The Physics of Sound



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Figure 3-19. Frequency response curves for three optical filters. The lowpass filter on the left allows low frequencies to pass through, while attenuating or blocking optical energy at higher frequencies. The highpass filter in the middle has the opposite effect, allowing high frequencies to pass through, while attenuating or blocking optical energy at lower frequencies. The bandpass filter on the right allows a band of optical frequencies in the center of the spectrum to pass through, while attenuating or blocking energy at higher and lower frequencies.


Figure 3-19 shows frequency response curves for several optical filters. Panel a shows a frequency response curve for the red optical filter discussed in the example above. If we put white light into the filter in panel a, the signal amplitude at the output of the filter will be high only when the frequency of the input signal is low. This is because the gain of the filter is high only in the low-frequency portion of the frequency-response curve. This is an example of a lowpass filter; that is, a filter that allows low frequencies to pass through. Panel b shows an optical filter that has precisely the reverse effect on an input signal; that is, this filter will allow high frequencies to pass through while attenuating low- and mid-frequency signals. A white surface viewed through this filter would therefore appear violet. This is an example of a highpass filter. Panel c shows the frequency response curve for a filter that allows a band of energy in the center of the spectrum to pass through while attenuating signal components of higher and lower frequency. A white surface viewed through this filter would appear green. This is called a bandpass filter.



Acoustic filters do for sound exactly what optical filters do for light; that is, they allow some frequencies to pass through while attenuating other frequencies. To get a better idea of how a frequency response curve is measured, imagine that we ask a singer to attempt to shatter a crystal wine glass with a voice signal alone. To see how the frequency response curve is created we have to make two rather unrealistic assumptions: (1) we need to assume that the singer is able to produce a series of pure tones of various frequencies (the larynx, in fact, produces a complex periodic sound and not a sinusoid), and (2) the amplitudes of these pure tones are always exactly the same. The wine glass will serve as the filter whose frequency response curve we wish to measure. As shown in Figure 3-20, we attach a vibration meter to the wine glass, and the reading on this meter will serve as our measure of output amplitude for the filter. For the purpose of this example, will assume that the signal frequency needed to break the glass is 500 Hz. We now ask the singer to produce a low frequency signal, say 50 Hz. Since this frequency is quite remote from the 500 Hz needed to break the glass, the output amplitude measured by the vibration meter will be quite low. As the singer gets closer and closer to the required 500 Hz, the measured output amplitude will increase systematically until the glass finally breaks. If we assume that the glass does not break but rather reaches a maximum amplitude just short of that required to shatter the glass, we can continue our measurement of the frequency response curve by asking the singer to produce signals that are increasingly high in frequency. We would
find that the output amplitude would become lower and lower the further we got from the 500 Hz natural vibrating frequency of the wine glass. The pattern that is traced by our measures of output amplitude at each signal frequency would resemble the frequency response curve we saw earlier for green sunglasses; that is, we would see the frequency response curve for a bandpass filter.



Figure 3-20. Illustration of how the frequency response curve of a crystal wine glass might be measured. Our singer produces a series of sinusoids that are identical in amplitude but cover a wide range of frequencies. (This part of the example is unrealistic: the human larynx produces a complex sound rather than a sinusoid.) The gain of the wine glass filter can be traced out by measuring the amplitude of vibration at the different signal frequencies.)




