Mass law calculator for double layer
Calculator finds out sound reduction in decibels (dB) for double layer wall with given density, thickness and air gap width using so-called mass law equation.

# Beta version#

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# Transmission loss at various frequencies#

 Frequency [Hz] Transmission loss [dB] 31.25 44.54 62.5 50.56 125 56.58 250 62.6 500 68.62 1000 74.64 2000 80.66 4000 86.68 8000 92.7 16000 98.72

# Sound insulation#

• When a sound wave moves through the air meets a barrier in the form of a wall, part of the acoustic energy is reflected, part is absorbed inside the wall (converted to heat), and the another part is transmited out (on the other side of the wall). We can write it mathematically as follows:
$\alpha + \beta + \tau = 1$
where:
• $\alpha$ - absorption coefficient (determines the part of the energy that was absorbed inside the wall),
• $\beta$ - reflection coefficient (defines the part of the energy remaining in the first room),
• $\tau$ - transmission coefficient (defines the part of the energy that was emitted to the second room).
• The transmission coefficient can be used as a measure of the acoustic insulation, because it determines the sound intensity ratio on both sides of the wall:
$\tau = \frac{I_t}{I_0}$
where:
• $I_t$ - intensity of the wave on the other side of the wall (sound intensity level audible in the second room),
• $I_0$ - incident wave intensity (sound intensity level audible in the first room).
• In practice, the transmission factor is most often given in the logarithmic scale. In this way, we obtain a decrease in sound intensity given in decibels, so-called transmission loss:
$\Delta TL = -10 ~ log (\tau) = 10 ~ log \left(\frac{1}{\tau}\right)$

# Some facts#

• The acoustic insulation of a single wall is limited by its thickness and density of the material used. We can overcome these limits using two walls separated by air gap. The resulting system is called depending on the source:
• wall-air-wall,
• mass-air-mass,
• mass-spring-mass (more often found in theoretical papers where such a system is modeled by two masses connected with a spring).
• Sound reduction index for double wall can be estimated using formulas introduced by London in 1950, updated later by Sharp in 1973:
$R(f) = \begin{cases} 20 ~ log\left[f \cdot (h_1 ~ \rho_1 + h_2 ~ \rho_2) \right] - 47, & \text{when } f f_l \end{cases}$
where:
• $R$ - decrease of the sound intensity level of a partition consisting of two walls in decibels,
• $R_1, R_2$ - sound level decrease calculated for the first and second walls separately,
• $h_1, h_2$ - thickness of the first and second walls,
• $\rho_1, \rho_2$ - material density of which the first and second walls are made,
• $d$ - distance between walls (cavity width),
• $f$ - frequency of the acoustic wave,
• $f_0$ - resonant frequency $f_0 = \sqrt{\frac{\rho_0 \cdot c_0^2}{d} \cdot \frac{h_1 ~ \rho_1 + h_2 ~ \rho_2} {h_1 ~ \rho_1 ~ h_2 ~ \rho_2}}$,
• $f_l$ - limit frequency $f_l \approx \sqrt{\frac{55}{d}}$,
• $c_0$ - speed of sound in the air,
• $\rho_0$ - air density

If you're interested in calculators related to acoustics, check out our other calculators:
• Sound intensity level (dB) - if you want to learn what is decibel and how the sound intensity level is measured,
• Sound velocity in materials - if you want to learn how the type of substance affects the speed of acoustic wave propagation,
• Acoustic impedance of substances - if you want to learn what is acoustic impedance and how it depends on the type of substance,
• Sound wave reflection - if you want to find out how an acoustic wave behaves when it encounters an obstacle in the form of media boundary,
• Mass law: single wall - if you're interested in building acoustics and would like to estimate the acoustic insulation of a single wall,
• Mass law: double wall - if you're interested in building acoustics and would like to estimate the acoustic insulation of a double wall with an air gap between the walls,
• Sound absorption coefficients - if you're interested in acoustic adaptation of room and you would like to learn how different materials absorb the acoustic wave,
• Noise propagation - if you want to learn how sound intensity level changes with distance from the source,
• Sound insulation countours - if you want to learn more about acoustic insulation assessment standards used over the world,
• Sound reduction index (SRI) - if you're searching for acoustic insulation of popular building materials expressed in the coefficient Rw,
• Sound transmission class (STC) - if you're searching for acoustic insulation of popular building materials expressed by the index STC.

# Room within the room#

• A room whose all walls and ceiling are surrounded by an empty air gap (the room has no common walls with others except a common floor) is often called a room-within-the-room.
• The disadvantage of this solution is high cost and permanent modification of the building.
• For example, in order to isolate a medium-sized live room with a usable area of ​​approx. 26 m2 we need an additional approx. 24 m2 of empty space for air-gap insulation (assuming a distance between the walls of 1 m). Additionally, it is necessary to raise the entire building or to give up part of the height of the separated room (to leave an empty space under the outer room ceiling).
• For this reason, classic room-within-the-room solutions are used only in specialized buildings, which of purpose is permanently related to the need of high acoustic insulation such as recording studios or sound laboratories.