Transmission Loss Prediction

**Strutt|Transmission Loss|TL Prediction** predicts the panel transmission loss using Davy's partition theory and inserts it into the current row of the worksheet.
The theory is broadly broken into two components:

- Single leaf theory
- Double leaf theory

- Airborne (including losses from absorbing media)
- Studborne (steel or timber studs in various configurations)

**Key Concepts**

**Wall Leaves**

The theory implemented in Strutt is based on the work of Davy with some additional elements and assumptions which are outlined in detail below. The theory is based on the concept of a *Single Leaf* or a *Double Leaf* (cavity wall) partition. In this implementation there are two leaves/panels, Panel 1 and Panel 2. Each leaf/panel can be comprised of multiple layers of a material and up to two different materials per leaf/panel - described here as the 'inner' and 'outer' layers of the single composite leaf. In practice the order of the materials in a given panel is irrelevant as they're combined into an equivalent single panel anyway.

Typical material properties for various construction materials are provided for convenience. The user is able to input their own custom material properties if they are known. Where these are different from the default material properties for the selected material, the input cell will turn orange to indicate that they've been changed from the default values.

**Equivalent Composite Panel**

The multiple layers and materials are composed into theoretical composite equivalent homogenous panels (one per side of the cavity). Validity of this approach has only been tested in limited studies which generally have considered multiple layers of plasterboard (gyprock). Application of this concept for composite leaves with concrete and lightweight cladding for example, is assumed to be reasonable although not explicity validated with published studies. It should be noted that the composite material properties are generally weighted against mass and on this basis, a composite material comprising of a heavy maonsry and a lightweight board are likely to be dominated by the masonry both in practice and using this theoretical methodology. On that basis it seems like a reasonable approach. The manner in which the various leaves are combined into a single equivalent panel are outlined below. The effective properties of importance are the mass per unit Area (`m`), Critical Frequency (`f_c`) and the internal damping (`\eta_i`), which are calcualted and displayed for both Panel 1 and Panel 2.

**Single Leaf Transmission**

For single leaf transmission loss either Panel 1 *or* Panel 2 can be used.

**Double Leaf Transmission (Cavity Walls)**

Selecting materials for both Panel 1 *and* Panel 2 will trigger the calculation to be considered as a 'Double Leaf' prediction and will also make the cavity details portion of the user input form active.

**Surface Density (Mass per Unit Area)**

The density (`\rho`) and thickness (`d`) of the leaf will dictate the mass per unit area (`m`). As can be seen below, `m` increases the non-resonant (forced) transmission coefficient `\tau_n` by 6 dB for each doubling of mass. This can be observed because `\tau_n` is inversely proportional to `z_m^2` - in other words this is the famous 'mass-law' of transmission loss theory. The resonant component also includes a mass term `z_m`, but the bending stiffness and damping also play a part in this part of transmission so it is more complicated to generalise.

**Internal Damping**

The internal damping (`\eta_i`) will influence the resonant transmission coefficient (`\tau_r`). In practice, increasing `\eta_i` will reduce the severity of the coincidence dip in the spectrum, and increase transmission loss near and above the critical frequency. It should be noted that two additional damping terms, the edge damping (`\eta_e`) and radiation damping (`\eta_r`) are also included in the total damping (as outlined below). There are optional tick boxes to turn off the radiation and edge damping (i.e. to set them equal to 0)

**Young's Modulus and Poisson's Ratio**

When combined with the thickness of the panel (`d`), Young's Modulus (`E`) and Poisson's Ratio (`\nu`) will determine the criticial frequency (refer to the critical frequency section below).

**Bond Type**

It should be noted that there are two selections for the Bond Type which are discussed in detail in the section on calculation of the Effective Composite Panel. In summary, by selecting 'Point Bonding', it is assumed that the individual layers of each leaf bend independantly with their own bending stiffness and may slip over each other. 'Whole Bonding' assumes that the layers are continuously bonded, meaning that they will be forced to bend together as a single panel of thickness equal to the total thickness. In practical terms, this will mean that point bonding will have a higher critical frequency than a panel which has whole bonding. Examples of materials which can be considered as having 'whole bonding' include laminated glass (e.g. two layers of 5 mm glass laminated into a 10.76 mm laminated pane). The bond type also affects the assumed internal damping.

