1.8. Delimitaciones y limitaciones
2.1.1. Generalidades de la CPEUM
In order to derive the relationship between sound pressure levels and distance, consider a noise source which emits noise with equal power in all directions. In this case, the relationship between the power of the source and the intensity is given by Equation (2–18). Further, substituting into the definition of sound power level, Equation (2–20), yields:
𝐿𝑤 = 10 log10(4𝜋𝑟
2𝐼
𝑊𝑟𝑒𝑓)
(2–23) Assuming that the distance between the noise source and receptor is sufficiently far apart so that the noise can be described as a far-field noise, the relationship between intensity and pressure is given by Equation (2–16) and substitution into (2–23) yields:
𝐿𝑤 = 10 log10( 4𝜋𝑟2 lim 𝑇→∞∫ 𝑝 2 𝑑𝑡 ∞ 0 𝜌𝑐𝑊𝑟𝑒𝑓 ) (2–24)
which can be rewritten in terms of root mean squared sound pressure:
𝐿𝑤 = 10 log10(
4𝜋𝑟2𝑝𝑟𝑚𝑠2
𝜌𝑐𝑊𝑟𝑒𝑓
) (2–25)
The root mean squared of the sound pressure in Equation (2–25) refers to the sound pressure at distance 𝑟 from the source of the noise. Hence, rearrangement of Equation (2–25) in terms of sound pressure level will yield the relationship between sound pressure levels, sound power levels and distance:
48 𝐿𝑤 = 10 log10( 𝑝𝑟𝑚𝑠2 𝑝𝑟𝑚𝑠,𝑟𝑒𝑓2 ) + 20 log10(𝑟) + 10 log10( 4𝜋𝑝𝑟𝑚𝑠,𝑟𝑒𝑓2 𝜌𝑐𝑊𝑟𝑒𝑓 ) 𝐿𝑤 = 𝐿𝑝+ 20 log10(𝑟) + 10 log10( 4𝜋𝑝𝑟𝑚𝑠,𝑟𝑒𝑓2 𝜌𝑐𝑊𝑟𝑒𝑓 ) 𝐿𝑝 = 𝐿𝑤 − 20 log10(𝑟) − 10 log10( 4𝜋𝑝𝑟𝑚𝑠,𝑟𝑒𝑓2 𝜌𝑐𝑊𝑟𝑒𝑓 ) (2–26)
The outcome of Equation (2–26) yields several insights. First, the sound pressure level is directly proportional to the sound power level. This implies that the louder the noise emitted at the source, the louder the noise perceived at the receptor. Further, the sound pressure level is inversely related to the distance from the source, i.e., as the distance between the noise source and receptor increases, the loudness of the sound perceived at the receptor falls.
Another formulation of Equation (2–26) is to consider two receptors of distance 𝑟1 and 𝑟2 from a noise source (Bies & Hansen 2009). The sound pressure levels experienced by these two receptors are 𝐿𝑝1 and 𝐿𝑝2 respectively. Since the noise sound, acoustic medium, reference power, and pressure are the same for the two receptors, Equation (2–26) can be simplified to:
𝐿𝑝1− 𝐿𝑝2 = 20 log10(𝑟2 𝑟1
) (2–27)
Equation (2–27) also illustrates the inverse square law relationship, which states that sound level decay by 6dB per doubling of distance from a point source (since 20 log10(2) ≈
6dB).
The results from Equation (2–26) illustrate the relationship between sound pressure levels, sound power levels, and distance for a non-directional spherical emitter of noise through a lossless medium. In reality, the source may be directional, i.e., there is an uneven distribution of the sound intensity as a function of direction (Beranek 1986). The medium that noise passes through may also absorb energy from the sound wave (Sutherland et al. 1974). For example, air could dissipate sound energy through friction between the air molecules as well as by molecules absorbing the sound energy and then re-radiating the sound at a later instant. The sound could also pass through noise barriers which seek to lower the loudness of the sound (Maekawa 1968). Hence, Equation (2–26) can be rewritten to take into account these factors:
𝐿𝑝 = 𝐿𝑤 − 20 log10(𝑟) − 10 log10(4𝜋𝑝𝑟𝑚𝑠,𝑟𝑒𝑓
2
𝜌𝑐𝑊𝑟𝑒𝑓 ) + 𝐷𝐼 − 𝐴𝑎𝑏𝑠− 𝐴𝐸
49
where 𝐷𝐼 is the directivity index, which is a function describing the difference between the actual sound pressure in each direction and the sound pressure from a non-directional point source with the same acoustic power. 𝐴𝑎𝑏𝑠 is a function describing the absorption from the air. 𝐴𝐸 is excess attenuation, which is the total attenuation in addition to the attenuation due to geometry and atmospheric absorption.
Consequently, to describe the transfer function, 𝜑(⋅), introduced in Section 2.1, it is possible to use Equation (2–28). Equation (2–28) provides insight into the key factors which affect the transfer function. First, lower levels 𝐿𝑤 will lead to lower levels of 𝐿𝑝, indicating that source control of noise will reduce the sound power level at the source which, in turn, reduces the sound pressure level at the receptor.
Second, an increase in the distance between the noise source and the receptor will reduce loudness at the receptor. Hence, zoning of activities, which physically separates noise sources and receptors, may reduce the loudness of sounds perceived at the receptor.
Third, path control of noise seeks to increase 𝐴𝐸, which also decreases the sound pressure level at the receptor (Maekawa 1968).
Fourth, air absorption, 𝐴𝑎𝑏𝑠, reduces the loudness of noise as it travels through the air. The effect of air absorption is defined to be:
𝐴𝑎𝑏𝑠 = 𝛼𝑟
100
(2–29) where 𝛼 is the absorption coefficient measured in dB per 100m and 𝑟 is measured in metres.
Piercy, Embleton & Sutherland (1977) found that the absorption coefficient is dependent on the frequency of the noise. Absorption tends to be higher for higher frequency noise and as the distance between the noise source and receptor increases, the noise tends to have lower frequencies, since higher frequencies would be absorbed by the air molecules. Piercy, Embleton & Sutherland analysed laboratory and field measurements of noise over a wide range of frequencies and estimated that the relationship between the absorption coefficient, measured in dB per 100m change in distance between source and receptor, and frequency is described by Figure 2-4. The researchers found that loudness decreased by less than 0.01dB for every 100m increase in distance if the frequency of the noise was less than 100Hz, compared to more than 100dB decrease if the frequency was more than 10kHz. Bass,
50
Sutherland & Zuckerwar (1990) extended the study by Piercy, Embleton & Sutherland to take into account the effect of humidity on the absorption coefficient.
Figure 2-4 Predicted atmospheric absorption at a pressure of one atmosphere, temperature of 20°C and relative humidity of 70%
Source: Piercy, Embleton & Sutherland 1977
In practice, solutions to Equation (2–28) may be difficult to estimate, given the large number of parameters and the complex functional forms of 𝐷𝐼, 𝐴𝑎𝑏𝑠, and 𝐴𝐸. The complexity of Equation (2–28) motivates the use of rule-of-thumb estimates of the transfer function, such as the inverse square law relationship described in Equation (2–27). The inverse square law relationship is commonly used by policy-makers when explaining noise regulations to the public (NSW-EPA 2013).