Reverberation Time

**Strutt|Building Acoustics|Reverberation Time** inserts a new Reverberation Time Calculation worksheet into the active workbook, and inserts a link from the results row of the new RT sheet to the row the user had selected when invoking this feature.

The RT calculation sheets (both Metric and Imperial) feature the following different formulae for calculating RTs:

- Sabine
- Eyring
- Millington-Sette
- Zhang
- Kuttruff
- Arau-Puchades
- Fitzroy (Eyring)
- Fitzroy (Sabine)
- Fitzroy-Kuttruff
- Fitzroy-Neubauer
- Kang-Orlowski

Some calculation methods use the X,Y and Z coordinates in empirical RT formulas designed for uneven sound absorption distribution. In these cases, all surfaces or items located in the plane which is normal to a particular dimension should be allocated to that dimension by selecting the correct X, Y, or Z axis in the corresponding row. Each axis is defined as being normal to a dimension as follows:

- X plane: surfaces who's normal is parallel to the length (L)
- Y plane: surfaces who's normal is parallel to the width (W)
- Z plane: surfaces who's normal is parallel to the height (H)

Some calculations also specifically use the Length, Width and Height of the room which are specified in the 'Room' section of the calculation sheet. For rooms which are not rectilinear some judgement as to the 'Characteristic' or 'Effective' length, width and height may be required.

A Target RT row is provided to allow the user to input the desired RT targets for the space, and a graph is placed to the right of the results row showing the difference between the current calculated RT and the RT targets, to allow a visual representation of the RT spectra of the space.

**Implementation in Strutt**

In the following equations the following definitions will apply:

`K = (4 \cdot 60) / (10 \log_{10}(e))` (approx 55.25)

`c_0` is the speed of sound in air

`V` is the volume of the space

`L` is the length of the room

`W` is the width of the room

`H` is the height of the room

`d` is the source-reciever distance

`alpha` is an absorption coefficient as per Sabine equation definition (ie minimum of 0 but not limited to 1)

`alpha_{st}` is the statistical absorption coefficient (ie, limited to between 0 and 0.999999999999999). Note that the maximum limit has been implemented wherever an `alpha_{st}` is used to avoid undefined errors in Excel.

`\hat{\alpha}` is the effective absorption coefficient

`\hat{\alpha_{st}}` is the effective statistical absorption coefficient

`\bar{\alpha}` is the average absorption coefficient

`\bar{\alpha_{st}}` is the average statistical absorption coefficient

`\alpha_{n}` is the absorption coefficient of the `n^{\text{th}}` surface

`\rho = 1 - \alpha_{st}` is the reflection coefficient

`S = S_0 + S_{xyz}` is the total surface area

`S_0 = \sum_{n}S_{0,n}` is the total surface area not assigned to a plane

`S_{0,n}` is the surface area of the `n^{\text{th}}` which is not assigned to a plane

`S_{xyz} = S_x + S_y + S_z` is the total surface area assigned to X, Y and Z

`S_x = \sum_{n}S_{x,n}` is the total surface area assigned to the X plane

`S_y = \sum_{n}S_{y,n}` is the total surface area assigned to the Y plane

`S_z = \sum_{n}S_{z,n}` is the total surface area assigned to the Z plane

`S_{x,n}` is the area of the `n^{\text{th}}` which is assigned to the X plane

`S_{y,n}` is the area of the `n^{\text{th}}` which is assigned to the Y plane

`S_{z,n}` is the area of the `n^{\text{th}}` which is assigned to the Z plane

`A_s = A_{s,0} + A_{s,x} + A_{s,y} + A_{s,z}` is the total absorption area of absorbing surfaces

`A_i = A_{i,0} + A_{i,x} + A_{i,y} + A_{i,z}` is the total absoption area of items

`A_0 = A_{s,0} + A_{i,0}` is the total absorption area of surfaces and items not assigned to a plane.

