chemical hazards and noise measure

19

After reading this chapter, you should be able to:

1. Describe how sound waves are produced, how they propagate, how fast they travel through air, and how they change with distance.

2. Dene frequency, period, amplitude, starting phase, and wave- length; interpret time-domain waveforms of pure tones with different frequencies, amplitudes, and starting phases.

3. Dene how intensity and pressure are related to each other; specify the minimum reference levels for intensity and pres- sure; specify the range of audibility for intensity (in watts/m2) and pressure (in µPa).

4. Understand why and how to use decibels to quantify intensity and pressure; describe the range of audibility of intensity and pressure using decibels; dene dB IL and dB SPL; describe the threshold of audibility across frequency.

5. Perform simple decibel calculations to compare the intensity and/or pressure of two sounds.

6. Explain the inverse square law and calculate how intensity or pressure changes with changes in distance.

7. Understand how to combine the outputs of two sounds and the resulting dB IL and dB SPL.

8. Describe periodic and aperiodic complex vibrations; interpret time-domain and spectral graphs of complex vibrations; describe the importance of Fourier analyses.

9. Describe the basic acoustic characteristics of speech and un- derstand how to read spectrograms.

10. Understand how ltering can be used to shape the spectrum of noise; recognize commonly used lter shapes.

11. Explain what is meant by resonance; calculate resonance frequencies for simple tubes of varying length (open at both ends or only at one end); know the difference between a half- wave resonator and quarter-wave resonator.

Properties of Sound3

C o p y r i g h t 2 0 1 9 . P l u r a l P u b l i s h i n g , I n c .

A l l r i g h t s r e s e r v e d . M a y n o t b e r e p r o d u c e d i n a n y f o r m w i t h o u t p e r m i s s i o n f r o m t h e p u b l i s h e r , e x c e p t f a i r u s e s p e r m i t t e d u n d e r U . S . o r a p p l i c a b l e c o p y r i g h t l a w .

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AUDIOLOGY: SCIENCE TO PRACTICE20

We live in a world of sounds, some of which are meaningful and some of which are just part of our noisy environment. We often take for granted the remarkable ability of the auditory system to extract meaningful sounds from the less mean- ingful so that we can sense danger, localize the source of a sound, communicate, learn, and even be entertained. Even when asleep we learn to tune out familiar sounds, but may wake up at an unfamiliar sound. At a noisy party, you can focus on a conversation with one person while ignor- ing the background conversations, but readily become aware when someone calls your name from across the room or your favorite song be- gins. When you listen to an orchestra or band you may find yourself listening to the whole song or picking out the various instruments. Our ability to hear in our everyday world requires the auditory system to process complex sounds from our envi- ronment. The process of hearing involves the gen- eration of sounds, their travels and interactions within the environment, physiological processing by the ear, neural processing in the nervous sys- tem, and psychological/cognitive processing by the brain. The sounds we hear have basic physi- cal properties that are processed by the auditory system into meaningful information.

Acoustics is the study of the physical prop- erties of sounds in the environment, how they travel through air, and how they are affected by objects in their environment. As you will see in this chapter, any simple vibration can be uniquely described by its frequency, amplitude, and starting phase. Complex vibrations can be described as combinations of simple vibrations. However, not all sounds generated in the envi- ronment are audible and the audible range may be different across species; for example, dogs and cats are more responsive to higher pitched sounds than are humans. The human ear is ca- pable of hearing a wide range of frequencies over an extensive range of amplitudes. But how

does frequency relate to our perception of pitch? How does amplitude relate to our perception of loudness? How do we compare the loudness of sounds across frequencies? How do we use our two ears to localize the source of sounds? These types of questions come under the area of psychoacoustics, which is the study of how we perceive sound. The psychoacoustic aspects of sound covered in this chapter include some basic perceptions of pitch, loudness, temporal integra- tion, and localization. After reading this chapter, perhaps you will be able to answer the age-old philosophical question that goes something like, “If a tree falls in the woods and there are no living creatures around, does it make a sound?”

