Loudness of Sound. The loudness L, in bels (after Alexander...

Loudness of Sound. The loudness $L$, in bels (after Alexander Graham Bell), of a sound of intensity $I$ is defined to be $$ L=\log \frac{I}{I_{0}} $$ where $I_{0}$ is the minimum intensity detectable by the human ear (such as the tick of a watch at 20 ft under quiet conditions). If a sound is 10 times as intense as another, its loudness is 1 bel greater than that of the other. If a sound is 100 times as intense as another, its loudness is 2 bels greater, and so on. The bel is a large unit, so a subunit, the decibel, is generally used. For $L$, in decibels, the formula is $$ L=10 \log \frac{I}{I_{0}} $$ Find the loudness, in decibels, of each sound with the given intensity. \begin{tabular}{ll} \hline Sounp & INTENSITY \\ \hline a) Jet engine at 100 ft & $10^{14} \cdot I_{0}$ \\ b) Loud rock concert & $10^{11.5} \cdot I_{0}$ \\ c) Train whistle at 500 ft & $10^{9} \cdot I_{0}$ \\ d) Normal conversation & $10^{6.5} \cdot I_{0}$ \\ e) Trombone & $10^{10} \cdot I_{0}$ \\ f) Loudest sound possible & $10^{19.4} \cdot I_{0}$ \end{tabular}

Loudness of Sound. The loudness $L$, in bels (after Alexander Graham Bell), of a sound of intensity $I$ is defined to be
$$
L=\log \frac{I}{I_{0}}
$$
where $I_{0}$ is the minimum intensity detectable by the human ear (such as the tick of a watch at 20 ft under quiet conditions). If a sound is 10 times as intense as another, its loudness is 1 bel greater than that of the other. If a sound is 100 times as intense as another, its loudness is 2 bels greater, and so on. The bel is a large unit, so a subunit, the decibel, is generally used. For $L$, in decibels, the formula is
$$
L=10 \log \frac{I}{I_{0}}
$$
Find the loudness, in decibels, of each sound with the given intensity.
\begin{tabular}{ll}
\hline Sounp & INTENSITY \\
\hline a) Jet engine at 100 ft & $10^{14} \cdot I_{0}$ \\
b) Loud rock concert & $10^{11.5} \cdot I_{0}$ \\
c) Train whistle at 500 ft & $10^{9} \cdot I_{0}$ \\
d) Normal conversation & $10^{6.5} \cdot I_{0}$ \\
e) Trombone & $10^{10} \cdot I_{0}$ \\
f) Loudest sound possible & $10^{19.4} \cdot I_{0}$
\end{tabular}

Key Concepts

Loudness Calculation

The formula for calculating the loudness of a sound in decibels is L = 10 log(I/I?), where I is the sound intensity and I? is the reference intensity. This calculation transforms the ratio of the actual sound intensity to the reference intensity into a decibel value, making it possible to quantify and compare sounds across a broad range of intensities using a scale that is consistent with human hearing behavior.

Sound Intensity

Sound intensity is a measure of the energy that a sound wave transports per unit area and per unit time. It is an important physical quantity in acoustics that determines how strong or powerful a sound is perceived. More intense sounds transfer more energy, and their measurement is fundamental in understanding the behavior and effects of sound waves in various media.

Reference Intensity

The reference intensity, often denoted as I?, is the threshold of human hearing. It represents the smallest sound intensity that the average human ear can detect under ideal conditions. Using a common reference point allows for a standardized way to compare different sound intensities, regardless of the absolute energy values involved.

Logarithmic Scale

The logarithmic scale is used to convert the wide range of sound intensities into a manageable scale for human perception. Because the human ear perceives sound in a multiplicative rather than linear manner, the logarithmic scale allows for a more intuitive comparison of how much louder one sound is compared to another. In logarithmic terms, each multiplicative increase in intensity corresponds to an additive increase in perceived loudness.

Bel and Decibel Units

The bel is a unit of measurement for sound level, where a change of one bel represents a tenfold increase in sound intensity. However, since the bel is a relatively large unit, the decibel, which is one-tenth of a bel, is more commonly used in practice. This subdivision makes it easier to express and compare the differences in sound levels that the human ear can perceive.

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