`A_{s,0} = \sum_{n}A_{s,0,n}` is the total absorption area of surfaces not assigned to an X, Y, or Z plane

`A_{s,x} = \sum_{n}A_{s,x,n}` is the total absorption area of surfaces assigned to the X plane

`A_{s,y} = \sum_{n}A_{s,y,n}` is the total absorption area of surfaces assigned to the Y plane

`A_{s,z} = \sum_{n}A_{s,z,n}` is the total absorption area of surfaces assigned to the Z plane

`A_{s,0,n} = S_{0,n}\alpha_{n}` is the absorption area of the `n^{\text{th}}` surface not assigned to a plane

`A_{s,x,n} = S_{x,n}\alpha_{n}` is the absorption area of the `n^{\text{th}}` surface assigned to the X plane

`A_{s,y,n} = S_{y,n}\alpha_{n}` is the absorption area of the `n^{\text{th}}` surface assigned to the Y plane

`A_{s,z,n} = S_{z,n}\alpha_{n}` is the absorption area of the `n^{\text{th}}` surface assigned to the Z plane

`A_{i,0} = \sum_{n}N_{i,n}A_{i,0,n}` is the total absorption area of items not assigned to a plane

`A_{i,x} = \sum_{n}N_{i,n}A_{i,x,n}` is the total absorption area of items assigned to the X plane

`A_{i,y} = \sum_{n}N_{i,n}A_{i,y,n}` is the total absorption area of items assigned to the Y plane

`A_{i,z} = \sum_{n}N_{i,n}A_{i,z,n}` is the total absorption area of items assigned to the Z plane

`A_{i,0,n}` is the absorption area of the `n^{\text{th}}` item not assigned to a plane

`A_{i,x,n}` is the absorption area of the `n^{\text{th}}` item assigned to the X plane

`A_{i,y,n}` is the absorption area of the `n^{\text{th}}` item assigned to the Y plane

`A_{i,z,n}` is the absorption area of the `n^{\text{th}}` item assigned to the Z plane

`N_{i,n}` is the number (multiplier) of the `n^{\text{th}}` item

`m\prime = (-m)/(\log_{10}(e))` is the air absorption per unit mean free path (MPF)

`m` is the air attenuation in dB per unit length

**The Sabine Equation**

The Sabine equation is the best known, and first to be derived, formula for estimating the reverberation time in a room. It tends to be more accurate for diffuse spaces, evenly distributed absorption and for rooms with low average absorption (< 0.4).

`T_{60,\text{Sabine}} = (K V)/(c_0 S \hat{\alpha})`, where

`\hat{\alpha} = (A_s + A_i + 4m\primeV)/S`

**The Eyring Equation**

Eyring derived this equation based on the principles of geomety. It assumes a perfectly diffuse soundfield and that the sound absorption is evenly distributed. It is generally considered to be more accurate in rooms with high absorption. Note thtat strictly speaking, statistical absorption coefficients should be used in this equation, not Sabine derived coefficients. However, these are not readily available. In many cases, the Sabine coefficient and the Statistical coefficient will be close.

`T_{60,\text{Eyring}} = (K V)/(-c_0S\ln(1-\hat{alpha_{st}}))`, where

`\hat{\alpha_{st}} = (A_s + A_i + 4m\primeV)/S`

**The Millington-Sette Equation**

The Millington-Sette equation is derived in a similar vein to the Eyring equation, but it focuses on the probability that a reflection will hit a portion of the wall `S_n` an exact number of times in the decay. The equation assumes a perfectly diffuse reflection process and results in the following equation.

`T_{60,\text{Millington-Sette}} = (K V)/(-c_0S\ln(\hat{rho}))`, where

`\hat{rho} = \prod_{n}\rho_{n}^(S_{n}/S)`

`rho_n = 1 - \hat{alpha_{st,n}}`

`\hat{alpha_{st,n}} = \alpha_n + (A_i + 4m\primeV)/S`

**The Zhang Equation**

Zhang proposed a correction to the Millington-Sette formulation as it was pointed out that in cases where a very high absorption coefficient was present that the reflection probability density no longer holds. It assumes a perfectly diffuse reflection process. Taking this into consideration, Zhang derived the following equation.

