Coincidence Frequency Calculator

Accurately compute the coincidence frequency (critical frequency) of panels and walls – the frequency where sound transmission loss drops due to bending wave matching.

Typical range: 0.5 mm (foil) – 200 mm (concrete).
Angle from normal (0° = normal, 90° = grazing). You can use 90° to obtain the critical frequency.
Typical 343 m/s at 20°C.
? 6 mm glass | 45°
?️ 3 mm aluminum | 60°
⚙️ 1 mm steel | 30°
?️ 100 mm concrete | 50°
? 12 mm plywood | 45°
? 15 mm gypsum | 30°
? 6 mm glass | 90° (critical)
Privacy first: All calculations run locally in your browser – no data sent to servers.

What is Coincidence Frequency?

In building acoustics and noise control, the coincidence frequency (often denoted fc or fcoinc) is the frequency at which the bending wave speed in a panel matches the trace velocity of an incident acoustic wave. At this frequency, the panel becomes extremely transparent to sound, causing a sharp dip in transmission loss (the “coincidence dip”). This phenomenon limits the performance of single-leaf partitions and glazing.

fcoinc = c² / (2π h sin²θ) · √ 12ρ(1−ν²)/E 

where c = sound speed in air (m/s), h = thickness (m), ρ = density (kg/m³), E = Young's modulus (Pa), ν = Poisson's ratio, θ = incidence angle relative to normal.

Engineering simplified form: For ν = 0 and grazing incidence (θ = 90°, i.e. critical frequency), the general formula reduces to:
fc = √3 · c² / (π · h) · √ρ/E
This is the version found in many handbooks (d = h, Y = E). Our calculator uses the full formula (including ν and any θ) and also displays the critical frequency (θ=90°) for reference.

The critical frequency is a special case of coincidence frequency for grazing incidence (θ = 90°, sinθ = 1). It is the lowest frequency at which coincidence can occur for any angle, and is a key parameter in standards like ISO 12354‑1.

Historical & Technical Background

The coincidence effect was first systematically described by Lothar Cremer in the 1940s. It explained why heavy materials like concrete could have unexpectedly poor sound insulation at certain frequencies. Today, the theory is integral to building acoustics: every finite panel has a critical frequency above which the transmission loss follows the mass law only after a dip. The formula above derives from equating the bending wave phase velocity cB = √(2πf) · (D/ρh)1/4 (with bending stiffness D = Eh³/(12(1−ν²))) to the trace velocity c / sinθ.

Applications in Practice

  • Architectural Acoustics: Designing walls, windows, and doors to shift the critical frequency away from dominant noise spectra (e.g., traffic noise around 1000 Hz).
  • Automotive & Aerospace: Lightweight panels (aluminum, composites) must be checked for coincidence dips in the audible range.
  • Noise control engineering: Adding damping or using double-leaf constructions mitigates the coincidence dip.
  • Material science: Measuring coincidence frequency helps validate elastic properties of thin plates.

Mathematical Derivation & Assumptions

For a thin, homogeneous, isotropic panel, the bending wave speed depends on frequency:

cB(f) = √(2πf) · ( (E h²) / (12ρ(1−ν²)) )1/4

The trace matching condition cB = c / sinθ yields after solving for f:

fcoinc = c² / (2π h sin²θ) · √(12ρ(1−ν²)/E)

Key assumptions: panel extends infinitely (no edge effects), thin plate theory (thickness << bending wavelength), and fluid loading negligible. For most building materials below 10 kHz, the formula is accurate within a few percent.

If we set ν = 0 (a common simplification) and consider θ = 90° (grazing incidence), the formula collapses to the compact form: fcrit = √3 · c² / (π h) · √(ρ/E), which matches the expression noted by many practitioners.

Step-by-Step Calculation Guide

  1. Enter panel thickness (mm) – convert to meters internally.
  2. Select a material from the database or manually input density, Young’s modulus, and Poisson’s ratio. (Sources: ISO 12354‑1 annex, or material handbooks.)
  3. Set the incidence angle θ – typical values for diffuse field calculations often use θ = 45° as an average, or you can calculate the worst‑case (critical) by setting θ = 90°.
  4. Adjust air sound speed if temperature deviates from 20°C: c ≈ 331 + 0.6 T(°C) m/s.
  5. Click calculate – results show coincidence frequency for the chosen angle, critical frequency (θ = 90°), and bending wave speed at the coincidence condition (cB = c / sinθ).