Additional Comments on Filters
Cutoff Frequency, Center Frequency, Bandwidth. The top panel of Figure 3-21 shows frequency response curves for two lowpass filters that differ in a parameter called cutoff frequency. Both filters allow low frequencies to pass through while attenuating high frequencies; the filters differ only in the frequency at which the attenuation begins. The bottom panel of Figure 3-21 shows two highpass filters that differ in cutoff frequency. There are two additional terms that apply only to bandpass filters. In our wineglass example above, the natural vibrating frequency of the wine glass was 300 Hz. For this reason, when the frequency response curve is measured, we find that the wine glass reaches its maximum output amplitude at 300 Hz. This is called the center frequency or resonance of the filter. It is possible for two bandpass filters to have the same center frequency but differ with respect to a property called bandwidth. Figure 3-22 shows two filters that differ in bandwidth. The tall, thin frequency response curve describes a narrow band filter. For this type of filter, output amplitude reaches a very sharp peak at the center frequency and drops off abruptly on either side of the peak. The other frequency response curve describes a wide band filter (also called broad band). For the wide band filter, the peak that occurs at the resonance of the filter is less sharp and the drop in output amplitude on either side of the center frequency is more gradual.


F
ixed vs. Variable Filters.
A fixed filter is a filter whose frequency response curve cannot be altered. For example, an engineer might design a lowpass filter that attenuates at frequencies above 500 Hz, or a bandpass filter that passes with a center frequency of 1,000 Hz. It is also possible to create a filter whose characteristics can be varied. For e
Figure 3-21. Lowpass and highpass filters differing in cutoff frequency.

xample, the tuning dial on a radio controls the center frequency of a narrow bandpass filter that allows a single radio channel to pass through while blocking channels at all other frequencies. The human vocal tract is an example of a variable filter of the most spectacular sort. For example: (1) during the occlusion interval that occurs in the production of a sound like /b/, the vocal tract behaves like a lowpass filter; (2) in the articulatory posture for sounds like /s/ and /sh/ the vocal tract behaves like a highpass filter; and (3) in the production of vowels, the vocal tract behaves like a series of bandpass filters connected to one another, and the center frequencies of these filters can be adjusted by changing the positions of the tongue, lips, and jaw. To a very great extent, the production of speech involves making adjustments to the articulators that have the effect of setting the vocal tract filter in differ modes to produce the desired sound quality. We will have much more to say about this in later chapters.



Figure 3-22. Frequency response curves for two bandpass filters with identical center frequencies but different bandwidths. Both filters pass a band of energy centered around 2000 Hz, but the narrow band filter is more selective than the wide band filter; that is, gain decreases at a higher rate above and below the center frequency for the narrow band filter than for the wide band filter

