The Human Auditory Response

One of the most frequent applications of the decibel scale is in the communication and entertainment industries. The human ear does not respond in a linear fashion to changes in source power level; that is, a doubling of the audio power level from $1/2 \mathrm{W}$ to $1 \mathrm{W}$ does not result in a doubling of the loudness level for the human ear. In addition, a change Rom $5 \mathrm{W}$ to $10 \mathrm{W}$ will be received by the ear as the same change in sound intensity as experienced from $1/2 \mathrm{W}$ to $1 \mathrm{W}$. In other words, the ratio between leveIs is the same in each case
$$(1\mathrm{W}/\mathrm{O}.5\mathrm{W}= 10\mathrm{W}/5\mathrm{W}=2), $$
$$\bbox[10px,border:1px solid grey]{\log _{10} a^{n}=n \log _{10} a} \tag{1}$$
resulting in the same decibel or logarithmic change defined by Eq. (1). The ear, therefore, responds in a logarithmic fashion to changes in audio power levels. To establish a basis for comparison between audio levels, a reference level of 0.0002 microbar $(\mu \mathrm{bar})$ was chosen, where 1 $\mu \mathrm{bar}$ is equal to the sound pressure of 1 dyne per square centimeter, or about 1 millionth of the normal atmospheric pressure at sea level. The $0.0002-\mu \mathrm{bar}$ level is the threshold level of hearing. Using this reference level, the sound pressure level in decibels is defined by the following equation:
$$\bbox[10px,border:1px solid grey]{\mathrm{dB}_{s}=20\log_{10}\frac{P}{0.0002\mu \mathrm{bar}} }\tag{2}$$
where $P$ is the sound pressure in microbars.
Typical sound levels and their decibel levels.
Fig. 1:Typical sound levels and their decibel levels.
The decibel levels of Fig. 1 are defined by Eq. (2). Meters designed to measure audio levels are calibrated to the levels defined by Eq. (1) and shown in Fig. 1. A common question regarding audio levels is how much the power level of an acoustical source must be increased to double the sound level received by the human ear. The question is not as simple as it first seems due to considerations such as the frequency content of the sound, the acoustical conditions of the surrounding area, the physical characteristics of the surrounding medium, and of course the unique characteristics of the human ear.
However, a general conclusion can be formulated that has practical value if we note the power levels of an acoustical source appearing to the left of Fig. 1. Each power level is associated with a particular decibel level, and a change of $10 \mathrm{dB}$ in the scale corresponds with an increase or a decrease in power by a factor of 10. For instance, a change from $90 \mathrm{dB}$ to $100 \mathrm{dB}$ is associated with a change in wattage from $ 3\mathrm{W}$ to $ 30 \mathrm{W}$.
Through experimentation it has been found that on an average basis the loudness level will double for every $10 \mathrm{dB}$ change in audio level--a conclusion somewhat verified by the examples to the right of Fig. 1. Using the fact that a $10 \mathrm{dB}$ change corresponds with a tenfold increase in power level supports the following conclusion (on an approximate basis): Through experimentation it has been found that on an average basis, the loudness level will double for every $10 \mathrm{dB}$ change in audio level.
To double the sound level received by the human ear, the power rating of the acoustical source (watts) must be increased by a factor of $10$.
In other words, doubling the sound level available from a 1-W acoustical source would require moving up to a 10-W source.

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