Sound intensity measures how much acoustic power passes through a unit area. Its SI unit is watts per square meter, written as
\(\text{W/m}^2\). For an ideal isotropic point source, the sound power spreads equally in all directions, so at distance \(r\) it is distributed over the
surface area of a sphere:
\[
A=4\pi r^2.
\]
Since intensity is power divided by area, we get
\[
I=\frac{P}{4\pi r^2}.
\]
This is the mathematical form of the inverse-square law: when you double the distance from the source, the same power is spread over four times the
area, so the intensity becomes one-fourth as large.
Human hearing responds to sound intensity over an enormous range. Because of that, it is not convenient to describe loudness using raw intensity values
alone. Instead, acoustics commonly uses the decibel scale, which is logarithmic:
\[
\beta = 10\log_{10}\left(\frac{I}{I_0}\right).
\]
Here \(I_0\) is a reference intensity, usually taken to be
\[
I_0 = 10^{-12}\ \text{W/m}^2,
\]
which is approximately the threshold of hearing for a healthy young person at around 1 kHz. The result \(\beta\) is measured in decibels (dB).
The logarithmic scale has two important consequences. First, equal ratios of intensity correspond to equal differences in decibels. For example,
if intensity becomes 10 times larger, the level increases by 10 dB. If intensity becomes 100 times larger, the level increases by 20 dB. Second, a
decrease in intensity with distance looks much less dramatic in decibels than in raw \(\text{W/m}^2\), even though the physical intensity may have changed
a lot.
Combining the inverse-square law with the decibel formula gives a practical way to estimate how loud a source sounds at a certain distance. For a point
source of power \(P\),
\[
I(r)=\frac{P}{4\pi r^2},
\qquad
\beta(r)=10\log_{10}\left(\frac{P}{4\pi r^2 I_0}\right).
\]
This helps explain why moving only a short distance away from a loudspeaker or machine can reduce the sound level noticeably.
For example, if \(P=1\ \text{W}\) and \(r=10\ \text{m}\), the intensity is
\[
I=\frac{1}{4\pi(10)^2}\approx 7.96\times 10^{-4}\ \text{W/m}^2,
\]
which corresponds to about
\[
\beta \approx 89\ \text{dB}.
\]
That is a loud level, much higher than everyday conversation.
This calculator assumes an ideal isotropic source in free space. Real sound sources may be directional, and real environments may include
absorption, reflections, atmospheric losses, and obstacles. At university level, the model is extended to directional emitters, intensity from distributed
sources, reverberant fields, sound pressure level (SPL), and acoustic impedance methods.
The plot in this calculator shows how intensity or decibel level changes with distance, and the animation provides an intuitive picture of why sound
weakens as it spreads outward. The power does not disappear; it is simply distributed over a larger and larger area.
