The spectrum of periodic sound waves (cont.)
A sine wave represents a single frequency and therefore a sine wave of a given amplitude can be plotted as a single line on a graph of amplitude versus frequency. The components of a square wave plotted in this form are shown in Figure 1.47, which clearly shows that the square wave consists of a set of progressively reducing discrete sine wave components at odd multiples of the lowest frequency.
Figure 1.47 The frequency domain representation, or spectrum, of a square wave.
This representation is called the frequency domain representation, or spectrum, of a waveform and the waveform's amplitude versus time plot is called its time domain representation.The individual sine wave components of the waveform are often called the partials of the waveform. If they are integer related, as in the square wave, then they can be called harmonics. The lowest frequency is called the fundamental, or first harmonic, and the higher frequency harmonics are labelled according to their frequency multiple relative to the fundamental. Thus the second harmonic is twice the frequency of the fundamental and so on.
Partials on the other hand need not be harmonically related to the fundamental, and are numbered in their order of appearance with frequency. However, as we shall see later, this results in a waveform that is aperiodic.
So for the square wave the second partial is the third harmonic and the third partial is the fifth harmonic. Other waveforms have different frequency domain representations, because they are made up of sine waves of different amplitudes and frequencies. Some examples of other waveforms in both the time and frequency domains are shown in Chapter 3.
1.6.2 The effect of phase
The phase, which expresses the starting value of the individual sine wave components, also affects the waveshape. Figure 1.48 shows what happens to a square wave if alternate partials are subtracted rather than added, and this is equivalent to changing the phase of these components by 180°. That is, alternate frequency components start from halfway around the circle compared with the other components, as shown by the dotted line in Figure 1.48. However, although the time domain waveform is radically different the frequency domain is very similar, as the amplitudes are identical, only the phase of some of the harmonics have changed.
Figure 1.48 The effect of adding harmonically related sine waves together with different phase shifts.
Interestingly, in many cases, the resulting wave is perceived as sounding the same, even though the waveform is different. This is because the ear, as we will see later, appears to be less sensitive to the phase of the individual frequency compared to the relative amplitudes.
However, if the phase changes are extreme enough we can hear a difference (see Schroeder 1975). Because of this, often only the amplitude of the frequency components are plotted in the spectrum and, in order to handle the range of possible amplitudes and because of the way we perceive sound, the amplitudes are usually plotted as decibels. For example, Figure 4.24 in Chapter 4 shows the waveform and spectrum plotted in this fashion for middle C played on a clarinet and tenor saxophone.