Problem 95 - Pearson Physics

Question

A particular sound has an intensity level of \(45 \mathrm{~dB}\). What is the intensity level of a second sound that is perceived to be twice as loud as the first sound?

Step-by-Step Solution

Answer

The intensity level of the second sound is 55 dB.

Step 1: Understanding Sound Intensity

The intensity level of sound in decibels (dB) is given by the formula \(L = 10 \log_{10}(I/I_0)\), where \(I\) is the intensity of the sound and \(I_0\) is the reference intensity level, usually \(10^{-12} \text{ W/m}^2\). A sound perceived to be twice as loud typically increases by 10 dB.

Step 2: Calculate New Intensity Level

The current sound has an intensity level of 45 dB. Since the second sound is perceived to be twice as loud, its intensity level will be \(45 + 10 = 55 \, \text{dB}\).

Step 3: Verify Using Perceptual Loudness

The rule of thumb for perceived loudness is that an increase of 10 dB is often experienced as doubling in loudness. The increase from 45 dB to 55 dB therefore aligns with the perception of the sound being twice as loud.

Key Concepts

Decibels

Decibels are a unit of measurement used to express the intensity level of sound. This scale is logarithmic, which means each increase of 10 units represents a tenfold increase in intensity. The use of a logarithmic scale helps manage the vast range of sound intensities our ears can detect, from the quietest whisper to the roar of a jet engine.
Decibels provide a convenient way to compare different sound intensities. For example, a sound that measures 40 dB is not twice as intense as a sound at 20 dB, but rather 100 times more intense. This is because each 10 dB increment is a tenfold increase in actual physical intensity.

• The formula for converting intensity to decibels is \(L = 10 \log_{10}(I/I_0)\), where \(L\) is the intensity level in decibels, \(I\) is the intensity, and \(I_0\) is a reference level, typically set at \(10^{-12} \text{ W/m}^2\).

Understanding decibels thus gives us a clearer picture of how sound intensity is quantified and perceived.

Perceptual Loudness

Perceptual loudness is how humans experience and interpret sound intensity. While the physical intensity of a sound can be quantified using decibels, how loud it seems to us can be quite different.
Due to the non-linear nature of our hearing, an increase of 10 dB is often perceived as a sound being twice as loud, even if the physical intensity is much more. This subjective aspect of loudness is key to understanding why some noises seem overwhelming while others are barely noticeable.

• This phenomenon is why, in the example problem, an increase from 45 dB to 55 dB results in the second sound being perceived as twice as loud as the first.
• It highlights the difference between physical measurements and personal experience of sound.

Learning about perceptual loudness helps demystify why our ears react to changes in sound in the peculiar way they do.

Sound Intensity Level

Sound Intensity Level (SIL) represents the power of the sound wave that passes through a given area. It is a crucial concept that connects the physical properties of sound with human auditory perception.
SIL is measured in watts per square meter (W/m²), and it describes how much sound energy reaches our ears. The reference intensity level in this context is very low, at \(10^{-12} \text{W/m}^2\), as it corresponds to the quietest sound that the average human ear can hear.

• As the sound intensity level increases, so does the sound pressure, creating a louder sensation.
• The logarithmic nature of the decibel scale allows for easier handling of the extremely wide range of sound intensities we encounter.

This concept is not only essential for scientific measurements but also in practical applications like setting regulated noise limits and designing pristine acoustics in venues.