`T_{60,\text{Zhang}} = (K V)/(-c_0 S \ln(\hat{\rho}))`, where

`\hat{rho} = \prod_{n}\rho_{n}`

`rho_n = 1 - \hat{alpha_{st,n}}S_{n}/S`

`\hat{alpha_{st,n}} = \alpha_n + (A_i + 4m\primeV)/S`

**The Kuttruff Equation**

Kuttruff introduced two new concepts into the calculation of the reverberaton time. The first is a correction due to shape (`\gamma`) and the second is a correction term `\Delta` which is largest when the variation between `\alpha` is high amongst the different surfaces. In other words he is trying to correct for shape and non-uniform distribution of absorption. The equation assumes a perfectly diffuse reflection. It tends to predict lower reverberation times than other equations.

`T_{60,\text{Kuttruff}} = (K V)/(c_0S\hat{\alpha})`, where

`\hat{\alpha} = -\ln(1 - \bar{\alpha_{st}})(1 + \gamma^2/2 \ln(1 - \bar{\alpha_{st}})) + \ln(1 + \Delta)`

`\bar{\alpha_{st}} = (A_s + A_i + 4m\primeV)/S`

`\Delta = (\sum_{n}\rho_{n}(\rho_n - \bar{\rho})^2)/((\bar{rho}S)^2 - \sum_{n}(\rho_nS_n)^2)`

`\bar{\rho} = 1 - \hat{\alpha_{st}}`

`\rho_n = 1 - \hat{\alpha_{st,n}}`

`\hat{\alpha_{st,n}} = \alpha_n + (A_i + 4m\primeV)/S`

`\gamma = \sqrt(0.0179(L + W)/H - 0.0001(L - W)/H - 0.0011((L - W)/H)^2 + 0.3025)`

The `\gamma^2` term in this equation is the variance of the distribution of path lengths between reflections divided by the square of the mean free path. In simplistic terms, it is a *shape factor*. It is typically only able to be found through a monte-carlo simulation. Zhang (2005), showed that an empirical formula was able to estimate gamma to within 98% accuracy for rooms with L:W:H ratios between 1:1:1 to 1:10:10. This equation has been implemented here. Bies and Hanson (2009) state that `\gamma^2` is typically around 0.4.

**The Arau-Puchades Equation**

The Arau-Puchades equation is based on the assumption of parallel opposed walls (rectilinear cuboid). The argument is that the decay will linger longer between such walls. This equation is most appropriate for rectangular rooms with orthogonal walls.

`T_{60,\text{Arau-Puchades}} = (K V)/(-c_0S\ln(1 - \hat{\alpha_{st}}))`, where

`\hat{\alpha_{st}} = (A_{xyz} + A_0 + 4m\primeV)/S`

`A_{xyz} = (1 - e^(-\hat{\alpha_{st,xyz}}))*S_{xyz}`

`\hat{\alpha_{st,xyz}} = (-\ln(1 - \bar{\alpha_{st,x}}))^(S_x/S_{xyz})(-\ln(1 - \bar{\alpha_{st,y}}))^(S_y/S_{xyz})(-\ln(1 - \bar{\alpha_{st,z}}))^(S_z/S_{xyz})`

`\bar{\alpha_{st,x}} = (A_{s,x} + A_{i,x})/S_x`

`\bar{\alpha_{st,y}} = (A_{s,y} + A_{i,y})/S_y`

`\bar{\alpha_{st,z}} = (A_{s,z} + A_{i,z})/S_z`

**The Fitzroy (Eyring) Equation**

The Fitzroy equation is an empirical equation based on the premise of 3 simultaneous decay processes occuring in the room, one for each orthogonal axis. The resulting reverberation times are summed in parallel, in a similar manner to the summation of parallel resistors in an electrical circuit.

