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Sound: Transmission

When a sound wave impacts upon the surface of a solid body, some portion of it's energy will be reflected, some absorbed and the rest transmitted through the body. The relative proportion of each depends on the nature of the material impacted. This topic concentrates on the transmitted component.

Transmission Loss

If we consider the transmission of sound through a partition, we can actually measure the sound energy on both the source side (Wsrc) and the receiving side (Wrec) to determine exactly what fraction of the sound is transmitted through. We can thus determine the transmission coefficient (t) for that partition as follows:

t  =  Wrec / Wsrc

The term Transmission Loss (TL), or more commonly Sound Reduction Index (SRI) are used to describe the reduction in sound level resulting from transmission through a material. This is given by:

SRI = 10 log (Wsrc / Wrec)
    = 10 log ( 1/t )
    = -10 log (t)

Composite Partitions

If a partition is composed of more than one element, for example a wall with a door and a window in it, then the effective transmission coefficient must be found as an average of the area weighted sum of each component's transmission loss. If the partition has n separate elements, then the average transmission is given by:

tAve = S (An tn) S An

Thus:

SRIAve = -10 log (tAve)

Frequency Dependence

Unfortunately, the SRI of nearly all materials varies with frequency. The main effect is the mass law, with the effects of resonance and coincidence also contributing. Thus, SRI values are normally shown as a curve within a graph, as shown in Figure 4 below. However, it is possible to use a single SRI value when dBA or dBB sound weighting curves have been applied.

Figure 1 - SRI curves for some example materials.
Figure 1 - SRI curves for some example materials.

The Mass Law

Obviously, the greater the mass of the wall, the greater the sound energy required to set it in motion. The mass law states that every doubling of the mass of a partition will result in a 6 dB reduction in the level of sound transmitted through it. It is given by;

R = 20 log (2p f m / roc) dB
  = 20 log (f m) - 47 dB
where
f = the frequency (Hz),
m
= the mass per unit area (kg/m²), and
roc = the characteristic impedance of air (basically, density times the speed of sound: taken to be between 410 and 420 rayls for 20°C and 1 atm).

The mass law applies strictly to limp, non-rigid partitions. However, most materials used in buildings possess some rigidity or stiffness. This means that other factors must really be considered, and that the mass law should only be taken as an approximate guide to the amount of attenuation obtainable.

Resonance and Coincidence Effects

Sound attenuation in ordinary building materials is the result of an interplay between mass, stiffness and damping. In addition, the mass law is affected by resonance at lower frequencies and coincidence at higher frequencies, as shown in Figure 1 below.

Figure 2 - Graph of resonance and coincidence effect.
Figure 2 - Graph of resonance and coincidence effect.

Stiffness Controlled Region

At low frequencies (for most building materials below 100Hz), transmission depends mainly on the stiffness of the wall, with damping and mass having little effect. The effectiveness of stiffness in the attenuation of sound transmission decreases by 6dB for every doubling of frequency (one octave).

Resonant Frequencies

At slightly higher frequencies the resonance of the wall begins to control its transmission behaviour. Because every panel has a finite boundary and edge fixings, it will have a series of natural frequencies at which it will vibrate more easily than others. These are called resonant frequencies and consist of a fundamental frequency (having the greatest effect), and integer multiples of this fundamental called harmonics (having less and less effect). The fundamental resonant frequency of a panel can be calculated as follows:

Fr = 0.45 * vL * b((1/l)² + (1/h)²)

and:

vl = sqrt(E / (p * (1 - s²)))

where:
b = the panel thickness (m),
l and h = length and height(m), and
vl = the longitudinal velocity of sound in the partition (m/s).

In the calculation of vl:
E = Young's modulus of elasticity,
s = it's Poisson ratio, and
p = density (kg/m³).

To calculate harmonic frequencies, simply replace the number 1 in the first equation with the required harmonic number.

Figure 3 - Resonance occurs when a stiff panel flexes as a result of incident sound waves.
Figure 3 - Resonance occurs when a stiff panel flexes as a result of incident sound waves.

Mass Controlled Region

At frequencies well above that of the lowest resonant frequency, the wall tends to behave as an assembly of much smaller masses and is then said to be mass controlled. It is within this range that the mass law directly applies.

Critical Frequency and Coincidence

High frequencies cause bending or ripple waves that travel longitudinally along a wall or panel. The wavelength of a bending wave is different from that of the incident sound wave which created it except at one frequency, the critical frequency. Unlike compressional waves, bending waves of different frequencies travel at different speeds. This means that for every frequency above the critical frequency, there will be an angle of incidence at which the wavelength of the bending wave is equal to the wavelength of the impacting sound. This condition is known as coincidence.

Figure 4 - The coincidence effect when 'ripples' in a material are created by incident sound waves.
Figure 4 - The coincidence effect when 'ripples' in a material are created by incident sound waves.

When coincidence occurs it gives rise to a far more efficient transfer of sound energy from one side of the panel to the other, hence the big coincidence-dip at the critical frequency. In many thin materials (such as glass and sheet-metal), the coincidence frequency begins somewhere between 1000 and 4000 Hz, which includes important speech frequencies.

The lowest frequency at which coincidence can occur is when the angle of incidence of the sound is at 90° (grazing incidence) and can be calculated from:

Fc = c² / (1.8 * h * vl * sin²(a))

where:
c = the speed of sound in air (m/s),
h = the panel thickness (m),
vl = the longitudinal velocity of sound in the partition (m/s), and
a = the angle of incidence.

