Question

Abdi incorrectly states, "A noise of \(20 \mathrm{dB}\) is twice as loud as a noise of \(10 \mathrm{dB}\) " Explain the error in Abdi's reasoning.

Short answer

Abdi's error is assuming a linear relationship in a logarithmic scale; 20 dB is 10 times more intense than 10 dB.

Step-by-step solution

Understanding Decibels

The decibel (dB) scale is logarithmic, not linear. This means that an increase in decibels represents a multiplicative, not additive, increase in intensity.

Logarithmic Nature of dB Scale

In the dB scale, each increase of 10 dB represents a tenfold increase in intensity. Therefore, 20 dB is not simply twice 10 dB in terms of loudness.

Comparing 10 dB and 20 dB

Given that each 10 dB increase results in a tenfold increase in intensity, a 20 dB noise is actually 10 times more intense than a 10 dB noise, not twice as loud.

Abdi's Error

Abdi's error is in assuming that the dB scale works additively. However, because it is logarithmic, doubling the number of dB does not double the loudness but increases it by a factor of ten.

Key concepts

Logarithmic Scale

The decibel scale is a logarithmic measurement, which is quite different from a linear one. In a linear scale, each unit increase adds a consistent amount. For example, moving from 1 to 2 is the same as moving from 2 to 3. However, in a logarithmic scale, each increment represents a multiplication of the previous value.
An example of a base-10 logarithmic scale, like the decibel (dB) scale, would show that increasing from 10 to 20 results in a tenfold increase rather than a doubling.
This property helps us to manage large ranges of values more effectively. It's particularly useful when dealing with phenomena that can vary by many orders of magnitude, like sound intensity.

Sound Intensity

Sound intensity refers to the amount of energy a sound wave carries per unit area, typically measured in watts per square meter (W/m²). It describes how much sound energy passes through a specified area.
The human ear perceives sound intensity logarithmically rather than linearly, which is why the decibel scale is used. This relationship allows us to better understand and quantify how loud or quiet a sound is, despite having varying intensities.

Decibel Comparison

When comparing sounds using the decibel scale, it's crucial to remember that each increase of 10 dB represents a tenfold increase in sound intensity.
For instance, reconsider the given example: a noise at 20 dB isn't merely twice as loud as one at 10 dB. Instead, it is 10 times more intense. In practical terms, this means that if you increase a sound from 10 dB to 20 dB, the sound intensity has increased dramatically, not just slightly.

Logarithmic vs Linear

Understanding the difference between logarithmic and linear scales is crucial in many scientific fields, including acoustics.
A linear scale, such as a ruler, adds the same amount for each step (1, 2, 3,...). In contrast, a logarithmic scale increases by a factor (10, 100, 1000,...).
This means that while a linear scale is straightforward and easy to understand, a logarithmic scale can more effectively represent vast ranges of values, such as those encountered in sound intensity. By recognizing this difference, one can better understand why the decibel scale is used for measuring sound.