The definitions and terminology reviewed in this chapter are necessary to be able to bet- ter understand topics that are covered in the fol- lowing chapters, including the physiology of the auditory system, the clinical procedures used to evaluate hearing loss, and the function of hear- ing aids. A thorough understanding of acoustics requires knowledge of some mathematical con- cepts and formulas; however, in this introductory text, only the basic concepts are presented and every attempt is made to keep the mathematics to a minimum. The interested reader is referred to other textbooks (Gelfand, 2009; Mullin, Gerace, Mestre, & Velleman, 2003; Speaks, 2017; Villchur, 2000) for a more thorough treatment of acoustics and psychoacoustics.

SIMPLE VIBRATIONS AND SOUND TRANSMISSION

Sounds are produced because of an object being set into vibration. Some familiar examples in- clude vibrations of tuning forks, guitar strings, other musical instruments, stereo speakers, en- gines, thunder, and the vocal cords while speak- ing. Almost any object can be made to vibrate,

12. Discuss and interpret graphs related to the psychoacoustic (perceptual) properties of loudness, pitch, temporal integration, and localization.

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3. PRoPERTIES oF SoUnD 21

but some objects vibrate more easily than other objects depending on their mass and elasticity. Although most sounds in our environment are complex vibrations, we begin by looking at very simple vibrations called pure tones. Pure tones are used by audiologists as part of the basic hearing evaluation. In addition, an understand- ing of pure tones is useful because all complex vibrations can be described as combinations of different pure tones, which was mathematically proven by a man named Fourier. Today, we have electronic instruments that can perform fast Fou- rier transforms (FFTs) to determine the different pure tones that comprise any complex vibration.

The vibrating sound source sets up sound waves that travel, called propagation ( propa- gate), through some elastic medium, such as air, water, and most solids. Propagation of sound through air occurs because of the back and forth movement of air molecules around their position of equilibrium in response to the back and forth vibration of an object. The air mole- cules closest to the vibrating object move back and forth first. Because of the inertial and elastic properties of the air molecules, the air molecules only move within a localized region, but as they push against adjacent air molecules the process repeats itself, which causes the pressure varia- tions to propagate through the medium. When the vibrating object moves outward, the air mol- ecules are pushed together causing an increase in the density of air molecules (more molecules per volume), called condensation, and this cor- responds to an increase in sound pressure. When the vibrating object moves in the opposite direc- tion, there is a decrease in the density of air mol- ecules, called rarefaction, and this corresponds to a decrease in sound pressure. Figure 3–1 il- lustrates how these increases and decreases in the density of air molecules occur in response to a simple vibrating object such as a tuning fork. When the vibration repeats itself over and over, as depicted in Figure 3–1, there are continuing cycles of condensation and rarefaction that pro- duce a continuous sound that can be measured at different points in the surrounding area. In Figure 3–1, you can see the areas in which the air molecules are more densely packed (conden- sations) and where the air molecules are less densely packed (rarefactions). The condensa-

tions and rarefactions reflect a repetitive pattern of increasing and decreasing air pressure. For un- obstructed sound waves in air, the air molecules move outward in a spherical direction and the actual size of the air pressure peak (amplitude) diminishes with distance because of friction, as well as because the pressure is being radiated in an increasing spherical pattern. At some distance from the source, the pressure will no longer be measurable because the energy is spread out over a large enough spherical area. The actual

A

B

Tuning Fork

rarefac on condensa on

rarefac on

condensa on

FIGURE 3–1. A and B. Illustration showing pro p- agation of air molecules to a vibrating sound source. A. Tuning fork vibration producing alternating areas of increased density of air molecules (condensation) and decreased density of air molecules (rarefaction) that are propagated across the air from its source. B. Sound waves as they propagated spherically away from the sound source with alternating condensation and rarefaction phases. As the distance from the sound source increases, the force is distributed over a wider area.

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AUDIOLOGY: SCIENCE TO PRACTICE22

amplitude of a sound at any point in space ob- viously depends on the original intensity level of the sound, that is, louder sounds will travel greater distances than softer sounds.

