KAHIBARO
Table of Contents
Measuring Sound Level
When we talk about sound in everyday life, we often say that one sound is louder than another. In physics, loudness is related to the intensity of a sound wave, but the range of sound intensities that the human ear can detect is enormous. Because of this huge range, physicists and engineers use a logarithmic scale called the decibel scale.
The decibel, abbreviated dB, does not measure intensity in a simple linear way. Instead, it compares one intensity to a reference intensity. This makes it much easier to describe very weak and very strong sounds with manageable numbers.
Why a Logarithmic Scale Is Used
The faintest sounds a healthy human ear can detect are extremely weak, while painful sounds are many trillions of times more intense. Writing all sound levels directly in watts per square meter would be inconvenient.
A logarithmic scale compresses this wide range. On such a scale, multiplying the intensity by a certain factor adds a fixed number of decibels.
For sound intensity level, the formula is
$$
\beta = 10 \log_{10}\left(\frac{I}{I_0}\right)
$$
where $\beta$ is the sound level in decibels, $I$ is the sound intensity, and $I_0$ is the reference intensity.
Important formula for sound level:
$$
\beta = 10 \log_{10}\left(\frac{I}{I_0}\right)
$$
For sound in air, the standard reference intensity is usually
$$
I_0 = 1.0 \times 10^{-12}\ \text{W/m}^2
$$
This is approximately the threshold of human hearing.
Reference Intensity
The reference intensity $I_0$ is chosen so that a sound at the threshold of hearing has a level of $0$ dB. This does not mean there is no sound. It means the sound intensity equals the reference value.
If $I = I_0$, then
$$
\beta = 10 \log_{10}(1) = 0\ \text{dB}
$$
So a sound can have a positive decibel value if it is stronger than the reference, or even a negative decibel value if it is weaker than the reference.
Interpreting Decibel Changes
Because the scale is logarithmic, equal changes in decibels do not mean equal changes in intensity.
If the intensity increases by a factor of $10$, then
$$
\beta = 10 \log_{10}(10) = 10\ \text{dB}
$$
So an increase of $10$ dB means the intensity becomes $10$ times larger.
If the intensity increases by a factor of $100$, then the sound level increases by $20$ dB.
If the intensity doubles, the increase is
$$
10 \log_{10}(2) \approx 3.0\ \text{dB}
$$
So doubling the intensity increases the level by about $3$ dB.
Key decibel rules:
A change of $+10$ dB means intensity is multiplied by $10$.
A change of $+20$ dB means intensity is multiplied by $100$.
A change of $+3$ dB means intensity is approximately doubled.
A change of $-10$ dB means intensity is divided by $10$.
Common Sound Levels
The decibel scale gives a convenient way to compare everyday sounds.
| Sound source | Approximate level |
|---|---|
| Threshold of hearing | $0$ dB |
| Whisper | $20$ dB to $30$ dB |
| Normal conversation | $60$ dB |
| Busy traffic | $70$ dB to $85$ dB |
| Rock concert | $110$ dB to $120$ dB |
| Threshold of pain | about $120$ dB to $130$ dB |
These values are approximate. Actual levels depend on distance from the source and the environment.
Example Calculation
Suppose a sound has intensity
$$
I = 1.0 \times 10^{-6}\ \text{W/m}^2
$$
Using the reference intensity $I_0 = 1.0 \times 10^{-12}\ \text{W/m}^2$, we find
$$
\beta = 10 \log_{10}\left(\frac{10^{-6}}{10^{-12}}\right)
= 10 \log_{10}(10^6)
= 10 \times 6
= 60\ \text{dB}
$$
So this sound level is $60$ dB, which is about the level of normal conversation.
Comparing Two Sounds Directly
Sometimes we want to compare two sounds without first comparing each one to the reference. If two sounds have intensities $I_1$ and $I_2$, then the difference in their sound levels is
$$
\Delta \beta = 10 \log_{10}\left(\frac{I_2}{I_1}\right)
$$
This tells us how many decibels louder one sound is than another.
For example, if $I_2 = 100 I_1$, then
$$
\Delta \beta = 10 \log_{10}(100) = 20\ \text{dB}
$$
So the second sound is $20$ dB higher.
To compare two sound intensities directly, use
$$
\Delta \beta = 10 \log_{10}\left(\frac{I_2}{I_1}\right)
$$
Do not subtract intensities first. The decibel scale depends on the ratio.
Decibels and Human Hearing
The decibel scale is useful because human hearing responds roughly logarithmically to changes in intensity. A tenfold increase in physical intensity does not feel like ten times the loudness, but it does correspond to a clear increase in perceived loudness.
This is why decibels are widely used in acoustics, engineering, medicine, and environmental noise studies.
Visual Idea of the Decibel Scale
This drawing shows that equal steps in decibels correspond to multiplying the intensity by equal factors, not adding equal amounts.
Practical Meaning
Decibels are especially useful for discussing safe and unsafe sound exposure. A small increase in decibel level can mean a large increase in physical intensity. For this reason, very loud environments can become dangerous quickly.
For example, an $80$ dB sound is not just a little more intense than a $70$ dB sound. It is
$$
10^{(80-70)/10} = 10
$$
times more intense.
Final Reminder
The decibel is a comparison scale for intensity. It is logarithmic, not linear, and always depends on a reference level.
Remember:
The decibel level is not the intensity itself.
It is a logarithmic measure of intensity relative to a reference:
$$
\beta = 10 \log_{10}\left(\frac{I}{I_0}\right)
$$
A higher dB value means a stronger sound, and even a small dB increase can represent a large physical increase in intensity.