Sound intensity from a localized source decreases as the inverse square of the distance, according to Equation \(14.8 .\) If the distance from the source doubles, what happens to (a) the intensity and (b) the decibel level?

Sound intensity, often referenced in acoustics, denotes the power per unit area carried by a sound wave. It's an objective measure, which means it does not depend on the listener's perception but rather on physical attributes such as the energy rate that is being transmitted through a certain area. More simply put, it indicates how much energy the sound wave transports across a unit surface.

Under the inverse square law, sound intensity decreases as the distance from the sound source increases. To visualize this, imagine a balloon inflating; as the surface area of the balloon grows, the gas inside becomes less dense. Similarly, as a sound wave expands outward from its source, the energy must cover a larger area, resulting in a decrease in sound intensity.

For example, if you were standing right next to a speaker at a concert, the sound intensity would be at its maximum because you are close to the source. As you walk away from the speaker, the intensity drops in accordance with this principle. This physical law is a crucial element in various fields, from audio engineering to architectural acoustics, to ensure sound is managed effectively for comfort and utility.

The decibel level is a logarithmic unit used to measure sound intensity and express it on a scale that mirrors the human ear's perception of sound loudness. Our ears perceive changes in sound intensity on a logarithmic scale rather than a linear one, which means that the sensation of loudness does not increase proportionally with the intensity. This is why we use the decibel scale; it better represents how we actually hear changes in volume.

Decibels indicate a relative measurement rather than an absolute one, based on a reference level. Commonly, the threshold of hearing (the quietest sound that can be heard by the average human ear) is used as this reference point. On the decibel scale, an increase of 10 decibels typically corresponds to a sound that is perceived as twice as loud.

When sound intensity is reduced, as in the exercise where the distance from the source is doubled, we can expect the decibel level to also decrease. However, due to the logarithmic nature, this decrease will not be a straightforward halving or quartering, but rather a specific decrease in decibels calculated by taking the logarithm of the ratio of the new intensity to the original.

A logarithmic scale is a way of displaying numerical data over a very wide range of values in a compact way. On this type of scale, equal distances represent equal ratios, which means that incrementing by one unit increases the value by a constant multiplier. This is especially useful in fields like acoustics, where the range of perceivable sound intensities spans several orders of magnitude.

For example, if the sound intensity level is increased by 3 decibels, this translates to a doubling of the sound intensity; if it's increased by 6 decibels, the intensity quadruples. This happens because a logarithmic scale compresses the scale’s length, and small increments can represent substantial changes in absolute terms.

The use of the logarithmic scale in calculating the decibel level allows us to compress a large range of intensities into a much smaller, more manageable range. This is critical when dealing with sound, as the human ear has a great ability to detect changes in sound levels across a vast range, from the barely audible whisper to the roar of a jet engine.