Sound propagation can also be influenced by how the waves are reflected or interfered with by objects or walls. Much of our real-world listening situations are in closed environments, whereby much of the sound energy does not penetrate the walls but instead bounces off or is absorbed by the walls. The angle at which a sound will bounce off a wall is similar to a ball bouncing off a wall. The angle of reflection will depend on the angle of incidence relative to the perpendic- ular. This becomes even more complicated when the encountered object is curved (convex or con- cave), or in a room with four walls, where the sound may bounce back and forth among the walls. How sound waves might interact with an object in its environment is also important. Some sounds will bounce off an object, whereas other sounds easily go around the object, and depends primarily on the sound’s wavelength (see section on wavelength). As you will learn in the follow- ing sections, there are also areas in which the con- densation phase of a wave meets up with another wave’s rarefaction phase, resulting in wave can- cellation (where no sound is present). In addition, materials have certain absorption characteristics that come into play in determining how sounds act in the real world. Understanding acoustics in these types of environments is especially im- portant when designing theater or music venues (something acoustic engineers are trained to do, but it is well beyond the scope of this textbook).

Another characteristic of sound waves is the speed or velocity with which they are prop- agated through the medium. Sound travels faster in water and most solids than it does in air. The speed of sound in air is about 343 m/s or 1126 feet/s,1 which is much slower than the 186,282 miles per second that light travels. You probably use this knowledge, maybe unknow-

1 The speed of sound in air is dependent upon both the temperature and the density. The value used in this textbook is an approximation for 68°F. The speed of sound in air slows down as temperature decreases, for example, it is about 341 m/s or 1086 feet/s at 32°F.

ingly, when you estimate how many miles away you are from a storm by counting the seconds be- tween seeing the lightning (seen instantaneously) and hearing the thunder (heard later). Your esti- mate of how far away the storm is will be more accurate if you divide the number of counted sec- onds by five to take into account that the speed of sound is about one-fifth of a mile per second.

When the increases and decreases in pres- sure occur in the direction of the vibrating ob- ject, as for sound waves, the sound is called a longitudinal wave. The process of localized back and forth movement of air molecules results in the propagation of a longitudinal sound wave through the air, more precisely in a spherical pat- tern. When this sound wave reaches the ear, the corresponding condensations and rarefactions in air pressure cause the tympanic membrane to move in and out, thus beginning the process of hearing. You will see in the next chapter how vibrations are received by the ear and how the ear transforms the incoming vibrations into audi- tory information. Before that, however, we need to turn our attention to understanding the basic physical parameters of sound, frequency, ampli- tude, and starting phase.

FREQUENCY

Pure tones are characterized by regular repeti- tive movements. Imagine holding a pencil in your hand and moving it up and down on a piece of paper at a consistent height and speed. As you are moving your hand up and down, begin to move the paper from right to left; you should see a pattern that looks something like those shown in Figure 3–2. The actual separation of the peaks that are produced will depend on the speed at which you move the paper (the slower the paper, the closer the peaks). To be able to quantify the pattern of vibratory movement, the motion is displayed as a function of time along the x-axis. The y -axis represents a measure of magnitude or amplitude of the vibrations (e.g., how far up and down you moved your hand). When the pattern of movement is displayed with amplitude as a function of time, it is called a time-domain wave- form or simply a waveform.

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3. PRoPERTIES oF SoUnD 23

A cycle of vibration describes the pattern of movement as the object goes through its full range of motion one time. In other words, one cycle represents the movement of an object from its starting point to its maximum peak, then to its negative peak, then back to its starting point. Figure 3–3 shows one cycle of a pure tone.

Most vibrations repeat themselves; therefore, pure tones are usually described by how many cycles occur in 1 second (s), called frequency of vibration. However, instead of using cycles per second as the unit of measure for frequency, the term hertz (Hz) is used to mean the same thing. For example, a vibration that repeats itself 100 cycles in 1 s is called a 100 Hz pure tone. Con- versely, a 100 Hz pure tone would complete 100

cycles in 1 s. An 8000 Hz pure tone completes 8000 cycles in 1 second. The frequency range of audibility for humans is from 20 to 20,000 Hz.

