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 particular lecture is going to concentrate in 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:

	Wreceiver
t =
	Wsource

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 (Wsource / Wreceiver)

　

= 10 log (1/t)

= -10 log (t)

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 (2pfm / roc) dB

= 20 log (fm) - 47 dB

where f is the frequency (Hz), m is the mass per unit area (kg/m²) and roc is 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).

Resonance and Coincidence effects

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.

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 (cf: Diagram 1.1).

Stiffness Controlled Region

At low frequencies (for most building materials below 10-20Hz), 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 = [E / (p * (1 - s^2))]^1/2

where b is the panel thickness (m), l and h it's length and height (m) and vL is the longitudinal velocity of sound in the partition (m/s) [where E is Young's modulus of elasticity, s is it's Poisson ratio and p it's density]. To calculate harmonic frequencies, simply replace the number 1 in the first equation with the required harmonic number.

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

The critical frequency is the frequency at which the wavelength of bending waves in the wall match those of the incident sound. Bending waves of different frequencies travel at different speeds, the velocity increasing with frequency.

This means that for every frequency above a certain critical frequency, there in an angle of incidence for which the wavelength of the bending wave can become equal to the wavelength of the impacting sound. This condition is known as coincidence (cf: Diagram 1.2).

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 is the speed of sound in air (m/s), h is the panel thickness (m), vL is the longitudinal velocity of sound in the partition (m/s) and a is the angle of incidence.

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

Altering the TL of a panel

• 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.

Composite 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 (think about it).

A situation approaching this is possible if the two walls are 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 mis-matching 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.

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...

Thus, it is better to have a well-fitting light door than a loose-fitting heavy one. In the next session we will discuss why the SRI of a composite panel is dominated by it's weakest element.

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;

(a) when W << Wn,  stiffness dominates,
(b) when W == Wn,  damping dominates and
(c) 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.

This topic concentrates on composite partitions and methods of rating the insulation of partitions, as well as some considerations when putting together various building elements.

Transmission Loss of Composite Partitions

If a partition is composed of more than one element, for example a wall and a door, then the effective transmission coefficient must be found as an average of the area weighted sum of each component's transmission loss.

	S (An tn)
tAve =
	S An

then: SRI = -10 log (tAve)

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) (cf: diagram 1.1).

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 (cf: example). As always, there are computer programs which calculate this value much faster and easier than hand calculations.

CONSTRUCTIONAL CONSEQUENCES

Doors

The insulation provided by a door does not follow the predictions of the mass law for two basic reasons:

　

a) there are nearly always small gaps between the door and door frame

　

b) the size of the door is usually much smaller than the wall in which it is placed so the resonant frequencies of the door occur at a higher frequency.

Therefore, when high insulation is required, the edges of the door should be sealed very carefully with gaskets of felt or rubber.

Values of insulation greater than those predicted by the mass law can be obtained through the use of double doors. For double doors separated by at least 8cm, the average insulation is usually 5 dB greater. If the two doors are separated by a short passageway, and possibly lined with absorbant material, then the increase can be as much as 10-12 dB.

Outer Walls and Windows

Most of the noises that disturb people occur out of doors. The noise from such sources enter buildings throught the outer walls, the windows and the roof. Many modern buildings have roofs made of concrete (or other comparably heavy materials) so these do not pose as much of a problem as the outer walls and windows.

The insulation of outer walls is usually (but not always) determined by the insulation of the windows. When a high degree of insulation is required, it is essential that fixed windows be used (ie: not openable), which may mean the use of mechanical ventilation. The insulation of windows is slightly more difficult to estimate than a solid wall because it is more dependant on the window's dimensions and coincidence plays a much more important role.

The insulation curve of a single glazed window is typically marked by a deep trough in the mid-frequency range, at which the ear is most sensitive. Whilst not a major problem with traffic noise (given the predominace of low frequencies), aircraft noise shows up a window's poor insulation properties at higher frequencies.

As with a wall, an improvement in the insulation of a window can be obtained through the use of double-leaf construction. As the mass of each glass pane is relatively low, the thickness of the air gap should be quite large in order to act as a spring, reducing the resonant frequency.

In some countries double glazing is often used for thermal reasons before acoustic. Unfortunately, such constructions usually have a very narrow cavity (only 10-12 mm). Consequently the two panes are closely coupled and the resonant frequency is around 300Hz.

For increased insulation, the gap should be at least 75 mm (preferably 100 mm), with sound absobant material placed around the perimeter and different thicknesses used in each layer (even angled slightly differently).

Given local conditions, it is often more appropriate to provide natural ventilation. This can cause problems when there are high levels of background noise immediately outside a window. To overcome this, acoustic baffles can be used.

Floating Floors

As well as normal transmission, floors require special consideration due to impact noise. Often a floor that is a good insulator (say a concrete slab) may be unacceptable when considering the transmission of impact noise. An obvious solution to the problem of impact insulation is to cover the floor with a resilient layer such as carpet or rubber tiling. Such floor coverings are most effective in reducing the higher frequencies of the impact noise but may conflict with other considerations such as durability and resistance to chemical attack.

To overcome this, it is possible to use a floor composed of a hard upper layer and a resilient lower layer which rests on the structural floor. In such a construction it is vitaly important that the resilient layer is not bridged at any point. This includes service pipes and conduits.

Floating Rooms

When very high insulation values are required, a discontinuous construction may be considered. This means that the entire room is completely separated from the main structure of the building, supported only by vibration-isolating mountings. Once again, small inadvertant sound bridges can undo a lot of money spent isolating a room. Therefore great care must be taken not to bridge the gap between the two rooms.

Ceilings

Ceilings can only really assist in the insulation of impact noise from above. This can be achieved to some degree using either a false or suspended ceiling. A false ceiling is one which is independant of the structure above it, supported by side walls whilst a suspended ceiling is hung from the structure by wires or resilient hangers.

Given that most proprietry ceiling systems are lightweight and modular, they provide flanking paths around each panel and suffer from a lack of mass. They can be made much more effective with the use of sealants and heavier materials, however, the additional structure required for support may make such a construction uneconomical.