Above the critical frequency, panel stiffness begins to play the most important role again.

Sound Transmission Class

To avoid the misleading nature of an average SRI value and to provide a reliable single-figure rating for comparing partitions, the sound transmission class rating procedure has been widely adopted. According to this procedure, the STC of a partition is determined by comparing the 16-frequency SRI curve with a standard reference contour. This contour consists of 3 segments with different vertical increments, 125-400Hz (15 dB), 400-1250Hz (5 dB) and 1250-4000Hz (0 dB), as shown in the Figure below.

images/stc.gif
Figure 5 - Sound Transmission Class (STC) curves.

The calculation of this value, whilst not necessarily complex, is quite laborious. It is found by shifting this contour vertically until some of the measured values fall below the STC curve and the following two conditions are met:

  1. The sum of all the deficiencies do not exceed 32 dB.
  2. The maximum deficiency at any frequency does not exceed 8 dB.

This shifting is always done in integer steps and, when a matching position is found, the final STC rating is given by the value of the reference curve at 500 Hz. The SoundTool is a software program which calculate this value much faster and easier than hand calculations.

Altering the Transmission Loss of a Panel

Resonance and coincidence effects cannot be eliminated. If the designer aims to create the maximum SRI, an attempt should be made to get resonant frequencies as low as possible (preferably well below the audible range) and the critical frequency as high as possible (preferably well above the audible range). Whilst it is not possible to apply a generic solution to all panels, the following general relationships do hold:

  • Reducing the stiffness of a panel lowers it's resonant frequency and raises it's critical frequency, basically increasing the region for which the mass law applies.
  • Increasing panel mass also lowers resonant frequencies and raises the critical frequency.
  • Decreasing panel thickness raises the critical frequency but generally reduces panel mass.
  • Increasing the amount of damping applied to the panel will not alter the frequencies of resonance and coincidence but will act to reduce their effect.

Good insulation is therefore a combination of low stiffness, high mass and high damping (given cost constraints).

NOTE: The most common method of adding damping is to apply a thick layer of mastic-like material to one side of the panel. This type of treatment is only effective on materials that have low mass and an inherent lack of damping. It would be useless on thick concrete walls, for example, but very effective on metal automobile panels.

Multi-Layer Partitions

As just discussed, the insulation of a single-leaf panel can be improved in a number of ways, but this process can only continue up to a certain point given the exponential increase in mass required. Consider the example of a single brick wall with an SRI of 22dB. To increase this to an overall 40dB in all regions, the mass must be increased to 8 times the original (2^3). This is clearly impractical from a building perspective.

Consider, on the other hand, the fact that the wall already has a 22dB SRI. If we were to build another brick wall right next to it, we could (in theory) achieve a further drop of 22dB. A situation approaching this is possible if the two walls were completely separated from each other with no common links, footings or edge supports, and an air gap greater than a metre between them.

Unfortunately, this is often just as impractical as vastly increasing the mass of the wall. In practice, walls do have common supports at the edges. It is also rare to find a cavity wall with more than few centimetres of air gap.

On the other hand, it is possible to create composite or sandwich panels whose total SRI does approach that of a double wall, if the following points are considered:

  • Well sealed cavities can result in an increase in sound insulation well above mass law (6-8dB), assuming the cavity is at least 100mm deep.
  • Use of layers of different thickness can greatly assists in mismatching resonant and critical frequencies across the panel.
  • The use of absorbent materials within the cavities can help to further reduce transmission.
  • Only resilient elastic materials should be used as wall ties and suspension members to reduce any direct connection between layers.
  • If required, only widely spaced and staggered studs should be used within partitions.
  • Caulking and sealants should be used to eliminate perimeter sound leaks.

NOTE: The very last point is quite important as it alludes to flanking. The highest achievable SRI value for a partition is about 55-60dB.

Above 45-50dB, flanking paths become more and more important. This explains why multiple-layer (three or more) partitions do not offer any significant improvement over double-leaf construction.

The following are some examples of different building sections and their corresponding transmission loss values. It is worth spending some time looking at these details as it will give you some idea as to the requirements to meet different values.

Flanking

There are often several other paths sound can follow apart from the direct path through the panel. These include air conditioning ducts, through ceiling spaces, around edge fixings, etc. As the designer, you must always be thinking about possible flanking paths whenever you are doing acoustically sensitive details.

This applies to air seals as well - it is often better to have a tight-fitting lightweight door than a loose-fitting heavy one.

Figure 6 - Two different partition details illustrating the effects of flanking.

For Those Interested

Some clarifying points [From Norton, M.P., Fundamentals of Noise and Vibrational Analysis for Engineers. Section 3.9].

  1. If Wn is the natural frequency of a panel and W is the frequency of excitation:
    • when W << Wn, stiffness dominates,
    • when W == Wn, damping dominates and
    • when W >> Wn mass dominates.
  2. If a panel is mechanically excited, most of the energy is produced by resonant panel modes irrespective of W.
  3. If a panel is acoustically excited by incidence, its vibrational response comprises both a forced vibrational response at W and a resonant response at all relevant natural frequencies which are excited by the interaction of the forced bending waves with the panel boundaries.

Related Links

Transmission Loss Explained
http://www.domesticsoundproofing.co.uk/tloss.htm
Sound: Propagation
Sound Transmission: Examples

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