PROVE: Inverse Square Law for Sound A law of physics states...

PROVE: Inverse Square Law for Sound 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 $B_{2}$ at distances $d_{1}$ 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 $10 \mathrm{m}$.

PROVE: Inverse Square Law for Sound 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 $B_{2}$ at distances $d_{1}$ 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 $10 \mathrm{m}$.

Key Concepts

Logarithmic Properties

Logarithmic properties, such as log(a/b) = log(a) - log(b) and the power rule log(a^b) = b log(a), are essential tools in manipulating logarithmic expressions. In the context of sound intensity, these properties are used to convert the inverse square law into an additive relationship when expressed in decibels, thereby simplifying the calculations of intensity differences over varying distances.

Application of Logarithms to Sound Intensity

By combining the inverse square law with the decibel equation, one can derive a formula that relates decibel levels at different distances. The process involves expressing the ratio of intensities gained from the inverse square law, then applying the logarithmic properties to convert this ratio into a difference in decibel levels. This results in a formula that distinctly shows how changes in distance affect sound level in a logarithmic manner.

Decibel Scale

The decibel scale is a logarithmic unit used to express the ratio of a physical quantity, typically sound intensity, relative to a reference value. In acoustics, it is defined by the formula B = 10 log(I/I?), where I is the intensity of the sound and I? is a reference intensity. This logarithmic approach allows a wide range of sound intensities to be represented in a more manageable form.

Inverse Square Law

The inverse square law states that the intensity of a physical quantity such as sound or light decreases in proportion to the square of the distance from the source. This means that if you double the distance, the intensity becomes one fourth its original value. This concept is fundamental in physics as it explains how a force or effect dissipates over an increasing area when moving away from its origin.

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