A law of physics states that the intensity of sound is...

A law of physics states that the intensity of sound is inversely proportional to the square of the distance $d$ from the source: $I=k / d^{2} .$ (a) Use this model and the equation $$B=10 \log \frac{I}{I_{0}}$$ (described in this section) to show that the decibel levels $B_{1}$ and $B_{B}$ at distances $d$ and $d_{2}$ from a sound source are related by the equation $$B_{2}=B_{1}+20 \log \frac{d_{1}}{d_{2}}$$ (b) The intensity level at a rock concert is 120 $\mathrm{dB}$ at a distance 2 $\mathrm{m}$ from the speakers. Find the intensity level at a distance of 10 $\mathrm{m} .$

A law of physics states that the intensity of sound is inversely proportional to the square of the distance $d$ from the source: $I=k / d^{2} .$
(a) Use this model and the equation
$$B=10 \log \frac{I}{I_{0}}$$
(described in this section) to show that the decibel levels $B_{1}$ and $B_{B}$ at distances $d$ and $d_{2}$ from a sound source are related by the equation
$$B_{2}=B_{1}+20 \log \frac{d_{1}}{d_{2}}$$
(b) The intensity level at a rock concert is 120 $\mathrm{dB}$ at a distance 2 $\mathrm{m}$ from the speakers. Find the intensity level at a distance of 10 $\mathrm{m} .$

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Key Concepts

Scaling Relationships in Acoustics

This concept involves understanding how changes in distance from the sound source affect the measured sound intensity levels. Using the inverse square law combined with logarithmic conversion, one can derive formulas that relate sound levels at different distances, facilitating practical applications such as noise control and audio system design.

Logarithmic Relationships

Logarithms are used to convert multiplicative relationships into additive ones. In the context of sound intensity, logarithmic properties enable the transformation of an inverse square law into a linear relationship with distance when expressed in decibels, making it easier to compare sound levels over varying distances.

Decibel Scale

The decibel scale is a logarithmic measure used to express the ratio of a physical quantity, typically sound intensity, relative to a defined reference level. The logarithmic nature of the scale allows it to handle the wide range of intensities that the human ear can perceive in a compact numerical format.

Inverse Square Law

This principle states that a physical quantity (such as sound intensity) decreases proportionally to the square of the distance from its source. It is fundamental in physics for understanding how energy spreads out over an area as distance increases.

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