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Sound: Propagation

This topic focuses on the calculations required to predict outdoor sound propagation. The rate of propagation will depend on several factors, the type of sound source, prevailing atmospheric conditions, the impedance of the surface over which it travels and the presence of obstructions.

images/propagation.gif
Figure 1 - The propagation characteristics of a point source (left), a linear source (middle) and a planar source (right).

Point Source

The effects of geometric spreading are well known for the three idealised sound sources; the point, line and plane. The behaviour of each is based solely on the assumption that, in a homogeneous medium, sound propagation from a single point source is purely spherical. Thus the sound energy in any particular direction is inversely proportional to the increasing surface area of the sphere. If SWL represents the continuous sound power output of the source measured at 1 metre, then at a distance of r metres (where r must always be divided by the measurement distance, which is usually 1m), the sound pressure level becomes:

SPL = SWLpt - 10log (4.pi.r²)

and

I = W / (4 p r²)

This is can be rewritten simply as:

SPL = SWLpt - 20log (r) - 11

which is known as the standard inverse square law for point sources.

It is most often referred to as a 6dB reduction in relative intensity per doubling of distance. If the ground is quite hard and reflective, compensation must be made for these ground reflections. In this case 11 is replaced by 8dB.

Linear Source

Having established this basis, line and plane sources can then be considered to consist of an infinite number of evenly distributed individual point sources. The overall behaviour is then found by integrating the individual effects of each point source over the full length or area. In the case of an ideal line of infinite length, the results approximate that of purely cylindrical propagation. Thus the sound energy in any perpendicular direction is inversely proportional to the increasing circumference of the cylinder.

Using the same nomenclature as above, the sound pressure level becomes:

SPL = SWLline - 10log(4.pi.r)

This results in only a 3dB reduction in relative intensity per doubling of distance.

Plane Source

For a plane source, integrating an infinite number of point sources distributed in two dimensions produces a flat surface. Thus, propagation away from a planar source approximates a plane wave. The sound energy of each point source is therefore assumed to propagate in a straight line perpendicular to the plane, meaning that no geometric spreading need be considered as there is no change in distributed surface area as the wave propagates.

Obviously there will be some at the edges of an finite planar source, however, at close range near the centre of the plane there is no diminution with distance. Therefore, the sound pressure level can be written as:

SPL = SWLplane

Unfortunately, most real line and plane sources are of a finite size. This means that their overall behaviour becomes a definite integral. Considering this, it is easy to imagine that, at a very great distance or very small size, both sources will ultimately approximate an ideal point source.

This suggests that for such sources, there is a gradual change in behaviour as a function of both size and distance. If a represents the size of the source, then when r << a the behaviour tends toward the ideal (known as the nearfield), whereas when r >> a the behaviour tends to that of a point source (known as the farfield).

Given the order of room dimensions and the type of sources most often found in room acoustics, it is considered reasonable to deal only with farfield effects. Thus the most often used are ideal point sources, which is convenient as dealing with non-ideal sources can be complex. However, as a common source of ambient background noise is an audience spread across the floor plane, this problem is briefly considered in the interest of completeness.

Non-ideal Sound Source

Using the work of Bloemhof as a base, the integral of a finite line source may be simplified and replaced with the following adapted equation:

SPL = SWLline - 10log(b 4 p a r)

Where b is the angle subtended by the receiver and the two line ends, in radians. Thus when r << a, b tends towards p which results in the logarithm of 4a (a constant) multiplied by r, approximating the nearfield solution. When r >> a, b approaches sin(b) which itself tends towards (a/r) at roughly the same rate. Therefore, if b is replaced by (a/r), then the equation approaches the farfield solution.