`T_{60,\text{Fitzroy (Eyring)}} = (K V)/(-c_0S\ln(1-\hat{alpha_{st}}))`, where

`\hat{alpha_{st}} = (A_{xyz} + A_0 + 4m\primeV)/S`

`A_{xyz} = (1 - e^(-\hat{\alpha_{st,xyz}}))S_{xyz}`

`\hat{\alpha_{st,xyz}} = 1/(-S_x / (S_{xyz}\ln(1-\bar{\alpha_{st,x}})) -S_y / (S_{xyz}\ln(1-\bar{\alpha_{st,y}})) -S_z / (S_{xyz}\ln(1-\bar{\alpha_{st,z}})))`

`\bar{\alpha_{st,x}} = (A_{s,x} + A_{i,x})/S_x`

`\bar{\alpha_{st,y}} = (A_{s,y} + A_{i,y})/S_y`

`\bar{\alpha_{st,z}} = (A_{s,z} + A_{i,z})/S_z`

**The Fitzroy (Sabine) Equation**

The Fitzroy (Sabine) equation is a modification of the Fitzroy equation based on the Sabine equation (rather than the Eyring equation, which is used in Fitzroy's original paper)

`T_{60,\text{Fitzroy (Sabine)}} = (K V)/(c_0S\hat{\alpha})`, where

`\hat{\alpha} = (A_{xyz} + A_0 + 4m\primeV)/S`

`A_{xyz} = S_{xyz}\hat{\alpha_{xyz}}`

`\hat{\alpha_{xyz}} = 1/(S_x / (S_{xyz}\bar{\alpha_x}) + S_y / (S_{xyz}\bar{\alpha_y}) + S_z / (S_{xyz}\bar{\alpha_z}))`

`\bar{\alpha_x} = (A_{s,x} + A_{i,x})/S_x`

`\bar{\alpha_y} = (A_{s,y} + A_{i,y})/S_y`

`\bar{\alpha_z} = (A_{s,z} + A_{i,z})/S_z`

**The Fitzroy-Kuttruff Equation**

The Fitzroy-Kuttruff formula is a correction to the Fitzroy formula using Kuttruff's `\Delta` term for uneven distribtion of finishes. It also modified specifically for the case where absorption is primarily on the ceiling/floor plane (e.g. open-plan offices).

`T_{60,\text{Fitzroy-Kuttruff}} = (KV)/(c_0S\hat{\alpha})`, where

`\hat{alpha} = (A_{wcf} + A_0 + 4m\primeV)/S`

`A_{wcf} = (S^2 (-\ln(\bar{\rho}) + \Delta_{w})(-\ln(\bar{\rho}) + \Delta_{cf}))/(-S\ln(\bar{\rho}) + S_{w}\Delta_{cf} + S_cf\Delta_{w})`

`\bar{\rho} = 1 - \bar{\alpha_{st}}`

`\bar{\alpha_{st}} = (A_w + A_{cf} + A_0 + 4m\primeV)/S`

`A_w = A_x + A_y`

`A_{cf} = A_z`

`\Delta_{w} = (\sum_{n}\rho_{w,n}(\rho_{w,n} - \bar{rho_w})S_{w,n}^2)/(\bar{\rho_w}S_w)^2`

`\Delta_{cf} = (\sum_{n}\rho_{cf,n}(\rho_{cf,n} - \bar{rho_{cf}})S_{cf,n}^2)/(\bar{\rho_{cf}}S_{cf})^2`

`S_w = \sum_{n}S_{x,n} + \sum_{n}S_{y,n}`

`S_{w,n} = S_{x,n}` and `S_{w,n} = S_{y,n}`

`S_{cf} = \sum_{n}S_{z,n}`

`S_{cf,n} = S_{z,n}`

`\bar{\rho_w} = 1 - (A_{s,x} + A_{i,x} + A_{s,y} + A_{i,y}) / (S_w)`

`\bar{\rho_{cf}} = 1 - (A_{s,z} + A_{i,z})/S_z`

`\rho_{w,n} = 1 - \hat{alpha_{st,w,n}}`

`\rho_{cf,n} = 1 - \hat{alpha_{st,cf,n}}`

`\hat{\alpha_{st,w,n}} = alpha_{x,n} + A_{i,x}/S_w` and `\hat{\alpha_{st,w,n}} = alpha_{y,n} + A_{i,y}/S_w`

`\hat{\alpha_{st,cf,n}} = alpha_{z,n} + A_{i,z}/S_{cf}`

**The Fitzroy-Neubauer Equation**

The Fitzroy-Neubauer formula is an alternate version of the Fitzroy equation for flat and long rooms. It is based on the mean free path of a 2D decaying sound field.

