The decibel level of a noise is defined in terms of the...

The decibel level of a noise is defined in terms of the intensity $I$ of the noise, with $\mathrm{dB}=10 \log \left(I / I_{0}\right) .$ Here, $I_{0}=10^{-12} \mathrm{W} / \mathrm{m}^{2}$ is the intensity of a barely audible sound. Compute the intensity levels of sounds with (a) $\mathrm{dB}=80,$ (b) $\mathrm{dB}=90$ and $(\mathrm{c})$ $\mathrm{dB}=100 .$ For each increase of 10 decibels, by what factor does I change?

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Key Concepts

Intensity Multiplication Factor

In the context of the decibel scale, an increase of 10 decibels always corresponds to a tenfold increase in the sound intensity. This property is a direct consequence of the logarithmic nature of the decibel definition.

Reference Intensity

The reference intensity (I?) is typically defined as 10?¹² W/m², representing the threshold of human hearing. By comparing measured intensities to this reference, the decibel system provides a standardized means of quantifying sound levels.

Logarithmic Relationship

The use of the logarithmic function in the decibel scale (dB = 10 log(I/I?)) accounts for the human ear’s response to sound. This function converts multiplicative differences in sound intensity into additive differences in decibels, making it easier to interpret changes in sound power.

Decibel Scale

The decibel, abbreviated as dB, is a logarithmic unit used to express the ratio of a given sound's intensity relative to a fixed reference intensity. This scale is particularly useful in acoustics for comparing sound levels because it compresses a wide range of values into a manageable format.

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