Sound Insulation Performance (Transmission Loss)
Sound Insulation Performance (Transmission Loss): Theoretical Foundations
What is Transmission Loss (TL)?
Professor, what is transmission loss?
It's an index that represents how much of the sound incident on a wall or panel can be prevented from transmitting through.
$W_{in}$: Incident sound power, $W_{tr}$: Transmitted sound power. A larger TL indicates higher sound insulation performance.
Mass Law
Why does making a wall heavier improve sound insulation?
The Mass Law is the most fundamental law of sound insulation.
$m_s$: Surface density [kg/mยฒ], $f$: Frequency [Hz]. Doubling surface density โ TL +6dB, Doubling frequency โ TL +6dB. This is the "6dB rule of the Mass Law".
So heavier walls are better, right?
That's true for low to mid frequencies, but there is a frequency where the Mass Law breaks down. That is the coincidence frequency.
Coincidence Effect
At the frequency where the wavelength of the panel's bending wave matches the wavelength of the incident sound wave, sound transmission becomes easier.
$c$: Speed of sound, $D$: Bending stiffness $D = \frac{Eh^3}{12(1-\nu^2)}$. At the coincidence frequency, TL dips significantly.
So thinner panels have a higher coincidence frequency, right?
Correct. Halving thickness $h$ โ $f_c$ doubles. For a 6mm steel plate, $f_c \approx 2\,\text{kHz}$; for a 3mm aluminum plate, $f_c \approx 4\,\text{kHz}$.
Sound Insulation of Double Walls
Using a double wall (double-leaf wall) can achieve sound insulation performance significantly exceeding the Mass Law.
- Resonant transmission frequency: $f_0 = \frac{1}{2\pi}\sqrt{\frac{\rho c^2}{d}\left(\frac{1}{m_1}+\frac{1}{m_2}\right)}$
- Below $f_0$: Same as a single wall
- Above $f_0$: Sound insulation improves at 12dB/oct (twice the 6dB/oct of a single wall)
$d$: Thickness of the air gap. Placing sound-absorbing material in the air gap can mitigate the dip at $f_0$.
Summary
The Mass Law is a simple yet powerful formula derived by Berger in 1923
The "Mass Law" for sound insulation was formulated in 1923 by the German acoustician E. Berger as TLโ20logโโ(mยทf)โ47.5dB (SI units). This formula is still used today as a first approximation in design, but deviations from its derivation assumptions (infinite flat plate, normal incidence) can cause differences of 5-10dB from measured values. The phenomenon where TL drops sharply near the coincidence frequency (coincidence effect) cannot be explained by this formula and was first quantified by wave theory by Cremer (1942).
Computational Methods for Sound Insulation Performance (Transmission Loss)
FEM Calculation Method for TL
Please teach me how to calculate transmission loss using FEM.
The basic approach is a coupled model of three domains: incident-side acoustic domain + structural panel + transmitted-side acoustic domain.
1. Incident side: Plane wave incidence. Acoustic FEM or analytical input.
2. Structure: Model the panel with shell/solid elements.
3. Transmitted side: Acoustic FEM. Non-reflecting boundary (PML or impedance boundary).
Calculation of Diffuse Field TL
In experiments, it's measured in a diffuse sound field, right?
Yes. To calculate diffuse field TL with FEM:
- Method 1: Calculate individually for multiple incidence angles (0ยฐ to 78ยฐ) and average using Paris's formula.
- Method 2: Directly model the diffuse sound field (randomly place acoustic sources).
Method 1 is common. Good agreement is obtained with $\theta_{max} = 78ยฐ$ (equivalent to ISO 15186).
SEA (Statistical Energy Analysis)
For high frequencies (above several hundred Hz), SEA is more efficient than FEM.
- Described by the energy balance of each subsystem (panel, air gap, room)
- Modal density and coupling loss factor are key parameters
- Computational cost is extremely low (algebraic equations per frequency band)
Summary
The 2011 revision of ISO 10140 standards revolutionized testing laboratories
The ISO 10140 series of standards for measuring sound insulation of building materials underwent a comprehensive revision in 2010-2011, significantly tightening the requirements for acoustic transmission and reception rooms. Before the revision, differences of up to 8dB for the same sample between different laboratories were problematic. After the revision, requirements for diffusivity in reverberation rooms were added (new standard: DIN EN ISO 10140-5). Prompted by this revision, Japan's JIS A 1416 was also reviewed, and a round-robin test among 8 domestic institutions confirmed measurement reproducibility within 3dB.
Sound Insulation Performance (Transmission Loss) in Practice
TL Analysis in Practice
Typical applications are automotive dash panels, architectural partition walls, and aircraft fuselage panels.
Analysis Flow
1. Identify Target Panel โ Shape, material, thickness, constraint conditions
2. Set Frequency Range โ Automotive NVH: 20-500Hz, Architectural: 125-4000Hz
3. Build FEM Model โ Acoustic mesh should be $\lambda_{min}/6$ or less
4. Set Incidence Conditions โ Normal incidence or diffuse incidence (multiple angles)
5. Calculate TL โ Ratio of incident power to transmitted power
6. Calculate STC/Rw โ Compare with standard values
Practical Checklist
Common Numerical Examples
| Panel | Surface Density [kg/mยฒ] | TL @500Hz [dB] | Coincidence [Hz] |
|---|---|---|---|
| Steel Plate 1.6mm | 12.5 |
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