**Edge Damping, Radiation Damping and Shear Wave Corrections**

The Edge Damping (`\eta_e`), Radiation Damping (`\eta_r`) and Shear Wave (`C_s`) corrections have tick boxes to allow the user to set these terms to zero. In general these should be kept on but for interest / comparison purposes they're able to be disabled.

The Edge Damping is an additional damping term which affects the resonant radiation efficiency. Radiation Damping affects the damping specifically in the coincidence region. The Shear Wave correction predominantly affects high frequency results and is more critical in thick panels such as concrete.

Note that the greyed out 'Sewell's Correction' is not relevant to Davy's theory. Sewell's Correction is a correction term which is sometimes applied to transmission loss predictions to account for the diffraction effects of low frequency sound. In Davy's theory this phenomenon is taken into account as part of the finite-size panel non-resonant radiation efficiency (`\sigma_n`). However, in other well known theories, such as Sharp's method (which is broadly what Insul is based on) a specific correction term described as 'Sewell's Correction' is usually applied. The setting is reserved for future use and has no effect on the prediction.

**Size of Partition**

The size of the partition will affect the transmission loss. Both The ISO and ASTM testing methodologies for transmission loss require a minimum partition size of `10 "m"^2` for a partition. The default dimensions for the partition are set such that the area is `10 "m"^2` and equivalent to a typical testing laboratory aperture. The reason the minimum areas of the partition is defined in the standards is that diffraction of sound affects the transmission loss. Generally the effect is only observable at low frequency, where the diffraction effects are most noticable. If the dimensions of the panel are changed to a lower size, one will observe an increase in low frequency transmission loss. The portion of the theory which dictates this behaviour is the non-resonant radiation efficiency (`\sigma_n`). It should be noted that chaning the size of the partition to very small values (`<1 "m"`) may cause strange results and is not recommended. However, for windows and doors this could be a consideration, noting that a larger partition size is more conservative as it will predict a lower transmission loss.

In addition to affecting the low-frequeny transmission loss, the width and height of the partition also affect the calculations of stud transmission - the width of the partition and the number of stud separation will determine the number of studs which radiate energy.

**Cavity Depth**

The cavity depth (`d_s`) primarily dictates the mass-air-mass resonant frequency (`f_0`). The larger `d_s` is, the lower `f_0` will be. The calculated `f_0` is displayed for reference. Note that in the implementation in strutt `f_0` is not adjusted for changes from adiabatic to isothermal wave propagation when absorption is located in the cavity. This is because Davy had initially not proposed this. In more recent work by Davy he has proposed that this be implemented and an adjustment to `f_0` due to absorption may be implemented in the future. `d_s` will also limit the maximum thickness of the absorber (`d_a`), if any and will influence the empirical `\alpha_"min"` values.

**Studs**

Studs are discussed in detail below. However, in summary there are steel and timber studs, and there are single, staggered and double stud options for both. Selecting 'Double Studs' of either timber or steel simply sets the stud transmission coefficient (`\tau_s`) to zero - i.e. the stud transmission path is ignored. It should be noted that flanking paths through shared floor slabs, walls facades etc may still exist but are not predicted with this tool.

The stud spacing is the distance between studs. A lower spacing will reduce transmission loss as there is more connection points and so more energy will be transferred through the studs per unit area of wall.

It is worth noting that double glazed unit's (DGU) frames have been shown by Davy to be important when predicting transmission loss of DGUs. To implement this using the transmission loss prediction tool, one should set the stud spacing equal to the width of the panel, and the width of the panel should be set to the width of the DGU. Davy recommends using timber studs. However, it is also noted that timber studs are known to be overly conservative. Steel studs could also be considered.

**Absorber**

Various materials are able to be selected which contain typical Flow Resistivities. The thickness of the absorber can be set. By default the thickness will follow a sensible typical dimension based on the cavity depth, and obviously is limited to the cavity depth. The absorption coefficients are not displayed, but are calculated from the flow resistivity

**Graph**

The graph displays the predicted transmission loss spectra of the total transmission loss, the transmission loss of each panel (assuming it were a single leaf panel), the transmission loss due to the airborne component only, and the transmission loss due to the studborne component only. These can be turned on and off using the tick-boxes at the bottom right of the figure.

