unit area (kg/m²)

Double Wall Construction

Resonant frequency of a double wall: f₀ = 60 × √(1/d × (1/m₁ + 1/m₂))

Where:

• d = cavity depth (m)
• m₁, m₂ = masses per unit area of the walls (kg/m²)

Improvements over single wall:

• Proper double wall with airgap: +8 to +10 dB
• Double wall with absorption in cavity: +12 to +15 dB
• Well-designed decoupled construction: +15 to +20 dB

Flanking Sound Transmission

Common flanking paths:

• Ductwork and HVAC penetrations
• Electrical outlets and fixtures
• Ceiling plenum pathways
• Structure-borne paths through floors/walls
• Window-to-window paths around barriers

Solutions:

• Staggered construction
• Resilient channels or clips
• Acoustic sealant at all penetrations
• Mass-loaded barriers in plenum spaces
• Floating floors/isolated ceiling systems

Noise Control

Noise Criteria (NC) Curves

HVAC Noise Control Techniques

1. Source treatment:

   • Vibration isolation of equipment
   • Equipment selection for lower noise
   • Proper equipment location

2. Path treatment:

   • Duct silencers/attenuators
   • Acoustic duct lining
   • Proper duct sizing and layout
   • Avoiding sharp bends

3. Receiver treatment:

   • Acoustic ceiling and wall treatment
   • Proper diffuser/grille selection
   • Background sound masking

Background Noise Calculations

A-weighted sound level (dBA): Weighting that corresponds to human hearing sensitivity

Equivalent continuous sound level (Leq): Leq = 10 log₁₀[(1/T) × ∫(10^(L(t)/10) dt)]

Day-night average sound level (Ldn): Ldn = 10 log₁₀[(15×10^(Ld/10) + 9×10^((Ln+10)/10))/24]

Electroacoustics

Loudspeaker Technical Parameters

• Sensitivity: dB SPL at 1m with 1W input (typically 85-95 dB)
• Impedance: Nominal resistance (typically 4-8 ohms)
• Power handling: Maximum input power (watts)
• Directivity (Q): Ratio of on-axis intensity to average intensity
• Directivity Index (DI): DI = 10 log₁₀(Q) (dB)
• Coverage angle: Angle where response is -6dB from on-axis

Microphone Technical Parameters

• Sensitivity: Output voltage per unit sound pressure (mV/Pa)
• Frequency response: Range of frequencies captured accurately
• Polar pattern: Directional sensitivity (cardioid, omni, figure-8, etc.)
• Self-noise: Equivalent noise level generated by the microphone (dBA)
• Maximum SPL: Highest sound level before distortion

Sound System Design Calculations

Required amplifier power: P_amp = 10^((SPL_desired – Sensitivity + Headroom)/10)

Where:

• SPL_desired = Target sound level (dB)
• Sensitivity = Speaker sensitivity (dB SPL/1W/1m)
• Headroom = Safety margin (typically 3-6 dB)

Critical distance (where direct and reverberant sound are equal): D_c = 0.141 × √(QV/RT60)

Where:

• Q = Directivity factor
• V = Room volume (m³)
• RT60 = Reverberation time (seconds)

Psychoacoustics

Hearing Sensitivity

• Audible frequency range: 20 Hz to 20,000 Hz (typical human)
• Most sensitive range: 1,000 Hz to 5,000 Hz
• Fletcher-Munson curves: Equal-loudness contours showing frequency-dependent sensitivity

Sound Perception Parameters

Critical Bands and Masking

• Critical bandwidth (CB) ≈ 100 Hz below 500 Hz
• Above 500 Hz: CB ≈ 0.2f
• Bark scale: Psychoacoustic scale dividing audible range into 24 critical bands
• Masking threshold: Sound level at which one sound masks another
• Temporal masking: Pre-masking (~20ms before) and post-masking (~200ms after)

Applied Acoustics

Concert Hall Design Principles

• Volume per seat: 7-10 m³ for symphonic music
• Width-to-height ratio: Typically 1.25-1.45:1
• Ceiling height: Typically 12-18m for symphonic halls
• Stage area: 150-200 m² for full orchestra
• First lateral reflections: Should arrive within 80ms of direct sound
• Bass ratio: Low-frequency reverberation should be 1.1-1.3 times mid-frequency reverberation

Critical Distance Calculations

Critical distance (distance where direct and reverberant sound are equal):

• D_c = 0.141 × √(QV/RT60)
• where Q = directivity factor, V = room volume, RT60 = reverberation time

