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{{#seo:
|title=TL Tool - Acoustic Formulas Reference | OROS
|keywords=acoustic formulas, transmission loss, absorption coefficient, transfer matrix, ASTM E2611, Delany-Bazley-Miki, impedance tube
|description=Complete reference of acoustic formulas used in the OROS TL Tool: air properties, plane wave decomposition, transfer matrix, TL, absorption, octave synthesis, DBM model.
}}

__TOC__

<div style="background:linear-gradient(120deg,#001F5B 0%,#0055A5 100%);color:white;padding:18px 24px;border-radius:10px;margin-bottom:18px;">
<span style="font-size:1.35em;font-weight:bold;">Acoustic Formulas Reference</span><br/>
<span style="opacity:0.85;">Mathematical foundations of the TL Tool &mdash; ASTM E2611 / ISO 10534-2</span>
</div>

&#8592; Back to [[TL_Tool_-_Sound_Transmission_Loss_Measurement|TL Tool main page]]

== 1. Air Properties ==

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'''Speed of sound''' (ISO 9613-1):

:<math>c = 20.05\,\sqrt{T_K} \quad [\text{m/s}]</math>

'''Air density:'''

:<math>\rho = 1.2929 \times \frac{273.15}{T_K} \times \frac{P}{1013.25} \quad [\text{kg/m}^3]</math>

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

'''Wave number:'''

:<math>k = \frac{\omega}{c} = \frac{2\pi f}{c} \quad [\text{rad/m}]</math>

'''Characteristic impedance of air:'''

:<math>Z_0 = \rho c \quad [\text{Pa\,s/m}]</math>

</div>

== 2. Valid Frequency Range ==

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The impedance tube supports plane-wave propagation only within:

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

:<math>f_{\min} = \frac{c}{10\,\Delta x} \quad \text{where } \Delta x = \min(x_2-x_1,\; x_4-x_3)</math>

'''Upper frequency limit''' (tube diameter must be smaller than 0.586 &lambda;):

:<math>f_{\max} = \frac{0.586\,c}{D}</math>

where D = internal tube diameter.

<div style="border-left:4px solid #e67300;background:#fff8f0;padding:10px 14px;border-radius:0 6px 6px 0;margin-top:10px;">
&#9888; At frequencies where k&middot;(&Delta;x) = n&pi; (half-wavelength resonance between two microphones),
the wave decomposition is singular. These frequencies are automatically masked with NaN.
</div>

</div>

== 3. Plane Wave Field ==

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In an impedance tube, the acoustic pressure field is:

:<math>P(x) = A\,e^{+jkx} + B\,e^{-jkx}</math>

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

The particle velocity is:

:<math>U(x) = \frac{1}{\rho c}\left(A\,e^{+jkx} - B\,e^{-jkx}\right)</math>

</div>

== 4. Wave Decomposition (Source Side) ==

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

:<math>\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}</math>

with determinant:

:<math>\Delta = 2j\,\sin\!\bigl(k(x_2-x_1)\bigr)</math>

In practice, pressures are obtained from the measured FRFs:

:<math>P(x_i) = H_{i1} \cdot \sqrt{S_{11}}</math>

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

</div>

== 5. Transfer Matrix &mdash; ASTM E2611 &sect;8 ==

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The sample is described by its 2&times;2 transfer matrix [T]:

:<math>\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}</math>

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:

:<math>\begin{bmatrix} T_{11} & T_{12} \\ T_{21} & T_{22} \end{bmatrix} = \begin{bmatrix} p^I_{\text{down}} & p^{II}_{\text{down}} \\ u^I_{\text{down}} & u^{II}_{\text{down}} \end{bmatrix} \begin{bmatrix} p^I_{\text{up}} & p^{II}_{\text{up}} \\ u^I_{\text{up}} & u^{II}_{\text{up}} \end{bmatrix}^{-1}</math>

This method does not require knowledge of the termination impedance.