The key information which is generally useful is to compare locations of coincidence dips from Panel 1 and Panel 2 so that they can (ideally) be offset sufficienty to improve Double Panel transmission loss in the coincidence region. This is generally the approach taken for DGU designs, but can also apply to light-weight stud partitions. The location of the mass-air-mass frequency is also sometimes important. The final key information which is easily observed is the contribution to the overall transmission loss due to the studs compared to the airborne transmission. This allows one to consider which component of your partition is constraining the performance. If the studborne transmission is significantly lower than the airborne transmission loss, for example, changing the stud type to an improved stud would be expected to yield significant improvements in the transmission loss. However, if the airborne transmission loss is lower, changing the stud type will not improve performance. In this case, one would need to focus on increasing the panel mass, cavity depth or absorption density.

For convenience, an 'Add Comparison' button is also available, which allows the user to select an existing row of transmission loss values in thier calculation sheet which will be plotted on the graph. This can be useful if you wish to compare the spectra of two design options.

**Values**

The Values tab allows the user to look at the numerical octave band values for the abovementioned transmission losses.

**Settings**

The settings are reserved for future use but are currently unused.

**Rw / STC**

An overall Rw (or STC if your settings are set to imperial units) is displayed. If metric units are selected the Ctr is also shown. Additionally the overall thickness of the partition is displayed for convenience.

**Implementation in Strutt**

In the following equations the following definitions will apply:

`R` is the transmission loss.

`\tau` is the transmission coefficient (ratio of transmitted to incidet power)

`(:\tau:)` is the diffuse field transmission coefficient

`\tau_r` is the transmission coefficient due to the resonant (or free) transmission

`\tau_n` is the transmission coefficient due to the non-resonant (or forced) transmission

`\tau_s` is the transmission coefficient due to the studs-borne transmission

`J` is the stud transmission ratio

`c_0` is the speed of sound in air

`\rho_0` is the density of air

`\sigma_{r}` is the resonant (or free) radiation efficiency

`\sigma_{n}(\theta)` is the non-resonant (or forced) radiation efficiency as a function of the angle of plane wave radiation

`\sigma_{c}` is the resonant (or free) radiation efficiency for the special case at coincidence

`(:\sigma:)` is the diffuse field radiation efficiency

`D` is the ratio of the total to non-resonant radiation

`\theta` is the angle of radiation or incidence

`\theta_c` is the coincidence angle

`W` is the width of the wall

`H` is the height of the wall

`U` is the perimeter of the wall

`S` is the area of the wall

`a = (2S/U)` is half the length of the characteristic wall dimension

`d_s` is the thickness of the stud (wall cavity thickness)

`d_a` is the thickness of the absorber

`C_M` is the mechanical compliance of the stud

`b` is the stud spacing

`rho` is the density (mass per unit volume) of the material

`m = \rho d` is the surface density (mass per unit area) of the material

`m_0 = (m_1m_2)/(m_1 + m_2)` is the reduced mass of the two sides of the wall

`d` is the thickness of the material

`E` is the Young's modulus of the materials

`G` is the shear modulus of the materials

`\nu` is Poisson's ratio

`"OUT"` denotes the outer leaves on one side of the partition

`"IN"` denotes the inner leaves on one side of the partition

`N` is the number of leaves on one side of the partition

`n` denotes the nth leaf.

