The loudness L, in bels (after Alexander Graham Bell), of a...

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. SOUND a) Jet engine at $100 \mathrm{ft}$ b) Loud rock concert c) Train whistle at $500 \mathrm{ft}$ d) Normal conversation e) Trombone f) Loudest sound possible INTENSITY $10^{14} \cdot I_{0}$ $10^{11.5} \cdot I_{0}$ $10^{9} \cdot I_{0}$ $10^{6.5} \cdot I_{0}$ $10^{10} \cdot I_{0}$ $10^{19.4} \cdot I_{0}$

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.
SOUND
a) Jet engine at $100 \mathrm{ft}$
b) Loud rock concert
c) Train whistle at $500 \mathrm{ft}$
d) Normal conversation
e) Trombone
f) Loudest sound possible
INTENSITY
$10^{14} \cdot I_{0}$
$10^{11.5} \cdot I_{0}$
$10^{9} \cdot I_{0}$
$10^{6.5} \cdot I_{0}$
$10^{10} \cdot I_{0}$
$10^{19.4} \cdot I_{0}$

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Key Concepts

Reference Intensity

Reference intensity, denoted as I?, is the minimum sound intensity detectable by the average human ear. It acts as a baseline in the formula, so that any measured sound's intensity is expressed as a multiple of this minimum threshold.

Decibel Unit

The decibel is a unit of measurement for sound intensity level that is one-tenth of a bel. It is used because a bel is relatively large, and the decibel provides a finer, more practical scale for everyday sound measurements. The formula L = 10 log(I/I?) converts the ratio of intensities into decibels, which is a more intuitive representation of perceived loudness.

Logarithmic Scale

The use of a logarithmic scale for measuring loudness reflects the human ear’s perception of sound, which can sense a vast range of intensities. By converting the ratio of an actual sound intensity to a reference intensity into a logarithm, even huge variations in intensity translate into manageable numerical values.

Sound Intensity

Sound intensity refers to the power per unit area carried by a sound wave. It is a physical measure of the energy of the sound and, when compared to the reference intensity, provides a basis for the logarithmic calculation of loudness in decibels.

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