1. The nature of sound

Sound is variations in air pressure detectable by the human ear. Pressure varies through time at a particular point, and over space at a particular time, as molecules of air collide (condensation) or move apart (rarefaction).


Fig. 1: Diagrammatic representation of fluctuations in air pressure such as those caused by a vibrating tuning fork.

2. Waves and energy

Movement, or variation at a particular point, can be plotted as a waveform on a graph, as in Fig. 1.

N.B. (i) The relation of waveform to molecule movement.

(ii) All complete periodic sounds can be conceived of as composed of sine waves.

(iii) Possibility of discrete sampling to represent waveform (see e.g. Johnson p. 23).

The variation in pressure at a given point gives a sine wave for pure sounds involving simple harmonic motion (SHM).

The amplitude (the amount of maximum displacement from zero) of the wave reflects the highest pressure involved, and therefore the acoustic energy. N.B. Amplitude is not connected to frequency.

Relations between frequency (F), period (P), wavelength (w) and speed of sound (c):

F = 1/P

For example, a wave with a period of 1/100th of a second has a frequency of F = 1/0.01 = 100 Hz (Hertz = cycles per second)

w = c/F

The standard speed of sound is 330 m/s.

For example, a wave with a frequency of 100Hz has a wavelength of 330/100 = 3.3 m

Fig. 2.

3. Types of sound sources

a. Tuning fork - periodic

b. Vowel sounds - quasi-periodic. Fig. 3:

c. Flow of water - continuous random noise

d. Fricative - random within certain constraints. Fig. 4:

e. Hammer hitting table - transient

f. Stop consonant - transient + noise. Fig. 5:

4. Harmonics and Spectra

(a) Complex waves can be mathematically analysed as being composed of different sine waves (Fourier analysis). Vibrating objects don't usually vibrate at a single frequency, they have harmonics.

(b) According to when each harmonic starts, the different elements of a complex wave can be in different phase relations (N.B. the human ear is not thought to be sensitive to phase relations).

(c) Very important: any complex wave can be analysed into the periodic elements of different frequencies of which it is composed. Result: the spectrum of sound. For example, the following spectrum is of the waveform produced at the glottis in a vowel. Note the series of peaks or harmonics, occurring at integer multiples of the fundamental frequency f₀.

107dB 5500Hz Fig. 6

(d) Non-periodic sounds are more adequately represented not by line spectra but by a continuous spectrum.

100dB 8000Hz

100dB 8000Hz

Fig. 7: Continuous spectra of non-periodic sounds: left [s] (peak at c. 5920 Hz); right [] (peak at c. 2700 Hz).

5. Resonators and Filters

(a) Natural frequencies and resonance - different objects more or less tuned to specific frequencies - can act as filters.

Filters have centre frequency and bandwidth - the range of frequencies passed by filter not more than 3dB down on its maximum amplitude. The bandwidth of a filter may be relatively narrow or broad.

6. Source + Filter theory: Vowel Sounds

(a) The vocal tract acts as a complex variable filter. Into this is input a signal from the glottal source (for voiced sounds). Fig 6 shows a glottal source wave for a vowel, i.e. a complex wave with numerous harmonics which rapidly decrease in amplitude as frequency increases (12 dB/octave).

(b) Vowel sounds seen as product of glottal source and variable filtering effect of supraglottal tract. So, the same vowels have the same gross spectral shape, irrespective of the fundamental frequency f₀ of the source.

Fig. 6 filtered once ...

filtered twice ...

and filtered 3 times

Fig. 8: Spectrum of a vowel sound shown as the product of the glottal source and the filtering effect of the supraglottal vocal tract.

Reading

Johnson, chapter 1.