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5 Acoustics in Biology and Medicine
angle. A laser source (near one frequency) passing through a thin slit (with a gap of
a tenth of a millimeter, or so) will generate an evident diffraction pattern on a far
wall.
Note that from Eq. (5.6) the pressure of a wave moving in one direction satisfies
p = p o ± ρ o v
∂ξ
∂t
.
(5.23)
We will use this relation when describing the energy carried by a wave.
5.2 Fourier Decomposition of Sound
Sound is often generated as a complicated wave, i.e. one that contains many more
than just one note. Your speech contains a variety of simultaneous waves at various
frequencies and amplitudes. Short pulses of sound contain a wide distribution
of frequencies. If all frequencies in a given range are present with nearly equal
amplitudes but random phases, we call such a wave ‘white noise’, in analogy to
white light, which contains all visible colors with about equal intensity. In contrast,
the sounds from which we get information and pleasure have temporal structure.
Musicians may even poetically call such sounds colorful.
In the analysis of sound, we can express properties of the sound according
to what frequencies are present, and with what amplitudes. Mathematically, the
frequency-amplitude analysis of a sound is a Fourier decomposition. A general
discussion of the Fourier decomposition of waves is given in Appendix H. The
idea is this: No matter how complicated a wave may be, that wave can always
be decomposed into a summation of simple harmonically-related sinusoidal waves
with various amplitudes. As the wave equation for sound is linear in the sound
amplitude, a linear combination of solutions will again be a solution. So, if we can
find simple sinusoidal waves which differ from each other only in frequency (with
a corresponding wavelength), then we have found the general solution to the wave
equation.
There are many practical examples of Fourier series. The decomposition of a
pure sound (or, in music, a pure note), will have just one non-zero term in the series.
In contrast, a repeating but short duration clicking sound will have numerous high
frequency terms in the harmonic analysis. The decomposition of a ‘square wave’
shows a dominant low frequency wave, but also many high frequency waves, which
are needed to make up the sharp rising and falling part of the wave.
In a Fourier series for a wave with period T , the amplitude of the harmonic
component cosine wave of frequency 2πj/T is usually called a j , while the
amplitude of the component sine wave is called b j . Figure 5.2 shows a graph of
a square wave and the result of adding the first five harmonics present within the
wave. If the line t = 0 in the graph is centered on a peak of the wave, then all
5 Acoustics in Biology and Medicine
angle. A laser source (near one frequency) passing through a thin slit (with a gap of
a tenth of a millimeter, or so) will generate an evident diffraction pattern on a far
wall.
Note that from Eq. (5.6) the pressure of a wave moving in one direction satisfies
p = p o ± ρ o v
∂ξ
∂t
.
(5.23)
We will use this relation when describing the energy carried by a wave.
5.2 Fourier Decomposition of Sound
Sound is often generated as a complicated wave, i.e. one that contains many more
than just one note. Your speech contains a variety of simultaneous waves at various
frequencies and amplitudes. Short pulses of sound contain a wide distribution
of frequencies. If all frequencies in a given range are present with nearly equal
amplitudes but random phases, we call such a wave ‘white noise’, in analogy to
white light, which contains all visible colors with about equal intensity. In contrast,
the sounds from which we get information and pleasure have temporal structure.
Musicians may even poetically call such sounds colorful.
In the analysis of sound, we can express properties of the sound according
to what frequencies are present, and with what amplitudes. Mathematically, the
frequency-amplitude analysis of a sound is a Fourier decomposition. A general
discussion of the Fourier decomposition of waves is given in Appendix H. The
idea is this: No matter how complicated a wave may be, that wave can always
be decomposed into a summation of simple harmonically-related sinusoidal waves
with various amplitudes. As the wave equation for sound is linear in the sound
amplitude, a linear combination of solutions will again be a solution. So, if we can
find simple sinusoidal waves which differ from each other only in frequency (with
a corresponding wavelength), then we have found the general solution to the wave
equation.
There are many practical examples of Fourier series. The decomposition of a
pure sound (or, in music, a pure note), will have just one non-zero term in the series.
In contrast, a repeating but short duration clicking sound will have numerous high
frequency terms in the harmonic analysis. The decomposition of a ‘square wave’
shows a dominant low frequency wave, but also many high frequency waves, which
are needed to make up the sharp rising and falling part of the wave.
In a Fourier series for a wave with period T , the amplitude of the harmonic
component cosine wave of frequency 2πj/T is usually called a j , while the
amplitude of the component sine wave is called b j . Figure 5.2 shows a graph of
a square wave and the result of adding the first five harmonics present within the
wave. If the line t = 0 in the graph is centered on a peak of the wave, then all