Page 37 - The Master Handbook Of Acoustics

12 CHAPTER ONE

Complex Waves

Speech and music waveshapes depart radically from the simple sine
form. A very interesting fact, however, is that no matter how complex the
wave, as long as it is periodic, it can be reduced to sine components. The
obverse of this is that, theoretically, any complex periodic wave can be
synthesized from sine waves of different frequencies, different ampli-
tudes, and different time relationships (phase). A friend of Napoleon,
named Joseph Fourier, was the first to develop this surprising idea. This
idea can be viewed as either a simplification or complication of the situ-
ation. Certainly it is a great simplification in regard to concept, but some-
times complex in its application to specific speech or musical sounds.
As we are interested primarily in the basic concept, let us see how even
a very complex wave can be reduced to simple sinusoidal components.

Harmonics
A simple sine wave of a given amplitude and frequency, f , is shown in
1
Fig. 1-11A. Figure 1-11B shows another sine wave half the amplitude and
twice the frequency (f ). Combining A and B at each point in time the
2
waveshape of Fig. 1-11C is obtained. In Fig. 1-11D, another sine wave half
the amplitude of A and three times its frequency (f ) is shown. Adding
3
this to the f f wave of C, Fig. 1-11E is obtained. The simple sine wave
1 2
of Fig. 1-11A has been progressively distorted as other sine waves have
been added to it. Whether these are acoustic waves or electronic signals,
the process can be reversed. The distorted wave of Fig. 1-11E can be dis-
assembled, as it were, to the simple f , f , and f sine components by either
1 2 3
acoustical or electronic filters. For example, passing the wave of Fig.
1-11E through a filter permitting only f and rejecting f and f , the origi-
1 2 3
nal f sine wave of Fig. 1-11A emerges in pristine condition.
1
Applying names, the sine wave with the lowest frequency (f ) of
1
Fig. 1-11A is called the fundamental, the one with twice the frequency
(f ) of Fig. 1-11B is called the second harmonic, and the one three
2
times the frequency (f ) of Fig. 1-11D is the third harmonic. The fourth
3
harmonic, the fifth harmonic, etc., are four and five times the fre-
quency of the fundamental, and so on.

Phase
In Fig. 1-11, all three components, f , f , and f , start from zero together.
1 2 3
This is called an in-phase condition. In some cases, the time relation-