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Frequency Response Curves vs. Amplitude Spectra. It is not uncommon for students to confuse a frequency response curve with an amplitude spectrum. The axis labels are rather similar: an amplitude spectrum plots amplitude on the y axis and frequency on the x axis, while a frequency response curve plots gain on the y axis and frequency on the x axis. The apparent similarities are deceiving, however, since a frequency response curve and an amplitude spectrum display very different kinds of information. The difference is that an amplitude spectrum describes a sound while a frequency response curve describes a filter. For any given sound wave, an amplitude spectrum tells us what frequencies are present with what amplitudes. A frequency response curve, on the other hand, describes a filter, and for that filter, it tells us what frequencies will be allowed to pass through and what frequencies will be attenuated. Keeping these two ideas separate will be quite important for understanding the key role played by filters in both hearing and speech science.
Resonance
The concept of resonance has been alluded to on several occasions but has not been formally defined. The term resonance is used in two different but very closely related ways. The term resonance refers to: (1) the phenomenon of forced vibration, and (2) natural vibrating frequency (also resonant frequency or resonance frequency) To gain an appreciation for both uses of this term, imagine the following experiment. We begin with two identical tuning forks, each tuned to 435 Hz. Tuning fork A is set into vibration and placed one centimeter from tuning fork B, but not touching it. If we now hold tuning fork B to a healthy ear, we will find that it is producing a 435 Hz tone that is faint but quite audible, despite the fact that it was not struck and did not come into physical contact with tuning fork A. The explanation for this "action-at-a-distance" phenomenon is that the sound wave generated by tuning fork A forces tuning fork B into vibration; that is, the series of compression and rarefaction waves will alternately push and pull the tuning fork, resulting in vibration at the frequency being generated by tuning fork A. The phenomenon of forced vibration is not restricted to this "action-at-a-distance" case. The same effect can be demonstrated by placing a vibrating tuning fork in contact with a desk or some other hard surface. The intensity of the signal will increase dramatically because the tuning fork is forcing the desk to vibrate, resulting in a larger volume of air being compressed and rarefied.3
Returning to our original tuning fork experiment, suppose that we repeat this test using two mismatched tuning forks; for example, tuning fork A with a natural frequency of 256 Hz and tuning fork B with a natural vibrating frequency of 435 Hz. If we repeat the experiment – setting tuning fork A into vibration and holding it one centimeter from tuning fork B – we will find that tuning fork B does not produce an audible tone. The reason is that forced vibration is most efficient when the frequency of the driving force is closest to the natural vibration frequency of the object that is being forced to vibrate. Another way to think about this is that tuning fork B in these experiments is behaving like a filter that is being driven by the signal produced by tuning fork A. Tuning forks, in fact, behave like rather narrow bandpass filters. In the experiment with matched tuning forks, the filter was being driven by a signal frequency corresponding to the peak in the filter's frequency response curve. Consequently, the filter produced a great deal of energy at its output. In the experiment with mismatched tuning forks, the filter is being driven by a signal that is remote from the peak in the filter's frequency response curve, producing a low amplitude output signal.
To summarize, resonance refers to the ability of one vibrating system to force another system into vibration. Further, the amplitude of this forced vibration will be greater as the frequency of the driving force approaches the natural vibrating frequency (resonance) of the system that is being forced into vibration.
Cavity Resonators
An air-filled cavity exhibits frequency selective properties and should be considered a filter in precisely the way that the tuning forks and wine glasses mentioned above are filters. The human vocal tract is an air-filled cavity that behaves like a filter whose frequency response curve varies depending on the positions of the articulators. Tuning forks and other simple filters have a single resonant frequency. (Note that we will be using the terms "natural vibrating frequency" and "resonant frequency" interchangeably.) Cavity resonators, on the other hand, can have an infinite number of resonant frequencies.


Figure 3-23. Frequency response curves for three uniform tubes open at one end and closed at the other. These kinds of tubes have an infinite number of resonances at odd multiples of the lowest resonance. As the figure shows, shortening the tube shifts all resonances to higher frequencies while lengthening the tube shifts all resonances to lower frequencies.





A simple but very important cavity resonator is the uniform tube. This is a tube whose cross-sectional area is the same (uniform) at all points along its length. A simple water glass is an example of a uniform tube. The method for determining the resonant frequency pattern for a uniform tube will vary depending on whether the tube is closed at both ends, open at both ends, or closed at just one end. The configuration that is most directly applicable to problems in speech and hearing is the uniform tube that is closed at one end and open at the other end. The ear canal, for example, is approximately uniform in cross-sectional area and is closed medially by the ear drum and open laterally. Also, in certain configurations the vocal tract is approximately uniform in cross-sectional area and is effectively closed from below by the vocal folds and open at the lips. The resonant frequencies for a uniform tube closed at one end are determined by its length. The lowest resonant frequency (F1) for this kind of tube is given by:
F1 = c/4L, where: c = the speed of sound

L = the length of the tube


For example, for a 17.5 cm tube, F1 = c/4L = 35000/70 = 500 Hz. This tube will also have an infinite number of higher frequency resonances at odd multiples of the lowest resonance:
F1 = F1 . 1 = 500 Hz

