[Part 1 discusses pressure waves and sound transmission. Part 2 covers sound intensity, power and pressure level. Part 3 looks at adding sounds together. Part 4 discusses the "inverse square law" for sound. Part 5 begins a look at sound interactions, including refraction, absorption and reflection. Part 6 continues a look at sound interactions with a discussion of sound interference, standing waves, diffraction and scattering.]
1.6 Time and frequency domains
So far we have mainly considered a sound wave to be a sinusoidal wave at a particular frequency. This is useful as it allows us to consider aspects of sound propagation in terms of the wavelength.
However, most musical sounds have a waveform which is more complex than a simple sine wave and a selection is shown in Figure 1.45. How can we analyse real sound waveforms, and make sense of them in acoustical terms? The answer is based on the concept of superposition and a technique called Fourier analysis.
Figure 1.45 Waveforms from musical instruments.
1.6.1 The spectrum of periodic sound waves
Fourier analysis states that any waveform can be built up by using an appropriate set of sine waves of different frequencies, amplitudes and phases. To see how this might work consider the situation shown in Figure 1.46.
Figure 1.46 The effect of adding several harmonically related sine waves together.
This shows four sine waves whose frequencies are 1F Hz, 3F Hz, 5F Hz, and 7F Hz, whose phase is zero (that is, they all start from the same value, as shown by the dotted line in Figure 1.46) and whose amplitude is inversely proportional to the frequency. This means that the 3F Hz component is 1/3 the amplitude of the component at 1F Hz and so on. When these sine waves are added together, as shown in Figure 1.46, the result approximates a square wave, and, if more high frequency components were added, it would become progressively closer to an ideal square wave.
The higher frequency components are needed in order to provide the fast rise, and sharp corners, of the square wave. In general, as the rise time gets faster, and/or the corners get sharper, then more high frequency sine waves are required to represent the waveform accurately. In other words we can look at a square wave as a waveform that is formed by summing together sine waves which are odd multiples of its fundamental frequency and whose amplitudes are inversely proportional to frequency.