`T_{60,\text{Fitzroy-Neubauer}} = (KV)/(-c_0S\ln(1-\bar{\alpha_{st}))`, where

`\bar{\alpha_{st}} = (A_{xyz} + A_0 + 4m\primeV)/S`

`A_{xyz} = S_{xyz}(1 - e^(-((KV)/(c_0S_{xyz}))((\pi c_0 \hat{\alpha_{xyz}})/(4K))^\eta))`

`\bar{\alpha_{xyz}} = 1/(-S_x/(P_x\ln(1 - \bar{\alpha_{st,x}})) - S_y/(P_y\ln(1 - \bar{\alpha_{st,y}})) - S_z/(P_z\ln(1 - \bar{\alpha_{st,z}})))`

`\bar{\alpha_{st,x}} = (A_{s,x} + A_{i,x})/S_x`

`\bar{\alpha_{st,y}} = (A_{s,y} + A_{i,y})/S_y`

`\bar{\alpha_{st,z}} = (A_{s,z} + A_{i,z})/S_z`

`P_x = 4(W\prime + H\prime)`, where `W\prime = \sqrt(S_x/2 W/H)` and `H\prime = S_x / (2W\prime)`

`P_y = 4(L\prime + H\prime)`, where `L\prime = \sqrt(S_y/2 L/H)` and `H\prime = S_y / (2L\prime)`

`P_z = 4(L\prime + W\prime)`, where `L\prime = \sqrt(S_z/2 W/L)` and `L\prime = S_z / (2W\prime)`

`\eta=1/2` in the normal case. In the explicit case of a cubic room (defined here as `S_x=S_y=S_z` and `L=W=H`) then `\eta=1/3`.

`P_x` is the total perimeter of the X plane walls

`P_y` is the total perimeter of the Y plane walls

`P_z` is the total perimeter of the Z plane ceiling/floor

`P_x`, `P_y` and `P_z` are calculated from `S_x`, `S_y` and `S_z` and scaled according to the `L`, `W`, `H` ratios provided in the calculation.

**The Kang-Orlowski Equation**

The Kang-Orlowski equation is empirically derived for very long rooms and tunnels (especially with absorptive end walls) and for source-receiver distances greater than the room width/height.

`T_{60,\text{Kang-Orlowski}}=150/(-850m - 10\log_{10}(d/(850 + d) (1 - \bar{\alpha_{st}})^(25.6 (1/H + 1/W) \sqrt(d + 425))))`, where

`\bar{\alpha_{st}} = (A_s + A_i)/S`

References:

- Sabine: Kuttruff, H.
*Room Acoustics*, Second Edition, p117 - Eyring: Kuttruff, H.
*Room Acoustics*, Second Edition, p116 - Millington-Sette: Bies and Hansen
*Engineeing Noise Control*, Fourth Edition, eq 7.58. - Zhang: Zhang, X. (2003),
*A new formula for reverberation time*, Atica Acustica United with Acustica,**89** - Kuttruff: Bies and Hanson
*Engineeing Noise Control*, Fourth Edition, eq 7.64. - Arau-Pachades, H. (1988)
*An improved reverberation formula*, Acoustica,**65**, 163-179. - Fitzroy: Davis and Davis,
*Sound System Engineering Second Edition*, p176 - Fitzroy-Kuttruff: Neubauer(2001)
*Estimation of Reverberation Time in Rectangular Rooms with Non-Uniformly Distributed Absorption Using a Modified Fitzroy Equation*,__Building Acoustics__**8**(2), pp115-137 - Fitzroy-Neubauer: Bies and Hansen
*Engineering Noise Control*, Third Edition, p295 - Kang-Orlowski: Kang, J.
*Acoustics of Long Spaces*, Thomas Telford Publishing, London

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