F2 = F1  3 = 1,500 Hz

F3 = F1 . 5 = 2,500 Hz

F4 = F1  7 = 3,500 Hz

The frequency response curve for this tube for frequencies below 4000 Hz is shown in the solid curve in Figure 3-23. Notice that the frequency response curve shows peaks at 500, 1500, 2500, and 3500 Hz, and valleys in between these peaks. The frequency response curve, in fact, looks like a number of bandpass filters connected in series with one another. It is important to appreciate that what we have calculated here is a series of natural vibrating frequencies of a tube. What this means is that the tube will respond best to forced vibration if the tube is driven by signals with frequencies at or near 500 Hz, 1500 Hz, 2500 Hz, and so on. Also, the resonant frequencies that were just calculated should not be confused with harmonics. Harmonics are frequency components that are present in the amplitude spectra of complex periodic sounds; resonant frequencies are peaks in the frequency response curve of filters.
We next need to see what will happen to the resonant frequency pattern of the tube when the tube length changes. If the tube is lengthened to 20 cm:
F1 = c/4L = 35,000/80 = 437.5 Hz

F2 = F1 3 = 1,312.5 Hz

F3 = F1  5 = 2,187.5 Hz

F4 = F1  7 = 3,062.5 Hz


It can be seen that lengthening the tube from 17.5 cm to 20 cm has the effect of shifting all of the resonant frequencies downward (see Figure 3-23). Similarly, shortening the tube has the effect of shifting all of the resonant frequencies upward. For example, the resonant frequency pattern for a 15 cm tube would be:
F1 = c/4L = 35,000/60 = 583.3 Hz

F2 = F13 = 1,750 Hz

F3 = F1  5 = 2,916.7 Hz

F4 = F1 7 = 4,083.3 Hz


The general rule is quite simple: all else being equal, long tubes have low resonant frequencies and short tubes have high resonant frequencies. This can be demonstrated easily by blowing into bottles of various lengths. The longer bottles will produce lower tones than shorter bottles. This effect is also demonstrated every time a water glass is filled. The increase in the frequency of the sound that is produced as the glass is filled occurs because the resonating cavity becomes shorter and shorter as more air is displaced by water. This simple rule will be quite

useful. For example, it can be applied directly to the differences that are observed in the acoustic properties of speech produced by men, women, and children, who have vocal tracts that are quite different in length.


