TL Tool - Acoustic Formulas Reference

Acoustic Formulas Reference
Mathematical foundations of the TL Tool — ASTM E2611 / ISO 10534-2

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1. Air Properties

Speed of sound (ISO 9613-1):

$c=20.05\,{\sqrt {T_{K}}}\quad [{\text{m/s}}]$

Air density:

$\rho =1.2929\times {\frac {273.15}{T_{K}}}\times {\frac {P}{1013.25}}\quad [{\text{kg/m}}^{3}]$

where T_K = temperature in Kelvin, P = atmospheric pressure in hPa.

Wave number:

$k={\frac {\omega }{c}}={\frac {2\pi f}{c}}\quad [{\text{rad/m}}]$

Characteristic impedance of air:

Failed to parse (syntax error): {\displaystyle Z_0 = \rho c \quad [\text{Pa\,s/m}]}

2. Valid Frequency Range

The impedance tube supports plane-wave propagation only within:

Lower frequency limit (microphone spacing must resolve the wavelength):

$f_{\min }={\frac {c}{10\,\Delta x}}\quad {\text{where }}\Delta x=\min(x_{2}-x_{1},\;x_{4}-x_{3})$

Upper frequency limit (tube diameter must be smaller than 0.586 λ):

$f_{\max }={\frac {0.586\,c}{D}}$

where D = internal tube diameter.

⚠ At frequencies where k·(Δx) = nπ (half-wavelength resonance between two microphones), the wave decomposition is singular. These frequencies are automatically masked with NaN.

3. Plane Wave Field

In an impedance tube, the acoustic pressure field is:

$P(x)=A\,e^{+jkx}+B\,e^{-jkx}$

• A = complex amplitude of the forward-travelling wave (incident)
• B = complex amplitude of the backward-travelling wave (reflected)

The particle velocity is:

$U(x)={\frac {1}{\rho c}}\left(A\,e^{+jkx}-B\,e^{-jkx}\right)$

4. Wave Decomposition (Source Side)

From pressures measured at x⊂1; and x⊂2;, the incident and reflected amplitudes are:

${\begin{bmatrix}A\\B\end{bmatrix}}={\frac {1}{\Delta }}{\begin{bmatrix}e^{-jkx_{2}}&-e^{-jkx_{1}}\\-e^{+jkx_{2}}&e^{+jkx_{1}}\end{bmatrix}}{\begin{bmatrix}P(x_{1})\\P(x_{2})\end{bmatrix}}$

with determinant:

$\Delta =2j\,\sin \!{\bigl (}k(x_{2}-x_{1}){\bigr )}$

In practice, pressures are obtained from the measured FRFs:

$P(x_{i})=H_{i1}\cdot {\sqrt {S_{11}}}$

where H_i1 is the FRF between microphone i and the reference (CH1), and S⊂11; is the auto-spectrum of CH1.

5. Transfer Matrix — ASTM E2611 §8

The sample is described by its 2×2 transfer matrix [T]:

${\begin{bmatrix}p_{\text{down}}\\u_{\text{down}}\end{bmatrix}}={\begin{bmatrix}T_{11}&T_{12}\\T_{21}&T_{22}\end{bmatrix}}{\begin{bmatrix}p_{\text{up}}\\u_{\text{up}}\end{bmatrix}}$

where (p_up, u_up) and (p_down, u_down) are the pressure and particle velocity on the upstream and downstream faces of the sample.

Two-Load Method

Two independent measurements (Load I and Load II, different tube terminations) give:

${\begin{bmatrix}T_{11}&T_{12}\\T_{21}&T_{22}\end{bmatrix}}={\begin{bmatrix}p_{\text{down}}^{I}&p_{\text{down}}^{II}\\u_{\text{down}}^{I}&u_{\text{down}}^{II}\end{bmatrix}}{\begin{bmatrix}p_{\text{up}}^{I}&p_{\text{up}}^{II}\\u_{\text{up}}^{I}&u_{\text{up}}^{II}\end{bmatrix}}^{-1}$

This method does not require knowledge of the termination impedance.