`\eta = \eta_i + \eta_e + 2\eta_r` is total damping of the materials

`\eta_i` is the material damping

`\eta_e = m / (485\sqrt(f))` is the edge damping

`\eta_r = (\sigma_{n}(\theta) \rho_0)/(k m)` is the radiation damping

`f` is the frequency of sound in air

`f_n` is the nth frequency in the calculation

`f_c` is the criticial frequency of the material

`f_0` is the mass-air-mass resonance frequency

`C_s` is the sheer wave correction factor

`f` is the excitation frequency

`\omega = 2\pi f` is the angular excitation frequency

`k = \omega/(c_0)` is the wavenumber in air

`k_M` is the corrected wave number of the free-bending wave

`k_T` is the wave number of a hypothetical corrected shear wave

`k_B` is the wave number of the free bending wave according to thin plate theory

`k_L` is the wave number of the quasilongitudinal wave

`G^(**)` is the corrected shear modulus

`r = f/f_c` is the ratio of the frequency to critical frequency

`Z_0 = 2 \rho_0 c_0` is the impedance of air

`z_m = (\omega m)/Z_0` is the mass law impedance

**Transmission Loss**

The transmission loss in Strutt is given as a negative number, in keeping with Strutt's convention that all decibel values for attenuation are presented as negative values. It is noted that this is opposite the the usual convention for Transmission Loss, which would be presented as a positive dB. The transmission loss in Strutt is calculated at multiple bands per octave then averaged down to either 1 or 1/3 octave as follows:

`R = -10 log_10(1/(N_f) sum_(n=1)^(N_f)(:\tau(f_n):))`, where

`N_f` is the number of frequencies per band to average. In Strutt `N_f` is set to 24/oct (8 per 1/3 octave)

**Single Leaf Partition**

A single leaf partition is calculated in two parts:

- Non-Resonant (or Forced) dominated region (below `f_c`), which broadly speaking follows the well known
*Mass-Law*. - Resonant (or Free) dominated region (above `f_c`), which is controlled by the resonant radiation efficiency, broadly following Cremer's theory of radiation.

The transmission ratio is given by:

`\tau(\theta) = {
(((\sigma_n(\theta))/z_m)^2, \if f < f_c),
(1/((1 + (z_m\eta r^2 sin^4\theta)/(\sigma_n(\theta)))^2 + (z_m/(\sigma_n(\theta)))^2(1 - r^2 \sin^4\theta)^2), \if f >= f_c):}`

The diffuse field transmission ratio is given by:

`(:\tau:) = {
(\tau_r + \tau_n, \if f < f_c),
(\tau_r, \if f >= f_c):}`

`\tau_r = C_s (\sigma_c ^ 2 / (2 z_m r (\sigma_c + z_m \eta)))
(\arctan((2 z_m) / (\sigma_c + z_m \eta)) -
\arctan((2 z_m (1 - r)) / (\sigma_c + z_m \eta)))`

`\tau_n = (2 (:\sigma_n:)) / (z_m^2)`

Definitions of these terms are outlined in the sections below.

**Double Leaf Partition**

The double leaf partition transmission loss is divided in to several parts as outlined below. An important aspect of the theory (to simplify the mathematics) is that the two sides of the wall are ordered such that panel 1 has the lowest `f_c` and panel 2 has the highest `f_c`. i.e. `f_(c,2) > f_(c,1)`.

- Airborne transmission
- Below `2/3f_0`
- Between `2/3f_0` and `f_0` (Linear interpolation in `\log_10(f)`)
- Between `f_0` and `f_(c,2)`
- Between `f_(c,2)` and `1.1f_(c,2)`
- Above `1.1f_(c,2)`

- Studborne transmission above `f_0`

The diffuse field transmission ratio of the double panel is predicted as follows. The definitions of various terms used in the following are outlined in the sections below.

`(:\tau(f):) = {
(\tau_A(f), if f <= 2/3f_0),
(\log_10(f/f_0) / \log_10(2/3) (\tau_A(2/3f_0) - \tau_B(f_0) - \tau_C(f_0) - \tau_s(f_0)) + \tau_B(f_0) + \tau_C(f_0) + \tau_s(f_0), if 2/3f_0 < f < f_0),
(\tau_B(f) + \tau_C(f) + \tau_s(f), if f_0 <= f < f_(c,2)),
(((1.1f_(c,2) - f)/(0.1 f_(c,2)))\tau_B(f) + \tau_C(f) + \tau_s(f), if f_(c,2) <= f <= 1.1 f_(c,2)),
(\tau_C(f) + \tau_s(f), if f_(c,2) < f):}`

Below the mass-air-mass frequency (`f_0`), the transmission coefficient is calculated from:

`\tau_A(f) = (1/(z_(m,1) + z_(m,2))^2) ln((1 + (z_(m,1) + z_(m,2))^2)/(1 + (z_(m,1) + z_(m,2))^2 cos^2(\theta_l)))`

Above the mass-air-mass frequency (`f_0`) the transmission coefficient is determined from `\tau_B`, `\tau_C` and `\tau_s`. The stud transmission (`\tau_s`) is defined in a subsequent section below.