Figure 3–4 shows some examples of differ- ent frequencies as they would appear on paper when graphed with a 1 s time scale. As you can notice, it is difficult to visually count the number of cycles as the frequency increases, and count- ing would be extremely difficult for much of the audible frequency range if graphed using a 1 s time scale. However, another way to graphically represent the different frequencies of pure tones is to change the time scale along the x-axis. In other words, only a few cycles (or even a single cycle) are plotted over a specified time scale. The actual frequency is calculated from knowing how long it takes to complete one cycle, called the pe- riod of the vibration. Figure 3–5 shows some ex- amples of how the period is related to frequency. In Figure 3–5A, you can see that the time it takes to complete the one cycle is equal to 0.01 s (one hundredth of a second), which means it would be able to complete 100 cycles in 1.0 s (100 Hz). In Figure 3–5B, the time it takes to complete the one cycle is 0.001 s, which means this vibration would be able to complete 1000 cycles in 1 s (1000 Hz). In Figure 3–5C, the time it takes to complete the

Time (arbitrary)

A m

pl itu

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ar bi

tr ar

y)

Time (arbitrary)

A m

pl itu

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ar bi

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A

B

FIGURE 3–2. A and B. Representations of two dif- ferent pure-tone vibration patterns as a function of time in arbitrary units. The vibration in (A) is slower than the vibration in (B) even though the time scales are equal.

FIGURE 3–3. Time domain waveform showing one cycle of vibration. The vibration moves from its start- ing point to its maximum peak (amplitude), then to its negative peak, then back to its starting point as a function of time.

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AUDIOLOGY: SCIENCE TO PRACTICE24

one cycle is 0.0001 s, which means this vibration would be able to complete 10,000 cycles in 1 s (10,000 Hz). You can see that there is a reciprocal trade-off between the period and the frequency. The following equation shows how you can cal- culate the period (T ) if you know the frequency,

or how you can calculate the frequency ( f ) if you know the period:

T (in seconds) = 1/f (in hertz)

f (in hertz) = 1/T (in seconds)

This inverse relation means that as the frequency increases, the period decreases and vice versa. It is also important to keep in mind that when frequency is described in hertz (Hz), the period would be calculated as seconds. However, other

Time (seconds)

0.0 0.2 0.4 0.6 0.8 1.0

A m

pl itu

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ar bi

tr ar

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Time (seconds)

0.0 0.2 0.4 0.6 0.8 1.0

A m

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ar bi

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Time (seconds)

0.0 0.2 0.4 0.6 0.8 1.0

A m

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A

B

C

FIGURE 3–4. A–C. Examples of three different fre- quencies as they would appear over a 1.0 s time scale. The number of cycles per second determines the fre- quency of vibration. The more cycles per second, the higher the frequency.

seconds milliseconds

.010000.007500.005000.0025000 10.007.505.002.500

A m

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seconds milliseconds

.001000.000750.000500.0002500 1.000.750.500.250

A m

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ar bi

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y)

seconds milliseconds

.000100.000075.000050.0000250 0.1000.0750.0500.0250

A m

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ar bi

tr ar

y)

A

B

C

FIGURE 3–5. A–C. one cycle of vibration for three different frequencies, each plotted with a different time scale. The time it takes to complete one cycle is the period. In (A) the period is equal to 0.01 s (one- hundredth of a second), which means the vibrating object would be able to complete 100 cycles in 1.0 s (100 Hz). In (B), the period is 0.001 s, which means this vibration would be able to complete 1000 cycles in 1 s (1000 Hz). In (C) the period is equal to .0001 s, which means this vibration would be able to complete 10,000 cycles in 1 s (10,000 Hz).

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3. PRoPERTIES oF SoUnD 25

units are often used, and you must be sure to use the appropriate units when making conver- sions between frequency and period. For exam- ple, frequency is often measured in units of kilo- hertz (kHz) (kilo means 1000), such that 1 kHz = 1000 Hz, 2 kHz = 2000 Hz, and so forth. In ad- dition, the period of pure tones is often mea- sured in units of milliseconds (ms) (milli means 1/1000), such that 1 ms = .001 s, 2 ms = .002 s, and so forth. Table 3–1 shows the relation be- tween period and frequency for pure tones com- monly used in studies of hearing and hearing tests. As the pattern in Table 3–1 shows, for each doubling of frequency, the period decreases by half; and for each halving of frequency, the pe- riod doubles. To help understand the relations in Table 3–1, try covering one column at a time and see if you can fill in the correct information by using the information in the other columns. Fortunately, there is an electronic instrument, a frequency counter, that can be used to measure the frequency of pure tones.

