The decibel is a logarithmic way to compare sound intensities. Instead of describing a sound only by its intensity in watts per square meter, acoustics often uses decibels because the range of sound intensities that humans can hear is extremely large. A logarithmic scale compresses this huge range into more manageable numbers.
The absolute sound level of an intensity \(I\) is defined by
\[
\beta = 10\log_{10}\!\left(\frac{I}{I_0}\right),
\]
where \(I_0\) is a reference intensity. In basic acoustics, the standard reference is
\[
I_0 = 10^{-12}\ \text{W/m}^2,
\]
which is often taken as the threshold of hearing.
Many practical questions do not ask for an absolute sound level but for the difference between two sound intensities. If the first sound has intensity \(I_1\) and the second has intensity \(I_2\), then the level difference is
\[
\Delta\beta = 10\log_{10}\!\left(\frac{I_2}{I_1}\right).
\]
This formula is especially useful because it compares the two sounds directly. If \(I_2>I_1\), then \(\Delta\beta\) is positive. If \(I_2
The logarithmic form means that equal decibel changes correspond to multiplicative intensity changes. For example:
\[
\Delta\beta = 10\ \text{dB} \quad \Rightarrow \quad \frac{I_2}{I_1}=10,
\]
\[
\Delta\beta = 20\ \text{dB} \quad \Rightarrow \quad \frac{I_2}{I_1}=100,
\]
\[
\Delta\beta = 30\ \text{dB} \quad \Rightarrow \quad \frac{I_2}{I_1}=1000.
\]
In general,
\[
\frac{I_2}{I_1} = 10^{\Delta\beta/10}.
\]
This inverse relation is important because it lets you convert back from decibel differences to physical intensity ratios.
Consider the sample values
\[
I_1 = 10^{-6}\ \text{W/m}^2,
\qquad
I_2 = 10^{-3}\ \text{W/m}^2.
\]
The ratio is
\[
\frac{I_2}{I_1} = \frac{10^{-3}}{10^{-6}} = 10^3 = 1000.
\]
Therefore,
\[
\Delta\beta = 10\log_{10}(1000)=10\cdot 3=30\ \text{dB}.
\]
So the second sound is 30 dB above the first one.
The calculator also reports the individual levels \(\beta_1\) and \(\beta_2\) relative to \(I_0\). That can help you understand not only the difference between the two sounds, but also where each one sits on the full decibel scale. For example, two sounds can differ by the same number of decibels even if their absolute levels are very different.
At more advanced level, acoustics often uses weighted scales such as dB(A), where the ear’s frequency sensitivity is taken into account. But for introductory physics and engineering problems, the unweighted intensity-level relation
\[
\Delta\beta = 10\log_{10}\!\left(\frac{I_2}{I_1}\right)
\]
is the standard starting point. This calculator uses that standard form, shows the ratio and inverse check clearly, and visualizes the result with both a graph and a dB-scale animation.