Acoustic Modeling Methods

Acoustic Materials and Constructions

Sound Absorbers by Type

1. Porous absorbers:

   • Mineral wool, fiberglass, acoustic foam
   • Work by viscous losses as sound waves move through material
   • Most effective at mid to high frequencies
   • Effectiveness increases with thickness and air space behind

2. Resonant absorbers:

   • Membrane absorbers (plywood over airspace)
   • Helmholtz resonators (perforated panels)
   • Tuned to specific frequency ranges
   • Most effective at low to mid frequencies

3. Diffusers:

   • QRD (Quadratic Residue Diffuser)
   • PRD (Primitive Root Diffuser)
   • Skyline diffusers
   • Scatter sound energy rather than absorbing

Acoustic Constructions STC Ratings

Impact Isolation

Impact Insulation Class (IIC): Rating system for impact sound transmission

Improvement strategies:

• Carpet and padding: +15-25 IIC points
• Floating floor: +10-20 IIC points
• Resilient ceiling: +5-15 IIC points
• Isolation mounts: +15-25 IIC points

Standards and Measurement Techniques

Key Acoustic Standards

Measurement Techniques

1. Reverberation time:

   • Impulse response method (balloon pop, starter pistol)
   • Interrupted noise method (cut off pink noise)
   • Integrated impulse response method (sine sweep)

2. Transmission loss:

   • Two-room method with sound intensity measurement
   • Requires average sound level in source and receiver rooms

3. Sound absorption:

   • Reverberation chamber method
   • Impedance tube method (for normal incidence)

4. Sound intensity:

   • Sound intensity probe (p-p probe)
   • Used for sound power and transmission loss measurements

Applications in Different Fields

Recording Studio Acoustics

• Control room:

  • RT60 = 0.3-0.4 seconds
  • Reflection-free zone around listening position
  • Symmetrical design
  • Bass trapping in corners
  • Neutral frequency response

• Live room:

  • Variable acoustics (0.4-1.2 seconds RT60)
  • Diffusion on walls and ceiling
  • Isolation from control room >60 dB
  • Low background noise (NC 15-20)

Performance Space Acoustics

• Classical music hall:

  • RT60 = 1.8-2.2 seconds
  • Strong early lateral reflections
  • 40-60% sound-reflective surfaces
  • Diffusion on rear walls
  • Adjustable elements for different performance types

• Theater/Speech venue:

  • RT60 = 0.8-1.1 seconds
  • Strong early reflections
  • Speech Transmission Index (STI) > 0.65
  • Diffusion on rear and side walls

Environmental Acoustics

• Outdoor sound propagation factors:

  • Geometric spreading (-6 dB per doubling of distance)
  • Atmospheric absorption (frequency dependent)
  • Ground effect (depends on ground type)
  • Wind and temperature gradients
  • Barriers and topography

• Community noise assessment:

  • Day-Night Level (DNL or Ldn)
  • Equivalent Continuous Level (Leq)
  • Statistical levels (L10, L50, L90)
  • Maximum levels (Lmax)

Resources for Further Learning

Software Tools

• Room Acoustics:

  • EASE
  • ODEON
  • CATT-Acoustic
  • REW (Room EQ Wizard) – Free

• Sound Insulation:

  • Insul
  • SoundFlow
  • dBKAisla

• Environmental Noise:

  • CadnaA
  • SoundPLAN
  • NoiseMap

Books and Publications

• “Acoustics: An Introduction to Its Physical Principles and Applications” by Allan D. Pierce
• “Architectural Acoustics” by Marshall Long
• “Master Handbook of Acoustics” by F. Alton Everest and Ken Pohlmann
• “Sound System Engineering” by Don Davis and Carolyn Davis
• “Music, Physics and Engineering” by Harry F. Olson
• “Acoustic Design for the Home Studio” by Mitch Gallagher
• “Environmental Noise Control” by Colin Hansen

Online Calculators and Tools

• Room Mode Calculator: https://amcoustics.com/tools/amroc
• Acoustic Impedance Calculator: https://www.mh-audio.nl/Calculators/Caac.html
• Reverb Time Calculator: https://www.acousticalsurfaces.com/acoustic_IOI/reverberation.htm
• Sound Absorption Calculator: https://www.acoustic.ua/st/web_absorption_data_eng.html
• Directivity Calculator: https://www.merlijnvanveen.nl/en/calculators

Conclusion

Acoustics is a multidisciplinary field requiring understanding of physics, mathematics, materials science, and human perception. The principles outlined in this cheatsheet provide a foundation for addressing acoustic challenges in various applications, from building design to sound system optimization, musical performance spaces to noise control. For specific applications, it’s recommended to consult with acoustic specialists and the resources listed above.