For a planar source, the contribution of a circular shape is used to allow direct integration. In this case, the finite integral can be simplified and replaced with:

SPL = SWLplane - 10log(4 p a² / ln(1+(a²/r²)))

Bloemhof shows that for r >> a, ln(1+(a²/r²)) approaches (a²/r²), which approximates the farfield. The nearfield of a plane source, on the other hand, becomes slightly more complex. In this case, ln(1+(a²/r²)) approximates 2ln(a/r). Thus the resulting nearfield equation becomes:

SPL = SWL* - 10log(1 / ln(a/r))

where:

SWL* = SWL - 10log(2 p a²)

Most of the interest in line and plane sources is in the area of outdoor acoustics and community noise. In these situations, incoherent sources are often of greater importance. Geometric spreading from incoherent sources is similar to that described above but the size of the nearfield is much more restricted and propagation far less directional. Examples of an incoherent acoustic line source in these situations may include busy railway lines or major roadways. Similarly, plane sources may be considered to be as diverse as audience chatter through to traffic noise from entire city blocks.

Atmospheric Effects

Molecular Absorption

Molecular absorption refers to the attenuation of sound intensity as a result of its passage through the medium, in this case air. The mechanisms of molecular absorption are quite complex, however, the overall effect can be considered as the product of three known factors; classical absorption, rotational relaxation and vibrational relaxation. Classical absorption and the rotational relaxation of oxygen molecules are considered together due to their linear relationship with frequency.

Classical Absorption

Classical absorption is termed such as it results from the transport processes of classical physics; namely shear viscosity, thermal conductivity, mass diffusion and thermal diffusion.

Rotational Absorption

Rotational absorption, however, results from the relaxation of the rotational energies within the molecule caused by pressure changes induced by the sound wave.

Vibrational Relaxation

Vibrational relaxation within the molecules of a gas results from the vibrational storage of incident energy within the molecule rather than translational storage though the physical displacement of that molecule.

Given that this energy converts to translational energy almost immediately, the finite time period taken to do so introduces a lag in the sound wave between changes in pressure and density. This lag, therefore, is the cause of a slight reduction in the intensity of the acoustic wave.

As is the nature of the two molecules, the main effects of vibrational relaxation in oxygen and nitrogen occur at different frequencies. The effects of nitrogen on the lower portion of the audible spectrum is only a recent addition to predictive formulae, thus, many earlier methods tend to under-predict absorption below 1 - 2kHz.

The vibrational relaxation frequencies of nitrogen and oxygen molecules are, to small extent, a function of atmospheric pressure and temperature, with the main determinant being the molar concentration of water vapour within the air. Though the amount of absorption is virtually unaffected by the water vapour content, it does significantly affect the relaxation times of the two molecules, thus shifting the vibrational frequencies within the audible spectrum. The actual molar concentration at any particular time is governed by both temperature and the ratio of partial pressure to the vapour pressure at saturation of any given air sample.

NOTE: The resulting coefficient represents a reduction in sound intensity per metre distance. The major point to be considered about molecular absorption is that it is linear with distance, not logarithmic. Thus, unlike geometric spreading, its effects tend to become much more important with increasing distance.

Prevailing Winds

Winds will increase sounds downwind from a source and reduce them upwind. This is not solely a result of the velocity effect, but also because the spherical wave-front is deformed by the prevailing wind.

Figure 2 - The effects of prevailing winds on the propagation of sound. Use the slider on the right to vary the wind intensity to see its changing effect.
Figure 2 - The effects of prevailing winds on the propagation of sound. Use the slider on the right to vary the wind intensity to see its changing effect.

Whilst you will not be called upon to actually calculate the effects of winds, the resulting radius of curvature of the sound rays can be derived as follows:

1/R = (((10/T1/2)(dT/dz)) + (du/dz)) / (c(1 + u/c)²)

Where:
R = the radius of curvature (m),
T = the temperature (K),
z = the elevation (m),
c = the speed of sound (m/s) and
u = the wind speed vector in the direction of propagation (m/s).

Temperature Gradients

As discussed previously, the speed of sound is dependant on temperature, the higher the temperature, the higher the speed. This means that when the temperature near the ground is higher than that of the upper air, sound rays tend to arc upwards slightly. Thus less energy will reach a listener some distance away at ground level (For a given amount of sound energy, the distribution area is increased).

images/temp-grad.gif
Figure 3 - The effect of temperature differentials in the air on sound propagation.

At night, when the ground surface is cooler than the upper air, the inverse occurs, sound energy tends to arc downwards. (For a given amount of sound energy, the distribution area is reduced).

Sound: Human Ear
Sound: Transmission

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