# Chapter 4: The physics and biology of audition

## Sound as a physical stimulus (pp. 81—89)

• Why do different instruments in an orchestra sound different even when they are playing the same note?
• What causes a sonic boom from an airplane?
• How does a microphone work?

### What you need to know

1. Simple sounds (pp. 81—83)
• Frequency
• Amplitude
• Phase
2. Complex sounds (pp. 82—89)
• Fourier theory
• Fourier spectrum
• Spectrogram
• Filters
• Transfer functions
• Linearity

#### Simple sounds

##### Demo: Pure tones

The graph plots the sinusoidal sound waveform of a pure tone. Use the slider to change the amplitude of the wave; use the buttons to change sound frequency, and to hear an audio clip of the tone.

Sound consists of pressure waves carried by vibrating air molecules. Parts of the wave where air pressure is increased are called compressions and parts where pressure is decreased are called rarefactions (see Fig. 4.2, FP p. 82). A sound wave's frequency corresponds to the number of alternations between compression and rarefaction in a 1-second period, i.e. the number of cycles per second (measured in Hz). High-frequency waves are perceived as high pitch and low-frequency waves as low pitch.

##### Research Study: Pitch

The amplitude of a sound wave corresponds to the amount of change in pressure created by it (see Fig. 4.3, FP p. 83) and is measured in decibels (dB) on the Sound pressure level (SPL) scale. SPL corresponds roughly with the loudness of a perceived sound.

One complete cycle of a sine wave spans 360° (see Fig. 4.3, FP p. 83). The phase of a wave describes the part of the cycle that a sound wave has reached at a specific point in time. Phase is often used to compare the timing of two sound waves (see Fig. 4.4, FP p. 84).

#### Complex sounds

##### Demo: Complex tones

This demonstration shows the composition of a clarinet note, as in FP Fig. 4.6 (p. 85). Press the "Add" button to add progressively more components to the note. The upper plot shows the shows the sound wave associated with each component; the lower plot shows the amplitude spectrum of the note. Press the "Play" button to hear the note defined by the components, and to see the resulting waveform.

Any complex sound can be treated as a collection of simple sine waves added together. The frequencies, amplitudes and phase of the individual waves determine the overall form of the complex wave and the sound it makes (see Fig. 4.5, FP p. 84 and Fig. 4.6, FP p. 85). The lowest frequency in a complex wave is called the fundamental frequency. Harmonic frequency are higher frequency components that are numbered according to their distance from the fundamental frequency, e.g. the fifth harmonic has a frequency that is five times greater than the fundamental frequency. Many natural sounds do not contain harmonics, but have a continuous "spectrum" of components in which all frequencies are represented.

Fourier analysis is a mathematical procedure that allows any complex sound to be broken down into its sine wave components. Fourier analysis produces a magnitude spectrum, or Fourier spectrum, containing information about the power in the original signal at each frequency and a phase spectrum, containing information about the phases of the sine waves that make up the complex signal. Together, the magnitude and phase spectrum provide a complete representation of the original signal, which can be recombined using Fourier synthesis. See the Tutorials section, FP pp. 107–115.

Fourier analysis assumes that the sound signal is unchanged over time; however, real signals do not remain unchanged. For example, human speech contains many frequency components that vary in amplitude over time. A simple magnitude spectrum cannot show these variations so, instead, the frequency content of the signal is analysed over a series of small time intervals and displayed graphically in a spectrogram (see Fig. 4.8, FP p. 87). The spectrogram clearly shows how the frequency content of the sound signal changes over time.

After Fourier synthesis, the spectrum of a recombined signal can be compared to the original signal to investigate how a transmitting device or medium has modified the sound signal. For example, when a sound passes over the human head it acts as a frequency filter, as it allows low-frequency components to pass but removes the higher frequency components from the signal (see Fig. 4.9, FP p. 88).

The transfer function of a filter describes the amount of attenuation applied by the filter at each sine wave frequency in the signal; for example, values close to 1 represent little attenuation, whereas values close to 0 indicate that little of the sound is transmitted through the filter. Filtering techniques are useful for describing the properties of filters (e.g. microphone) and predicting their effects on sound signals.

Using Fourier theory to analyse the properties of an acoustic filter assumes that the filter is linear. Linear filters have three essential properties:

1. The output of the filter never contains a frequency component that was not present in the input signal.
2. If the amplitude of the input filter is changed, the output should change by the same factor.
3. If a number of sine wave inputs are applied to the filter at the same time, the output must match the output that would be produced if the inputs were applied separately and their outputs summed.

If any one of these rules is broken, the filter is non-linear. A non-linear filter often adds distortions, as the output contains frequency components that were not present in the input signal. As a result, the filter's response to complex signals cannot be predicted straightforwardly.

### So What Does This Mean?

Sounds are vibrations. Simple sounds are in the form of a sine wave in which frequency corresponds to the number of cycles of the wave per second and relates perceptually to pitch. Amplitude corresponds to the change in pressure and is perceived as loudness. Phase refers to the part of the cycle the wave has reached, and is useful for comparing the timing of two waves.

Fourier analysis breaks complex waves into their sine wave components through the generation of a magnitude and a phase spectrum. Linear filters modify the signal components' amplitudes, but do not introduce new elements. Non-linear filters distort the signal.