Resonant Frequencies and Formant Frequencies
The term "resonant frequency" refers to natural vibrating frequency or, equivalently, to a peak in a frequency response curve. For reasons that are entirely historical, if the filter that is being described happens to be a human vocal tract, the term formant frequency is generally used. So, one typically refers to the formant frequencies of the vocal tract but to the resonant frequencies of a plastic tube, the body of a guitar, the diaphragm of a loudspeaker, or most any other type of filter other than the vocal tract. This is unfortunate since it is possible to get the mistaken idea that formant frequencies and resonant frequencies are different sorts of things. The two terms are, in fact, fully synonymous.
The Decibel Scale
The final topic that we need to address in this chapter is the representation of signal amplitude using the decibel scale. The decibel scale is a powerful and immensely flexible scale for representing the amplitude of a sound wave. The scale can sometimes cause students difficulty because it differs from most other measurement scales in not just one but two ways. Most of the measurement scales with which we are familiar are absolute and linear. The decibel scale, however, is relative rather than absolute, and logarithmic rather than linear. Neither of these characteristics is terribly complicated, but in combination they can make the decibel scale appear far more obscure than it is. We will examine these features one at a time, and then see how they are put together in building the decibel scale.
Linear vs. Logarithmic Measurement Scales
Most measurement scales are linear. To say that a measurement scale is linear means that it is based on equal additive distances. This is such a common feature of measurement scales that we do not give it much thought. For example, on a centigrade (or Fahrenheit) scale for measuring temperature, going from a temperature of 90o to a temperature of 91o involves adding one 1o. One rather obvious consequence of this simple additivity rule is that the difference in temperature between 10o and 11o is the same as the difference in temperature between 90o and 91o. However, there are scales for which this additivity rule does not apply. One of the best known examples is the Richter scale that is used for measuring seismic intensity. The difference in seismic intensity between Richter values of 4.0 and 5.0, 5.0 and 6.0, 6.0 and 7.0 is not some constant amount of seismic intensity, but rather a constant multiple. Specifically, a 7.0 on the Richter scale indicates an earthquake that is 10 times greater in intensity than an earthquake that measures 6.0 on the Richter scale. Similarly, an 8.0 on the Richter scale is 10 times greater in intensity than a 7.0. Whenever jumping from one scale value to the next involves multiplying by a constant rather than adding a constant, the scale is called logarithmic. (The multiplicative constant need not be 10. See Box 3-2 for an example of a logarithmic scale – an octave progression – that uses 2 as the constant.) Another way of making the same point is to note that the values along the Richter scale are exponents rather than ordinary numbers; for example, a Richter value of 6 indicates a seismic intensity of 106, a Richter value of 7 indicates a seismic intensity of 107, etc. The Richter values can, of course, just as well be referred to as powers or logarithms since both of these terms are synonyms for exponent. The decibel scale is an example of a logarithmic scale, meaning that it is based on equal multiples rather than equal additive distances.
Absolute vs. Relative Measurement Scales
A simple example of a relative measurement scale is the Mach scale that is used by rocket scientists to measure speed. The Mach scale measures speed not in absolute terms but in relation to the speed of sound. For example, a missile at Mach 2.0 is traveling at twice the speed of sound, while a missile at Mach 0.9 is traveling at 90% of the speed of sound. So, the Mach scale does not represent a measured speed (Sm) in absolute terms, but rather, represents a measured speed in relation to a reference speed (Sm/Sr). The reference that is used for the Mach scale is the speed of sound, so a measured absolute speed can be converted to a relative speed on the Mach scale by simple division. For example, taking 783 mph as the speed of sound, 1,200 mph = 1200/783 = Mach 1.53. The decibel scale also exploits this relative measurement scheme. The decibel scale does not represent a measured intensity (Im) in absolute terms, but rather, represents the ratio of a measured intensity to a reference intensity (Im/Ir).
The decibel scale is trickier than the Mach scale in one important respect. For the Mach scale, the reference is always the speed of sound, but for the decibel scale, many different references can be used. In explaining how the decibel scale works, we will begin with the commonly used intensity reference of 10-12 w/m2 (watts per square meter), which is approximately the intensity that is required for an average normal hearing listener to barely detect a 1,000 Hz pure tone. So, for our initial pass through the decibel scale, 10-12 w/m2 will serve as Ir, and will perform the same function that the speed of sound does for the Mach scale. Table 3-1 lists several sounds that cover a very broad range of intensities. The second column shows the measured intensities of those sounds, and the third column shows the ratio of those intensities to our reference intensity. Whispered speech, for example, measures approximately 10-8 w/m2, which is 10,000 times more intense than the reference intensity (10-8/10-12 = 104 = 10,000). The main point to be made about column 3 is that the ratios become very large very soon. Even a moderately intense sound like conversational speech is 1,000,000 times more intense than the reference intensity. The awkwardness of dealing with these very large ratios has a very simple solution. Column 4 shows the ratios written in exponential notation, and column 5 simplifies the situation even further by recording the exponent only. The term exponent and the term logarithm are synonymous, so the measurement scheme that is expressed by the numbers in column 5 can be summarized as follows: (1) divide a measured intensity by a reference intensity (in this case, 10-12 w/m2), (2) take the logarithm of this ratio (i.e., write the number in exponential notation and keep the exponent only). This method, in fact, is a completely legitimate way to represent signal intensity. The unit of measure is called the bel, after A.G. Bell, and the formula is:
bel = log10 Im/Ir, where: Im = a measured intensity

Ir = a reference intensity


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Table 3-1. Sound intensities and intensity ratios showing how the decibel scale is created. Column 2 shows the measured intensities (Im) of several sounds. Column 3 shows the ratio of these intensities to a reference intensity of 10-12 w/m2. Column 4 shows the ratio written in exponential notation while column 5 shows the exponent only. The last column shows the intensity ratio expressed in decibels, which is simply the logarithm of the intensity ratio multiplied by 10.

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