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.