</div>

== 6. Transmission Loss ==

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From the transfer matrix coefficient T&sub;12;:

:<math>\text{TL} = 20\,\log_{10}\!\left(\left|\frac{T_{12}}{2\,\rho c}\right|\right) \quad [\text{dB}]</math>

For a homogeneous sample of surface area S (normalized to S = 1 m&sup2;):

:<math>\text{TL} = 20\,\log_{10}\!\left(\frac{|T_{12}|}{2\,\rho c}\right)</math>

<div style="border-left:4px solid #17a2b8;background:#e8f7fa;padding:10px 14px;border-radius:0 6px 6px 0;margin-top:10px;">
The term T&sub;12; has units of acoustic impedance [Pa&middot;s/m]. The factor 2&rho;c normalizes it
to a dimensionless transmission coefficient &tau;, from which TL = &minus;10&middot;log&sub;10;(&tau;).
</div>

</div>

== 7. Absorption Coefficient ==

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=== Reflection Coefficient ===

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

:<math>R = \frac{B}{A}</math>

=== Normal-Incidence Absorption Coefficient ===

:<math>\alpha(f) = 1 - |R(f)|^2</math>

&alpha; = 0: total reflection (rigid wall) &mdash; &alpha; = 1: total absorption (anechoic).

=== ISO 11654 Weighted Coefficient &alpha;_w ===

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

</div>

== 8. Octave Band Synthesis ==

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

:<math>f_{\text{low}} = f_c \cdot 2^{-1/(2N)}, \qquad f_{\text{high}} = f_c \cdot 2^{+1/(2N)}</math>

=== Energy Averaging (TL) ===

Correct for quantities in dB (ASTM E2611):

:<math>\text{TL}_{\text{oct}} = -10\,\log_{10}\!\left(\frac{1}{n}\sum_{i=1}^{n} 10^{-\text{TL}_i/10}\right)</math>

=== Arithmetic Averaging (&alpha;) ===

Correct for linear quantities (&alpha; &isin; [0, 1]):

:<math>\alpha_{\text{oct}} = \frac{1}{n}\sum_{i=1}^{n} \alpha_i</math>

{| class="wikitable" style="width:80%;margin-top:10px;"
! style="background:#003F87;color:white;" | Quantity
! style="background:#003F87;color:white;" | Method
! style="background:#003F87;color:white;" | Reason
|-
| TL [dB] || Energy || Power averaging in linear domain (ASTM E2611)
|-
| &alpha; [0&ndash;1] || Arithmetic || Linear quantity, not logarithmic
|}

</div>

== 9. Delany-Bazley-Miki Model ==

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For a homogeneous porous layer with flow resistivity &sigma; [Pa&middot;s/m&sup2;], thickness d, the Miki (1990) model gives:

'''Characteristic impedance:'''

:<math>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]</math>

'''Complex wave number:'''

:<math>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]</math>

'''Transfer matrix of the porous layer:'''

:<math>[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}</math>

=== Model Fitting ===

The TL Tool minimizes:

:<math>\hat{\sigma} = \arg\min_{\sigma} \sum_f \bigl|\text{TL}_{\text{meas}}(f) - \text{TL}_{\text{DBM}}(f,\sigma)\bigr|^2</math>

Validity range: 0.01 &le; &rho;f/&sigma; &le; 1.0

</div>

== 10. Phase Calibration ==

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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 &middot; H_c &mdash; both mismatches present
* Position 2: H_ij^(2) = H_true / H_c &mdash; microphones swapped

The phase correction is extracted as:

:<math>H_c = \sqrt{\frac{H_{ij}^{(1)}}{H_{ij}^{(2)}}}</math>

Applied during calculation:

:<math>H_{ij,\text{corrected}} = \frac{H_{ij,\text{measured}}}{H_c}</math>

</div>

----

&#8592; [[TL_Tool_-_Sound_Transmission_Loss_Measurement|Back to TL Tool main page]]

== References ==

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