`\tau_B(f) = C_(s,1)C_(s,2)(1 - cos^2(\theta_l))/((1/2(z_(m,2)/(z_(m,1)) + z_(m,1)/(z_(m,2))) + (z_(m,2) z_(m,1) \alpha) cos^2(\theta_l))(1/2(z_(m,2)/(z_(m,1)) + z_(m,1)/(z_(m,2))) + (z_(m,2) z_(m,1) \alpha))`

In the following, the subscript `n` denotes the panel leaf number (1 or 2)

`\tau_C(f) = C_(s,1)C_(s,2)(I / (S_1^2 S_2^2 \alpha^2))`, where

`I = {
((A_I + B_I + C_I)/D_I, if Q_1 != Q_2 and P_1 != P_2),
((Q^2 - P(P - 1)) + (\arctan(P/Q) - \arctan((P - 1)/Q)), if Q_1 = Q_2 = Q and P_1 = P_2 = P):}`, where

`A_I = Q_1Q_2(P_2 - P_1) \ln(([Q_1^2 + (P_1 - 1)^2][Q_2^2 + P_2^2])/([Q_2^2 + (P_2 - 1)^2][Q_1^2 + P_1^2]))`

`B_I = Q_1[(P_1 - P_2)^2 + Q_1^2 - Q_2^2][\arctan(P_2/Q_2) - \arctan((P_2 - 1)/Q_2)]`

`C_I = Q_2[(P_2 - P_1)^2 + Q_2^2 - Q_1^2][\arctan(P_1/Q_1) - \arctan((P_1 - 1)/Q_1)]`

`D_I = Q_1Q_2[(P_2 - P_1)^2 + (Q_2 + Q_1)^2][(P_2 - P_1)^2 + (Q_2 - Q_1)^2]`

`P_n = 1 - 1/r_n`

`S_n = (2z_(m,n)r_n)/\sigma_c`

`Q_n = ((1 + (z_(m,n)\eta_n/sigma_c)))/(S_n)`

**Critical Frequency & Coincidence**

The Critical Frequency is the lowest frequency at which Coincidence can occur. Coincidence describes the condition when the induced (forced) bending wave in a panel is equal to the natural bending wave in the panel. Above the Critical Frequency, assuming a diffuse field and broad-band noise excitation, there will always be a frequency at coincidence. In other words, above the Critical Frequency, there would always be some resonant excitation of the panel. The for coincidence to occur, the right combination of frequency and angle of incidence is required, as the angle of incidence moves from grazing incidence (`90^(@)`) to normal incidence (`0^(@)`) (in an infinitely sized panel) the frequency at which coincidence occurs moves from critical frequency to infinite frequency. The critical frequency is calculated as follows:

`f_c = c_0^2/(2 \pi d) sqrt((12 \rho (1 - \nu^2))/E)`

**Radiation Efficiency**

Radiation efficiency describes the ratio of the real part of the fluid-wave impedance (the impedance of exciting a plane wave in air from a panel) to the characteristic impedance of air. It describes how efficiently airborne sound energy is transmitted to/from the panel.

The resonant radiation efficiency only considers bending waves which are bending at the natural resonances of the panel - ie it is not forced. An example of such bending waves would be mechanical excitation from an impulse. The resonant (free) radiation efficiency is given by Ver and Holmer (1971) in Chapter 11 of Beranek's book "Noise and Vibration Control". Davy (2009c) modifies this slightly to limit the maximum radiation efficiency to that of the forced radiation efficiency at coincidence. In this implementation, the Davy Resonant Radiation Efficiency is given by:

`\sigma_r = {
(c_0 / (f_c S) (U g_2 + (c_0 g_1) / f_c), \if f <= f_c),
(sigma_c, \if f > f_c ):}`, and

`\sigma_r \leq \sigma_c`, where

`\g_1 = {
((8(1 - 2r)) / (\pi^4 \sqrt(r(1 - r))), \if f <= f_c/2),
(0, \if f > f_c/2):}`

`g_2 = ((1 - r) \ln((1 + \sqrt(r)) / (1 - \sqrt(r))) + 2 \sqrt(r)) / (4\pi^2 (1 - r)^(3/2))`