PHASE

Pure tones are also called sine waves or sinu- soids because of their relationship to a sine func- tion. As illustrated in Figure 3–6, one cycle of a pure tone is the equivalent of making a full revolution around a circle, where each point on the waveform can be described by its sine func- tion relative to its phase angle (sin θ). You can think of a vibration starting at the object’s resting (non-vibratory) state, designated as zero degrees [sin (0) = 0], then reaching its maximum positive

peak at 90° [sin (90°) = 1)], returning to its initial point at 180° [sin (180°) = 0], reaching its max- imum negative peak at 270° [sin (270°) = −1], and finally returning to its starting point at 360° [sin (360°) = 0]. As Figure 3–6 shows, any point on the waveform can be found using the rela- tionship sin θ = x/r. For example, if θ = 45º, then:

x = r [sin (45º)]

x = r (0.707)

Starting phase refers to the point along the waveform’s cycle where the vibration begins, and is expressed in degrees relative to the angle around the circle. In other words, does the vi- bration first begin to move in the condensation direction or the rarefaction direction, and from what point does it begin? The waveforms shown in the previous figures have been plotted with a 0° starting phase, which means that the vibration begins from its equilibrium point and first moves toward the condensation peak, conventionally plotted as positive amplitude in the upward di- rection. Waveforms can begin at any point in their range of movement, and initially move to- ward the condensation peak or rarefaction peak. Figure 3–7 shows an example of a sinusoid with a 180° starting phase. In this case, the vibration

TABLE 3–1. Relationship between Frequency and Period (in Seconds and Milliseconds) for Commonly Used Frequencies

Frequency (Hz) Period (s) Period (ms)

250 0.004 4.0

500 0.002 2.0

1000 0.001 1.0

2000 0.0005 0.5

4000 0.00025 0.25

8000 0.000125 0.125

FIGURE 3–6. The projection of one cycle of a pure tone as it would appear relative to its position on a circle. one cycle of a waveform is the equivalent of making a full revolution around a circle. For example, the peak positive (condensation) point is equivalent to a 90º angle relative to the beginning point. The peak negative (rarefaction) point is equivalent to 270º (three-quarters around the circle). Equilibrium points occur at 0º, 180º, and 360º. These simple vibrations are often called sine waves because each point on the waveform can be expressed as a sine function (sin θ = x/r) relative to its angle (θ).

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AUDIOLOGY: SCIENCE TO PRACTICE26

begins at its equilibrium point, but first moves toward the rarefaction peak and continues its full cycle until it ends up back in equilibrium at the 180° starting point. Figure 3–8 shows an exam- ple of two waveforms with the same frequency, but with different starting phases, one with a 90° starting phase and one with a 270° starting phase. A 90° starting phase means that the vibration be- gins from its point of maximum condensation, moves to its equilibrium point (amplitude = 0), continues to its point of maximum rarefaction, back to its equilibrium point, and finally ends its cycle at the point of maximum condensation (where it began). A 270° starting phase means that the vibration begins from its point of maxi- mum rarefaction, moves to its equilibrium point, continues to its point of maximum condensation, back to its equilibrium point, and finally ends its cycle back at the point of maximum rarefaction (where it began).

Our ears are not sensitive, per se, to the start- ing phase; a pure tone with a starting phase of 0° will sound the same as with a starting phase of 270°. However, starting phase, or phase in general, has more relevance when two or more sounds interact with each other acoustically, be- fore reaching the ear. For the example in Fig- ure 3–8, can you predict what the resulting sound would be? If you answered, “no sound,” you

would be correct, since in this example the two waveforms would cancel each other out due to the condensation in one wave offset by the same amount of rarefaction in the other wave. Fig- ure 3–9 again demonstrates this phase inter- action with two relatively simple examples, in which two tones of the same frequency, but with opposite starting phases, are combined. For the two examples in Figure 3–9, the two tones are 180° out-of-phase with each other. Notice that the 180° out-of-phase relation between these (or any) two pure tones of the same frequency is maintained at all points in the waveform. Again, for these examples there would be no resulting sound pressure (and no sound) generated be- cause each condensation point would be can- celled out by an equal rarefaction point, and the net displacement would be zero. Figure 3–10 shows two examples of what happens when you combine two tones of the same frequency that are not 180° out of phase. In these examples, the phase relations of the two waves are more com- plicated and can produce places of cancellation when the points are in opposite phase directions or produce places of enhancement when the points are in the same phase direction. The in- teraction of two pure tones becomes even more complicated when they are of different frequen- cies as shown in Figure 3–11. In these relatively

