Amplitude Perception

Perception of Sound Intensity

The perceived increase in volume of a particular sound is logarithmically related to the amplitude of the sound wave. This is known as the Weber-Fechner Law, and it's not necessarily intuitive: each time you turn the volume up a notch on your speakers, you exponentially increase the amplitude. The same thing applies when visualizing the sound waves. Module 1 shows three scenarios.

• The first pair of waves differ in amplitude by 1 and have a ratio of amplitudes of 2.

• The second pair of waves differ in amplitude by 1 and have a ratio of amplitudes of 1.1.

• The third pair of waves differ in amplitude by 10 and have a radio of amplitudes of 2.

Module 1

Image 1

Two waves, one with an amplitude of 1, the other with an amplitude of 2.

Image 2

Two waves, one with an amplitude of 1, the other with an amplitude of 2.

Image 3

Two waves, one with an amplitude of 1, the other with an amplitude of 2.

Code

The code used to produce the waves.

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Note how the pictures for the first and third pairs are identical except for the values on the axes. This shows that the ratios of the amplitudes are more important than the differences between the amplitudes. Also notice in the second pair that you can barely tell the waves apart if you are standing far away from your screen.

Module 2 shows three sounds, with amplitudes in the ratio of 1.0 to 1.1 to 2.0. Note that the pair 1.0 and 1.1 sound almost indistinguishable, whereas the pair 1.0 and 1.2 clearly have different volume levels.

Module 2

Audio

Tone with a 1.0x multiplier to amplitude.

Tone with a 1.1x multiplier to amplitude.

Tone with a 2.0x multiplier to amplitude.

Code

The code used to produce the audio.

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From these demonstrations it is clear that to compare the volumes of two sound waves we must at some point calculate the ratio between the amplitudes of the waves.

Sound Pressure Level (SPL)

Sound pressure measures the local variation in atmospheric pressure caused by sound waves. When a sound wave passes through a medium, it causes small fluctuations in pressure compared to the ambient atmospheric pressure.

The sound pressure level (SPL) is a logarithmic measure of sound pressure relative to a reference pressure level, typically $20 \mu Pa$, which corresponds to the threshold of human hearing. Pascals are a measure of pressure that is force divided by area (Newtons per square meter, or $N/m^2$). SPL is calculated using the following formula:

$$SPL = 20 \log_{10} \left( \frac{p}{p_0} \right)$$

In the above equation,

• $p$ is the root mean square (RMS) sound pressure, representing the effective value of the pressure variations.

• $p_0$ is the reference sound pressure, $20 \mu Pa$.

SPL is measured in decibels (dB), a unit that quantifies the intensity of sound on a logarithmic scale. The SPL formula can also be derived in terms of the amplitude ($A$) of the sound wave because sound pressure is proportional to amplitude related.

$$SPL = 20 \log_{10} \left( \frac{A}{A_0} \right)$$

Sound Energy and Intensity

Sound energy is produced by vibrating objects, which causes the molecules in a medium (such as air, water, or solids) to vibrate. These vibrations travel through the medium as sound waves, which are classified as mechanical waves because they involve the transfer of mechanical energy through the medium.

Sound intensity is the power carried by sound waves per unit area. It is measured in Watts per square meter ($W/m^2$). It describes how much sound energy is passing through a given area per second. Intensity is is proportional to the square of the amplitude of the sound wave. This means that the SPL formula can also be expressed in terms of sound intensity.

$$\begin{aligned} SPL &= 20 \log_{10} \left( \frac{A}{A_0} \right) \\ SPL &= 20 \log_{10} \left( \frac{\sqrt{I}}{\sqrt{I_0}} \right) \\ SPL &= 20 \log_{10} \left( \left( \frac{I}{I_0} \right)^{1/2} \right) \\ SPL &= 10 \log_{10} \left( \frac{I}{I_0} \right) \\ \end{aligned}$$

Many sources report the formula for "decibels" using a coefficient of 10, while others use a coefficient of 20. This is because the sources sometimes use intensity instead of amplitude in the formula.

The Inverse Square Law

The Inverse Square Law describes how sound intensity decreases with distance from the source. As a sound wave spreads out from its source, its intensity diminishes according to the square of the distance ($d$) from the source.

$$I \propto \frac{1}{d^2}$$

As you move further away from a sound source, intensity decrease rapidly, which explains why sounds become quieter as distance increases.

Energy in Sound Waves

Since sound intensity is the power carried per unit area, and power is energy per unit time, then the equation for the mechanical energy carried by a sound wave is the following.

$$E = I * \text{(area)} * t$$

This now begs the question, what is the formula for the intensity of a sound wave? This involves some complex math using partial differential equations, but the important missing piece so far is that intensity is proportional to the square of the wave's frequency.

We can summarize all the information we've seen so far in one proportionality equation.

$$E \propto \frac{A^2 f^2}{d^2}$$