The non-resonant (forced) radiation efficiency is the radiation efficiency experienced under airborne excitation and is given by

`\sigma_n(g) = {
(1/sqrt(g^2 + q^2), \if 1 >= |g| >= p),
(1/(sqrt((h - (h/p - 1) g)^2 + q^2)), \if p > |g| >= 0):}`, where

`g = cos(\theta)`

and the diffuse field non-resonant radiation efficiency is given by

`(:sigma_n:) = ln((1 + sqrt(1 + q^2))/(f sqrt(p^2 + q^2))) + 1/\alpha ln((h + sqrt(h^2 + q^2))/(p + sqrt(p^2 + q^2)))`, where

`q = \pi/(2k^2a^2)`

`p = {
("w" sqrt(\pi / (2ka)), \if "w" sqrt(\pi / (2ka)) <= 1),
(1, \if "w" sqrt(\pi / (2ka)) > 1):}`

`h = 1 / (2/3 sqrt((2ka)/\pi) - \beta)`

`"w" = 1.3` is an empirical correction.

`\beta = 0.124` is an empirical correction.

The radiation efficiency at coincidence (resonant and non-resonant bending is indistinguishable at coincidence) is given as follows. It is a special case of the above radiation efficiency equations.

`\sigma_c = \sigma_r(\theta_c)`.

`\sigma_c = {
(1 / sqrt(g^2 + q^2), \if 1 >= g >= p),
(1 / sqrt((h - \alpha g)^2 + q^2), \if p > g >= 0):}`, where

`g = {
(sqrt(1 - f_c / f), \if f >= f_c),
(0, \if f < f_c):}`

In the special case of a double panel transmission loss, the radiation efficiency at coincidence is modified empirically to give better agreement with experiment. In this case it is defined as follows:

`\sigma_(c,"2 panel") = {
(\sigma_c, if f < 0.9f_(c,1)),
((f - 0.9f_(c,1))/(0.1f_(c,1)) (1 - \sigma_(n,0.9f_(c,1))) + \sigma_(n,0.9f_(c,1)), if 0.9f_(c,1) <= f <= f_(c,1)),
(1, if f_(c,1) < f):}`, where

`\sigma_(c,0.9f_(c,1))` is the non-resonant radiation efficiency at coincidence for `f = 0.9f_(c,1)`.

**Shear Wave Correction**

As the thickness of the panel increases, the bending wave speed changes, asymptoting at the shear wave speed. To take this into account, Davy implements a method derived by Ljunggren (1991) which uses Mindlin Plate Theory to add a correction term to the transmission coefficient to account for the transition from bending waves to shear waves. The effect is more noticable high frequency and in thick panels, such as concrete. The correction is defined as follows:

`C_s = A_s/(B_s D_s)`, where

`A_s = [1 + d^2/12((k_M^2 k_T^2)/k_L^2 - k_T^2)]^2`

`B_s = 1 - (k_T^2 d^2)/12 + (k_M^2 k_S^2)/k_B^4`

`D_s = sqrt(1 - (k_T^2 d^2)/12 + (k_S^4)/(4k_B^4))`

`k_T = sqrt((\omega^2 \rho)/(G^(**)))`

`k_S = sqrt(k_T^2 + k_L^2)^2`

`k_L = sqrt((\omega^2 \rho (1 - \nu^2))/E))`

`k_B = root(4)((12\omega^2 \rho(1 - \nu^2))/(Ed^2))`

`k_M = sqrt((k_L^2 / 2)(1 + (2\chi) /(1 - \nu) + sqrt((1 + (2\chi) / (1 - \nu))^2 + 48 / (d^2 k_L^2))))`

`G^(**) = G / \chi`

`\chi = ((1 + \nu)/(0.87 + 1.12\nu))^2`

**Mass-Air-Mass Resonance**

When a cavity is introduced into the partition, a resonance due to the air space surrounded by the two wall leaves is introduced into the transmission path. The air acts as a spring, and so the resonance can be modelleled as a mass-spring system. The mass-air-mass resonance is the fundamental resonance which occurs, and harmonics of this resonance occur at higher frequencies. Adding sound absorptive material to the cavity is able to reduce the harmonics, but the fundamental resonance is largely unchanged by the presence of absorption. The frequency at which this occurs is calculated as follows:

`f_0 = sqrt((rho_0c_0^2(m_1 + m_2))/(d_sm_1m_2))`

**Combining Multiple Leaves into Composite Equivalent Single Leaf**

The transmission loss theory is based on the concept of a single homogeneous isotropic panel (and optionally with a cavity and stud connections). To take multiple materials and leaves per side of the partition into consideration an equivalent single leaf (per side) is calculated using empirical methods outlined below.

