KAHIBARO
Table of Contents
Energy flow in sound
Sound intensity tells us how much sound energy passes through an area each second. It is a measure of the strength of a sound wave at a location. When a source such as a speaker, bell, or vibrating string produces sound, the wave carries energy through the medium. The intensity describes the rate of that energy transfer per unit area.
Mathematically, sound intensity is defined as
$$
I = \frac{P}{A}
$$
where $I$ is the intensity, $P$ is the sound power passing through an area, and $A$ is the area through which the energy flows.
The SI unit of intensity is watts per square meter, written as $\mathrm{W/m^2}$.
Sound intensity is the sound power transmitted per unit area.
$$
I = \frac{P}{A}
$$
Unit,
$$
1\ \mathrm{W/m^2} = 1\ \frac{\mathrm{J}}{\mathrm{s \, m^2}}
$$
Physical meaning
A larger intensity means more energy is being delivered by the sound wave every second through each square meter. A faint whisper has very small intensity. A loud siren has much greater intensity.
Intensity is not exactly the same as loudness. Intensity is a physical quantity that can be measured. Loudness is the way humans perceive sound, and it also depends on frequency and the sensitivity of the ear. Here we focus only on the physical quantity.
Intensity and direction of wave travel
Sound energy moves in the direction the wave propagates. If we imagine a surface placed perpendicular to the wave motion, the intensity measures how much power crosses that surface.
For a plane wave, the energy spreads uniformly across parallel surfaces, so the intensity can remain nearly constant if there is little energy loss.
Inverse square behavior
If a sound source radiates equally in all directions, the sound spreads over larger and larger spherical surfaces as it moves away from the source. The same total power is then shared over a larger area, so the intensity decreases with distance.
For a source with power $P$, the area of a sphere of radius $r$ is
$$
A = 4 \pi r^2
$$
so the intensity at distance $r$ is
$$
I = \frac{P}{4 \pi r^2}
$$
This is called the inverse square law. If the distance is doubled, the intensity becomes one fourth.
For a point source radiating uniformly in all directions,
$$
I = \frac{P}{4 \pi r^2}
$$
Therefore,
$$
I \propto \frac{1}{r^2}
$$
If $r$ doubles, $I$ becomes $\frac{1}{4}$ of its original value.
Relation to wave amplitude
A stronger sound wave usually has a larger amplitude. In a sound wave, larger pressure variations and larger particle motion mean greater energy transport. For many situations, the intensity is proportional to the square of the wave amplitude.
If the amplitude doubles, the intensity becomes four times as large.
This square dependence is very important because small changes in amplitude can produce much larger changes in intensity.
For sound waves, intensity is proportional to the square of amplitude.
$$
I \propto A^2
$$
If amplitude doubles, intensity becomes four times larger.
Here the symbol $A$ in the proportionality means wave amplitude, not area. The meaning is determined by context.
Average intensity
In many sound waves, especially sinusoidal waves, the pressure and particle speed change continuously with time. Because of this, the energy flow can also vary during a cycle. In practice, we often use average intensity, which means the average rate of energy transfer per unit area over time.
For most everyday sound measurements, intensity means average intensity.
Intensity level and reference intensity
Because the range of audible intensities is enormous, it is often more convenient to compare a sound to a standard reference intensity instead of using raw $\mathrm{W/m^2}$ values.
A commonly used reference intensity is
$$
I_0 = 1.0 \times 10^{-12}\ \mathrm{W/m^2}
$$
which is approximately the threshold of human hearing for certain frequencies.
The sound intensity level $\beta$ is defined by
$$
\beta = 10 \log_{10}\left(\frac{I}{I_0}\right)
$$
The unit is the decibel, abbreviated dB.
Sound intensity level in decibels is
$$
\beta = 10 \log_{10}\left(\frac{I}{I_0}\right)
$$
with reference intensity
$$
I_0 = 1.0 \times 10^{-12}\ \mathrm{W/m^2}
$$
Understanding decibels
The decibel scale is logarithmic. This means equal changes in decibels correspond to multiplicative changes in intensity, not additive ones.
If intensity increases by a factor of 10, the intensity level increases by 10 dB. If intensity increases by a factor of 100, the level increases by 20 dB.
This follows directly from the formula:
$$
\beta_2 - \beta_1 = 10 \log_{10}\left(\frac{I_2}{I_1}\right)
$$
So a difference in decibels depends on the ratio of intensities.
| Intensity ratio $\frac{I_2}{I_1}$ | Change in level |
|---|---|
| $2$ | about $3\ \mathrm{dB}$ |
| $10$ | $10\ \mathrm{dB}$ |
| $100$ | $20\ \mathrm{dB}$ |
| $1000$ | $30\ \mathrm{dB}$ |
A doubling of intensity gives
$$
10 \log_{10}(2) \approx 3\ \mathrm{dB}
$$
Common examples
The table below gives rough values for sound intensity levels. These are approximate and can vary.
| Sound | Intensity level |
|---|---|
| Threshold of hearing | $0\ \mathrm{dB}$ |
| Quiet room | $30\ \mathrm{dB}$ |
| Normal conversation | $60\ \mathrm{dB}$ |
| Busy traffic | $80\ \mathrm{dB}$ |
| Rock concert | $110\ \mathrm{dB}$ |
| Threshold of pain | $120\ \mathrm{dB}$ |
These values show how broad the range of human hearing is. A sound at $120\ \mathrm{dB}$ is not just twice as intense as one at $60\ \mathrm{dB}$, it is vastly more intense.
Example calculation
Suppose a sound wave carries power $2.0 \times 10^{-4}\ \mathrm{W}$ through an area of $5.0 \times 10^{-2}\ \mathrm{m^2}$. The intensity is
$$
I = \frac{P}{A}
= \frac{2.0 \times 10^{-4}}{5.0 \times 10^{-2}}
= 4.0 \times 10^{-3}\ \mathrm{W/m^2}
$$
Now find the intensity level:
$$
\beta = 10 \log_{10}\left(\frac{4.0 \times 10^{-3}}{1.0 \times 10^{-12}}\right)
$$
$$
\beta = 10 \log_{10}(4.0 \times 10^9)
$$
$$
\beta = 10 \left(\log_{10}4.0 + 9\right)
\approx 10(0.602 + 9)
\approx 96.0\ \mathrm{dB}
$$
So this sound has an intensity level of about $96\ \mathrm{dB}$.
Why intensity matters
Sound intensity is important because it connects wave motion to energy transport. It helps us compare weak and strong sounds, understand how sound fades with distance, and measure sound in practical settings such as classrooms, factories, and concert halls.
It is one of the key physical quantities used to describe sound waves quantitatively.
Key formulas for sound intensity:
$$
I = \frac{P}{A}
$$
$$
I = \frac{P}{4\pi r^2} \quad \text{for uniform spherical spreading}
$$
$$
\beta = 10\log_{10}\left(\frac{I}{I_0}\right)
$$
$$
I_0 = 1.0\times10^{-12}\ \mathrm{W/m^2}
$$