2.3 Amplitude and sound intensity

Amplitude and Sound Intensity

Amplitude and sound intensity

Amplitude and intensity describe two related but distinct properties of a sound wave. Amplitude tells you how far the medium moves; intensity tells you how much energy that movement carries.

• Amplitude is the maximum displacement of a wave from its equilibrium (rest) position, measured in meters. On a waveform diagram, it's the distance from the center line to the peak or trough.
• Sound intensity is the rate of energy transfer through a unit area, measured in watts per square meter ($\text{W/m}^2$). Think of it as how much acoustic power is hitting a given surface.

The critical relationship between them is:

$I \propto A^2$

This means intensity is proportional to the square of amplitude. If you double the amplitude, intensity doesn't just double; it quadruples. Triple the amplitude, and intensity increases by a factor of nine. This square relationship shows up constantly in acoustics problems, so make sure it's second nature.

Decibel calculations

Raw intensity values span an enormous range, so acoustics uses the decibel (dB), a logarithmic unit, to express sound levels in more practical numbers.

There are two main formulas you need to know:

Sound Intensity Level (SIL):

$SIL = 10 \log_{10}\left(\frac{I}{I_0}\right) \text{ dB}$

where $I_0 = 10^{-12} \text{ W/m}^2$ is the reference intensity (roughly the quietest sound a healthy human ear can detect).

Sound Pressure Level (SPL):

$SPL = 20 \log_{10}\left(\frac{P}{P_0}\right) \text{ dB}$

where $P_0 = 2 \times 10^{-5} \text{ Pa}$ is the reference pressure.

Notice the coefficient is 10 for intensity but 20 for pressure. That factor of 2 comes from the relationship between intensity and pressure:

$I = \frac{P^2}{\rho c}$

where $\rho$ is the density of air and $c$ is the speed of sound. Because intensity depends on pressure squared, the log formula for pressure picks up that extra factor of 2.

Worked example (SIL): A sound has an intensity 100 times the reference intensity. What is its SIL?

1. Plug into the formula: $SIL = 10 \log_{10}\left(\frac{100 \cdot I_0}{I_0}\right)$
2. Simplify: $SIL = 10 \log_{10}(100)$
3. $\log_{10}(100) = 2$, so $SIL = 10 \times 2 = 20 \text{ dB}$

Worked example (SPL): A sound pressure is 10 times the reference pressure. What is its SPL?

1. $SPL = 20 \log_{10}\left(\frac{10 \cdot P_0}{P_0}\right)$
2. $SPL = 20 \log_{10}(10) = 20 \times 1 = 20 \text{ dB}$

Both examples give 20 dB, which makes sense: a pressure ratio of 10 corresponds to an intensity ratio of $10^2 = 100$.

Logarithmic nature of decibels

The decibel scale is logarithmic for a good reason: human hearing itself responds logarithmically to changes in intensity. A sound that carries 10 times more energy doesn't sound 10 times louder to you. Instead, perceived loudness roughly doubles with every 10 dB increase.

This means 80 dB sounds about twice as loud as 70 dB, not eight-sevenths as loud. And 90 dB sounds about four times as loud as 70 dB.

The logarithmic scale also compresses a huge range of values into manageable numbers. The dynamic range of human hearing spans from about 0 dB (threshold of hearing) to around 120–140 dB (threshold of pain). In raw intensity terms, that's a range from $10^{-12} \text{ W/m}^2$ to about $1 \text{ W/m}^2$, a factor of a trillion. The decibel scale turns that trillion-fold range into a 0–120 dB span you can actually work with.

Typical sound intensity levels

Having a mental map of common dB values helps you check whether your calculations are reasonable and gives you a feel for what the numbers actually mean.

A few useful benchmarks to memorize: normal conversation sits around 60–70 dB, prolonged exposure above about 85 dB can cause hearing damage, and the threshold of pain begins around 120–140 dB.