`N = N_("OUT") + N_("IN")`

`d = d_("OUT")N_("OUT") + d_("IN")N_("IN")`

`m = \rho_("OUT")d_("OUT")N_("OUT") + \rho_("IN")d_("IN")N_("IN")`

`\nu = \nu_("OUT")d_("OUT")N_("OUT") + v_("IN")d_("IN")N_("IN")`

Nakanishi et al. (2011) derived an empirical correction to both Young's Modulus and Internal Damping when multiple layers of plasterboard were bonded in various ways. In Strutt, the option to select 'Whole Bonding' and 'Point Bonding' changes the way the materials are treated with respect to the Young's Modulus and Internal Damping. When there are multiple leaves of material bonded with point connections (either screws or spot glued, for example) on one side of a partition, the individual leaves are able to essentially bend independantly from one another. This means that the critical frequency of the composite leaf is largely unchanged but the mass is increased. Conversely with whole bonding (such as in laminated glass), the two leaves of the panel are forced to bend together with increased damping from the bonding agent. In Nakashini's work, there are different empirical corrections based on different amounts of point connection. In Strutt a single correction based on the average correction has been used. The implementation in Strutt is outlined below:

`E = W_B (\Sigma E_n d_n^3)/((\Sigma d_n)^3) + (1 - W_B)(\Sigma E_n d_n)/(\Sigma d_n)`

`\eta_i = H_B (\Sigma\eta_(i,n) d_n)/(\Sigma d_n)`

`W_B = {
(1, if N = 1),
(0.95, if "Point Bonding" and N > 1),
(0.3, if "Whole Bonding" and N > 1):}`

`H_B = {
(1, if N = 1),
(1.8, if "Point Bonding" and N > 1),
(1, if "Whole Bonding"):}`

**Stud Transmission**

The studs are modelled as linear springs with a mechanical compliance of `C_M`. `C_M = 0` is a rigid stud. The compliance is the force per unit length. The proportion of the transmission ratio due to the transmission via the studs is defined as follows:

`\tau_s(f) = (C_(s,1)C_(s,2)32 \rho_0^2 c_0^3 D J)/(4 g_s^2 b \pi^2 f^2)`

`J = 2 / (1 + (1 - (4(2 \pi f)^(3/2)m_1m_2cC_M)/g_s))`

`g_s = m_1 \sqrt(2 \pi f_(c,2)) + m_2 \sqrt(2 \pi f_(c,1))`

`D = (1 + (\pi \sigma_(r,1))/(4r\eta_1))(1 + \sigma_(r,2)/(2 \eta_2) sqrt(1/r))`

For double studs (both timber and steel) `\tau_s(f) = 0` i.e. the stud is ignored and the transmission loss is only determined from the airborne path. It should be noted that flanking noise (e.g. though the floor slab) is not taken into consideration and this would need to be taken into account seperately.

For steel and timber studs, the stud spacing is provided as an input (600 mm by default). When staggered studs are selected the stud spacing ignored and set internally to be equal to the height of the wall (i.e. a 'stud' at the head and foot track).

Emperical values for `C_M` are determined from comparison of experimental data and the `C_M` required to make the theory match experiment. `C_M` values for timber studs are set to 0. However, in a subsequent publication by Davy, he notes that the assumption of a perfectly rigid stud is overly conservative. As such, timber studs are expected to predict slighltly lower transmission loss than one might achieve in practice. Emperical frequency dependent `C_M` values have been determined for steel studs.

