A rock group is playing in a club. Sound emerging outdoors...

A rock group is playing in a club. Sound emerging outdoors from an open door spreads uniformly in all directions. If the decibel level is $70 \mathrm{dB}$ at a distance of $1.0 \mathrm{m}$ from the door, at what distance is the music just barely audible to a person with a normal threshold of hearing? Disregard absorption.

Key Concepts

Threshold of Hearing

The threshold of hearing is the minimum sound intensity level that the average human ear can detect. It is commonly defined as 0 dB, corresponding to an intensity of approximately 10^-12 watts per square meter. This threshold serves as a baseline for determining when sounds become just audible.

Inverse Square Law

The inverse square law explains how the intensity of sound diminishes with distance. For a point source emitting sound uniformly in all directions, the intensity decreases in proportion to the square of the distance from the source. This means that as you move away, the power per unit area spreads over an increasingly larger spherical surface.

Decibel Scale and Logarithmic Measurement

The decibel (dB) is a logarithmic unit used to express the ratio of a given sound intensity to a reference intensity. This scale compresses the wide range of sound intensities into a manageable numerical format, where each 10 dB change represents a tenfold change in intensity.

Sound Intensity and Power Distribution

Sound intensity is the amount of power that passes through a unit area. When sound propagates from a point source, the total emitted power is distributed over the surface area of a sphere, which increases with the square of the distance from the source. This concept is essential for analyzing how sound decreases as it travels through space.

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