6. Transmission Loss

From the transfer matrix coefficient T⊂12;:

${\text{TL}}=20\,\log _{10}\!\left(\left|{\frac {T_{12}}{2\,\rho c}}\right|\right)\quad [{\text{dB}}]$

For a homogeneous sample of surface area S (normalized to S = 1 m²):

${\text{TL}}=20\,\log _{10}\!\left({\frac {|T_{12}|}{2\,\rho c}}\right)$

The term T⊂12; has units of acoustic impedance [Pa·s/m]. The factor 2ρc normalizes it to a dimensionless transmission coefficient τ, from which TL = −10·log⊂10;(τ).

7. Absorption Coefficient

Reflection Coefficient

At the sample face (x = x⊂2;), from upstream wave decomposition:

$R={\frac {B}{A}}$

Normal-Incidence Absorption Coefficient

$\alpha (f)=1-|R(f)|^{2}$

α = 0: total reflection (rigid wall) — α = 1: total absorption (anechoic).

ISO 11654 Weighted Coefficient α_w

α(f) is averaged in 1/3 octave bands, then compared to a reference curve to obtain α_w and the absorption class (A to E).

8. Octave Band Synthesis

Band limits for centre frequency f_c at resolution 1/N:

$f_{\text{low}}=f_{c}\cdot 2^{-1/(2N)},\qquad f_{\text{high}}=f_{c}\cdot 2^{+1/(2N)}$

Energy Averaging (TL)

Correct for quantities in dB (ASTM E2611):

${\text{TL}}_{\text{oct}}=-10\,\log _{10}\!\left({\frac {1}{n}}\sum _{i=1}^{n}10^{-{\text{TL}}_{i}/10}\right)$

Arithmetic Averaging (α)

Correct for linear quantities (α ∈ [0, 1]):

$\alpha _{\text{oct}}={\frac {1}{n}}\sum _{i=1}^{n}\alpha _{i}$

Quantity	Method	Reason
TL [dB]	Energy	Power averaging in linear domain (ASTM E2611)
α [0–1]	Arithmetic	Linear quantity, not logarithmic

9. Delany-Bazley-Miki Model

For a homogeneous porous layer with flow resistivity σ [Pa·s/m²], thickness d, the Miki (1990) model gives:

Characteristic impedance:

$Z_{c}=\rho c\left[1+0.0571\left({\frac {\rho _{0}f}{\sigma }}\right)^{-0.754}-j\,0.0870\left({\frac {\rho _{0}f}{\sigma }}\right)^{-0.732}\right]$

Complex wave number:

$k_{c}={\frac {\omega }{c}}\left[1+0.0978\left({\frac {\rho _{0}f}{\sigma }}\right)^{-0.700}-j\,0.1890\left({\frac {\rho _{0}f}{\sigma }}\right)^{-0.595}\right]$

Transfer matrix of the porous layer:

$[T]={\begin{bmatrix}\cos(k_{c}d)&j\,Z_{c}\,\sin(k_{c}d)\\{\dfrac {j\,\sin(k_{c}d)}{Z_{c}}}&\cos(k_{c}d)\end{bmatrix}}$

Model Fitting

The TL Tool minimizes:

${\hat {\sigma }}=\arg \min _{\sigma }\sum _{f}{\bigl |}{\text{TL}}_{\text{meas}}(f)-{\text{TL}}_{\text{DBM}}(f,\sigma ){\bigr |}^{2}$

Validity range: 0.01 ≤ ρf/σ ≤ 1.0

10. Phase Calibration

For each microphone pair (i, j), two measurements are made with the microphones at position x_a then swapped to x_b:

• Position 1: H_ij^(1) = H_true · H_c — both mismatches present
• Position 2: H_ij^(2) = H_true / H_c — microphones swapped

The phase correction is extracted as:

$H_{c}={\sqrt {\frac {H_{ij}^{(1)}}{H_{ij}^{(2)}}}}$

Applied during calculation:

$H_{ij,{\text{corrected}}}={\frac {H_{ij,{\text{measured}}}}{H_{c}}}$

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References

• ASTM E2611 — Standard Test Method for Normal Incidence Determination of Porous Material Acoustical Properties Based on the Transfer Matrix Method
• ISO 10534-2 — Determination of sound absorption coefficient and impedance in impedance tubes
• ISO 9613-1 — Attenuation of sound during propagation outdoors
• ISO 11654 — Sound absorbers for use in buildings — Rating of sound absorption
• Miki Y. (1990) — Acoustical properties of porous materials: modifications of Delany-Bazley models, J. Acoust. Soc. Jpn.
• Allard & Atalla (2009) — Propagation of Sound in Porous Media, Wiley