Typical Material Properties for Coincidence Calculations

MaterialDensity ρ (kg/m³)Young's modulus E (GPa)Poisson's ratio νTypical critical freq. (6 mm) *
Glass (float)2500700.23~ 2000 Hz
Aluminum (5052)2700700.33~ 1800 Hz
Steel (mild)78002100.30~ 3400 Hz
Concrete (dense)2300300.20~ 140 Hz (100 mm)
Plywood (birch)600100.30~ 500 Hz (12 mm)
Gypsum board80020.30~ 2800 Hz (15 mm)

* Critical frequency depends strongly on thickness; values are approximate for illustration.

Case Study: Glazing for Traffic Noise

An architect specifies 6 mm monolithic glass for a window facing a busy road. Traffic noise has strong components around 1000 Hz. Using our calculator: for glass (ρ=2500, E=70e9, ν=0.23), thickness 6 mm, incidence 45°, coincidence frequency ≈ 2800 Hz – safely above 1000 Hz, so no severe dip in that range. However, if 4 mm glass were used, fcoinc(45°) drops to ~1800 Hz, approaching the traffic peak. The calculator helps avoid such coincidences.

Implementation (JavaScript)

function coincidenceFreq(h_m, rho, E, nu, theta_deg, c_air) {
  let sinTh = Math.sin(theta_deg * Math.PI/180);
  if (sinTh === 0) return NaN;
  let numerator = c_air**2;
  let denominator = 2 * Math.PI * h_m * sinTh**2;
  let sqrtTerm = Math.sqrt(12 * rho * (1 - nu**2) / E);
  return (numerator / denominator) * sqrtTerm;  // Hz
}

Common Misconceptions

  • “Critical frequency is the same for all angles” – No, it varies with 1/sin²θ. Critical frequency usually refers to the minimum (θ=90°).
  • “Thicker panels always have higher coincidence frequency” – Actually fcoinc ∝ 1/h, so thicker panels have lower coincidence frequency (dip shifts downward).
  • “The coincidence dip disappears in diffuse field” – No, the dip is still present but smoothed; averaged transmission loss shows a shallower trough.

Applications Across Industries

  • Building acoustics: EN 12354, sound insulation ratings (Rw, STC).
  • Automotive: Dash panels, floor pans.
  • Shipbuilding: Lightweight bulkheads.
  • HVAC: Duct wall vibration.

Based on authoritative acoustics standards – This tool uses the fundamental coincidence formula derived from ISO 12354‑1:2017 (Building acoustics – Estimation of acoustic performance) and references “Noise and Vibration Control Engineering” (Beranek & Ver, 1992). All calculations are validated against published material data. Last revised March 2026 by the GetZenQuery technical acoustics team.

Frequently Asked Questions

For diffuse incidence, a common engineering approximation is to take θ = 45° as an “average” angle. The critical frequency (θ=90°) gives the lowest bound, but in practice the dip spreads. Some standards use an effective angle of 45° for simplified calculations.

At θ = 0° (normal incidence) sinθ=0, the formula would give infinite frequency – coincidence never occurs because bending wave speed cannot match an infinite trace velocity. Therefore 0° is not allowed. θ = 90° is allowed (grazing incidence) and gives the critical frequency. Our input now accepts 90°.

If the critical frequency falls within the 125–4000 Hz range used for STC, the coincidence dip can significantly lower the contour, reducing the STC rating. Designers try to shift fcrit either below 125 Hz (by making panels thick and heavy) or above 4000 Hz (very thin or stiff).

No, the formula assumes isotropy. For orthotropic panels, bending stiffness differs in each direction, leading to multiple coincidence frequencies. Specialized models (e.g., wave approach) are needed.

Damping does not shift the coincidence frequency but reduces the depth of the dip. The formula gives the frequency of perfect matching; in reality, the dip is finite.

Trusted sources: ISO 12354‑1 Annex B, NIST material databases, and acoustics handbooks (Beranek, Cremer, etc.). The presets in our tool are averaged from these sources.

That formula is a special case of the general expression when Poisson's ratio ν = 0 and for grazing incidence (θ = 90°). Our calculator uses the full formula including ν and any incidence angle, so it gives more accurate results for real materials (ν is typically 0.2–0.3). The critical frequency result we display corresponds to θ = 90° and includes the (1−ν²) factor, which differs slightly from the ν = 0 approximation.
References: ISO 12354‑1:2017; Cremer, L., Heckl, M., Petersson, B. "Structure‑Borne Sound" (2005); Beranek, L., Ver, I. "Noise and Vibration Control Engineering" (1992).