Reverberation time, denoted as T, is a crucial metric in architectural acoustics, quantifying the time it takes for sound to decay by 60 decibels (dB) after the sound source has stopped. The calculation of reverberation time primarily relies on two fundamental formulas: the Sabine formula and the Eyring formula, with the latter offering enhanced accuracy, particularly in highly absorptive spaces or when considering air absorption.
Understanding Reverberation Time
Reverberation time directly impacts the clarity and perceived quality of sound within a space. A longer reverberation time can lead to muddy speech and music, while a very short time can make a space sound "dead" or unnaturally dry. Optimal reverberation times vary significantly depending on the room's intended use, from short times for speech intelligibility in classrooms to longer times for musical performance in concert halls.
The Sabine Formula
Developed by Wallace Clement Sabine in the late 19th century, the Sabine formula is one of the oldest and most widely used equations for calculating reverberation time. It provides a good approximation for spaces with relatively low sound absorption (e.g., concert halls, churches).
The Sabine formula is expressed as:
$$T = \frac{0.161 \cdot V}{A}$$
Where:
- T: Reverberation time in seconds (s).
- V: Volume of the room in cubic meters (m³).
- A: Total sound absorption in Sabine units (m²-Sabine). This is calculated as the sum of all surface areas multiplied by their respective sound absorption coefficients: $A = \sum (S_i \cdot \alpha_i)$.
- $S_i$: Area of the i-th surface in square meters (m²).
- $\alpha_i$: Sound absorption coefficient of the i-th surface (unitless, ranging from 0 to 1).
Key Considerations for Sabine:
- Assumption: Assumes a diffuse sound field (sound energy is evenly distributed) and a relatively low average absorption coefficient ($\alpha_{avg} < 0.2$).
- Limitations: Tends to overestimate reverberation time in rooms with high absorption.
- Applications: Widely used for initial estimations and in large, less absorptive spaces like auditoriums and gymnasiums.
The Eyring Formula
For spaces with higher average sound absorption coefficients ($\alpha_{avg} > 0.2$), the Eyring formula, also known as the Norris-Eyring formula, provides a more accurate prediction of reverberation time. It accounts for the non-diffuse sound field that occurs when a significant amount of sound is absorbed upon reflection.
The basic Eyring formula is:
$$T = \frac{0.161 \cdot V}{-S{total} \cdot \ln(1 - \alpha{avg})}$$
Where:
- T: Reverberation time in seconds (s).
- V: Volume of the room in cubic meters (m³).
- $S_{total}$: Total internal surface area of the room in square meters (m²).
- $\alpha_{avg}$: Average sound absorption coefficient of the room (unitless). This is calculated as $A / S_{total}$, where A is the total absorption from the Sabine formula.
- $\ln$: Natural logarithm.
Key Considerations for Eyring:
- Accuracy: More accurate for highly absorptive rooms, such as recording studios, anechoic chambers, or very dead rooms.
- Assumption: Assumes sound reflections undergo multiple absorptions, making it suitable for spaces where sound is absorbed more readily.
Advanced Eyring Formula with Air Absorption
For very large spaces, especially at higher frequencies (e.g., above 2000 Hz), the absorption of sound by the air itself becomes significant and must be included in reverberation time calculations for greater accuracy.
When accounting for air absorption, the Eyring equation, in metric units, can be expressed as:
$$T = \left[ \frac{0.161 \cdot V}{S{total} \cdot (-\ln(1 - \alpha{ey}))} \right] + (4 \cdot m \cdot V)$$
An equivalent form using the base-10 logarithm is:
$$T = \left[ \frac{0.161 \cdot V}{S{total} \cdot (-2.30 \cdot \log{10}(1 - \alpha_{ey}))} \right] + (4 \cdot m \cdot V)$$
Where:
- T: Reverberation time in seconds (s).
- V: Volume of the room in cubic meters (m³).
- $S_{total}$: Total internal surface area of the room in square meters (m²).
- $\alpha_{ey}$: The average absorption coefficient used in the Eyring formula (unitless).
- $m$: The atmospheric attenuation constant (air absorption coefficient), typically measured in Neper per meter (Np/m). This value is dependent on frequency, temperature, and relative humidity.
- $4 \cdot m \cdot V$: This term represents the additional time contribution due to sound energy loss in the air.
Practical Insight: The air absorption coefficient ($m$) is typically obtained from tables based on temperature, humidity, and specific frequency bands. For instance, at 20°C and 50% relative humidity, the value of $m$ for 4000 Hz would be significantly higher than for 500 Hz, illustrating its greater impact at higher frequencies.
Comparison of Reverberation Time Formulas
| Feature | Sabine Formula | Eyring Formula |
|---|---|---|
| Applicability | Low absorption ($\alpha_{avg} < 0.2$) | High absorption ($\alpha_{avg} > 0.2$) |
| Accuracy | Good for less absorptive rooms | More accurate for highly absorptive rooms |
| Theoretical Basis | Assumes diffuse sound field | Accounts for sound being absorbed upon reflection |
| Air Absorption | Not explicitly included in basic form | Can be incorporated for large volumes and high frequencies |
| Calculation | Simpler: $A = \sum S_i \alpha_i$ | More complex: Involves natural or base-10 logarithm |
Practical Applications and Design Considerations
- Architectural Design: Architects and acousticians use these formulas to predict and optimize the acoustic environment of spaces during the design phase.
- Material Selection: Understanding the impact of different materials' absorption coefficients is crucial. For example, porous materials like acoustic panels increase absorption, while hard, reflective surfaces like concrete decrease it.
- Room Volume: The larger the room's volume, the longer the reverberation time, all else being equal.
- Frequency Dependence: Reverberation time is not a single value but varies with frequency. Calculations are often performed across different octave bands (e.g., 125 Hz, 250 Hz, 500 Hz, 1000 Hz, 2000 Hz, 4000 Hz) to ensure a balanced acoustic response.
- Software Tools: While manual calculations are possible, specialized acoustic modeling software often uses these formulas (and more advanced algorithms) to simulate and predict reverberation time with high precision.
For further exploration of acoustic principles and design, you can consult resources from reputable organizations like the Acoustical Society of America or academic texts on architectural acoustics.
