The loudness L(x), measured in decibels (d B), of a sound of...

The loudness $L(x),$ measured in decibels $(d B),$ of a sound of intensity $x,$ measured in wans per square meter, is defined as $L(x)=10 \log \frac{x}{I_{0}},$ where $I_{0}=10^{-12}$ watt per square meter is the least intense sound that a human ear can detect. Determine the loudness, in decibels, of each of the following sounds. Diesel truck traveling 40 miles per hour 50 feet away: intensity 10 times that of a passenger car traveling 50 miles per hour 50 feet away, whose loudness is 70 decibels.

The loudness $L(x),$ measured in decibels $(d B),$ of a sound of intensity $x,$ measured in wans per square meter, is defined as $L(x)=10 \log \frac{x}{I_{0}},$ where $I_{0}=10^{-12}$ watt per square meter is the least intense sound that a human ear can detect. Determine the loudness, in decibels, of each of the following sounds.
Diesel truck traveling 40 miles per hour 50 feet away: intensity
10 times that of a passenger car traveling 50 miles per hour
50 feet away, whose loudness is 70 decibels.

Key Concepts

Reference Intensity

Reference intensity is the baseline or threshold value of sound intensity that is used in calculations involving the decibel scale. It represents the lowest intensity that the average human ear can detect, making it essential for standardizing loudness measurements.

Intensity Multiplication Factor

Intensity multiplication factors are used to compare the loudness of two sounds. When one sound's intensity is a multiple of another, the corresponding decibel difference can be computed using the properties of logarithms. This concept is key in understanding how changes in intensity translate to changes on the decibel scale.

Sound Intensity

Sound intensity refers to the power carried by sound waves per unit area, typically measured in watts per square meter. It quantifies how much sound energy passes through a specific area and is crucial for assessing the loudness of a sound.

Decibel Scale

The decibel scale is a logarithmic unit used to express ratios of power or intensity, particularly in acoustics. It is designed to mimic the human ear's perception of sound, where increases in loudness are perceived logarithmically rather than linearly.

Logarithmic Function

Logarithmic functions are mathematical functions that help transform multiplicative relationships into additive ones. In the context of acoustics, they are used to relate sound intensity, which can vary over many orders of magnitude, to a more manageable scale such as decibels.

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