`C_m = {
(C_(m1), if C_(m1) < C_(m2)),
(C_(m2), if C_(m1) >= C_(m2)):}`, where

`C_(m1) = A_1 f^(x_(f1)) m_0^(x_(m1)) b^(x_(b1)) d_s^(x_(d1))`

`C_(m2) = A_2 f^(x_(f2)) m_0^(x_(m2)) b^(x_(b2)) d_s^(x_(d2))`

`A_1`, `A_2`, `x_(f1)`, `x_(f2)`, `x_(m1)`, `x_(m2)`, `x_(b1)`, `x_(b2)`, `x_(d1)` and `x_(d2)` are empirical values, defined as follows:

- Timber Studs
- `C_m = 0`

- Staggered Timber Studs
- `C_m = 0`
- `b = H`

- Steel Studs
- `A_1 = 9.3xx10^(-5)`
- `x_(f1) = 0`
- `x_(m1) = -1.09`
- `x_(b1) = 0`
- `x_(d1) = 0.8`
- `A_2 = 1.74`
- `x_(f2) = -1.81`
- `x_(m2) = -1.4`
- `x_(b2) = -0.75`
- `x_(d2) = 0.28`

- Steel Staggered Studs
- `b = H`
- `A_1 = 9.3xx10^(-5)`
- `x_(f1) = 0`
- `x_(m1) = -1.09`
- `x_(b1) = 0`
- `x_(d1) = 0.8`
- `A_2 = 1.74`
- `x_(f2) = -1.7`
- `x_(m2) = -1.4`
- `x_(b2) = -0.75`
- `x_(d2) = 0.28`

**Limiting Angle**

In double panels the effect of sound at grazing indcidence and coincidence is considerably more sensitive to the angle of incidence. As a consequence, for double panels and empirical limit on the forced radiation efficiency for angles greater than a 'limiting angle' is introduced. Some people may be familiar with a similar concept that was historically used for single panels, often described as a 'field correction'. However, for single panels, it has been shown that the previously applied limiting angle was determinable and dependant on the size of the panel. Unfortunately for double panels the complexity and sensitivity of the theoretical result to this parameter has lead to the implementation of a limiting angle approach. When using the Double Panel Theory, the following apply:

`\cos^2(\theta_l) = {
(0.9, if 1/(2ka) > 0.9),
(1/(2ka), if 0.9 >= 1/(2ka) >= cos^2(61^(@))),
(cos^2(61^(@)), if cos^2(61^(@)) > 1/(2ka)):}`

**Cavity Absorption**

Absorption in the cavity is limited by the wavelength across the cavity and a minimum empirical absorption for empty or low absorption cavities as follows:

`\alpha = {
(\alpha_("min"), if \alpha < \alpha_("min")),
(\alpha, if \alpha_("min") <= \alpha <= \alpha_("max")),
(\alpha_("max"), if \alpha_("max") < \alpha):}`, where

`\alpha_("min") = {
(0.05 + 0.35d_s, if 0.05 + 0.35d_s < alpha_("max")),
(alpha_max, if 0.05 + 0.35d_s >= alpha_("max")):}`

`\alpha_("max") = {
(kd_s, if kd_s < 1),
(1, if kd_s >= 1):}`

`\alpha` is determined according to the methods outlined in Absorption or Trasmission Coefficients of Porous Absorber, calculated from the flow resistivity and thickness.

**References**

Davy, J. L. (**2009a**). "Predicting the sound insulation of walls", Build. Acout. **16**, 1-20.

Davy, J. L. (**2009b**). "The forced radiation efficiency of finite sized flat panels which are excited by incident sound", J. Acoust. Soc. Am. **126**, 694-702.

Davy, J. L. (**2009c**). "Predicting the sound insulation of single leaf walls - Extension of Cremer's model", J. Acoust. Soc. Am. **126**, 1871-1877.

Davy, J. L. (**2010a**). "The improvement of a a simple theoretical model for the prediction of the sound insulation of double leaf walls", J. Acoust. Soc. Am. **127**, 841-849.

Davy, J. L., Catherine, G., Villot, M. (**2010b**). "The equivalent translational compliance of steel studs", *Proceedings of the 20th International Congregss on Acoustics, ICA 2010*, 23-27 August 2010, Sydney Australia

Nakanishi, S., Yairi, M., Minemura, A. (**2011**). "Estimation method for parameters of construction on predicting transmission loss of double leaf dry partition", Applied Acoustics, **72**, 364-371.

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