Time (arbitrary)

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180º start phase

FIGURE 3–7. one cycle of a pure tone beginning at 180º starting phase. In this example, the movement begins in the direction of the rarefaction phase of vibration.

Time (arbitrary)

A m

pl itu

de (

ar bi

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90º start phase

270º start phase

FIGURE 3–8. Two pure tones at the same frequency, but with different starting phases. In this example, the two waveforms are 180º out of phase with each other.

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3. PRoPERTIES oF SoUnD 27

simple examples, the pairs of pure tones have the same 0° starting phase, but because they are of different frequencies the phase relation between the two pure tones changes at different points in time. The phase of an individual pure tone or

resultant combination of pure tones at any point in time is called the instantaneous phase. When a sound is made up of more than one pure tone, the resultant waveform no longer matches the pattern of a sine wave (i.e., simple wave) and is considered to be a complex vibration. As more pure tones are combined, with or without the same starting phases, the less the waveform re- sembles a sinusoid (complex vibrations are dis- cussed more in a later section of the chapter).

AMPLITUDE

Amplitude is a general term to describe the mag- nitude of a sound; the larger the magnitude, the higher the amplitude. Figure 3–12 shows the waveforms of pure tones with the same frequency and starting phase, but with different maximum amplitudes along the y -axis.2 For a vibrating

2 One can describe any sinusoidal vibration by the fol- lowing equation: a(t) = A sin (2π ft + θ) where a(t) is the instantaneous amplitude as a function of time, A is the maximum amplitude, 2πf (also called angular velocity, ω) is a measure of revolutions around a circle, and θ is the starting phase in radians.

A m

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Time (arbitrary)

A m

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Time (arbitrary)

450 start phase

900 start phase

250 start phase

2700 start phase

Sum 450 and 900 waves Sum 250 and 2700 waves

A B

FIGURE 3–10. A and B. Examples of how two pure tones of the same frequency, but with different start- ing phases, can result in different patterns of vibra- tion resulting from the summation of the two original waveforms.

A m

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ar bi

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Time (arbitrary)

A m

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ar bi

tr ar

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Time (arbitrary) A B

FIGURE 3–9. A and B. An illustration of how two pure tones of the same frequency, but 180º out of phase to each other, will cancel each other out. In (A) the solid curve represents a sound with a 0º starting phase and the dashed curve represents a sound with a 180º starting phase. In (B) the solid curve represents a sound with a 270º starting phase and the dashed curve re p- resents a sound with a 90º starting phase.

A m

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ar bi

tr ar

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Time (arbitrary)

A m

pl itu

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ar bi

tr ar

y)

Time (arbitrary) A B

FIGURE 3–11. A and B. Examples of how two pure tones with different frequencies, but with the same starting phase, combine to give different patterns of vibration resulting from the summation of the two orig- inal waveforms.

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AUDIOLOGY: SCIENCE TO PRACTICE28

object, maximum amplitude is related to how far the object moves back and forth. For sounds propagated through air, larger amplitudes create greater amounts of condensation and rarefaction of the air molecules. The amplitude scale along the y-axis is often expressed in units of displace- ment, intensity, or pressure.

Amplitudes vary across the waveform; there- fore, we need to have a way to specify the over- all amplitude of a waveform. Because pure tones have equal positive and negative amplitudes, tak- ing an average of the amplitudes at all points would result in zero amplitude and would not be useful at all. Instead, it is common to use the root-mean-square (RMS) amplitude (Arms) to ob- tain an average amplitude for the waveform. As shown in Figure 3–13, to obtain the RMS ampli- tude: (1) square each of the instantaneous ampli- tudes to eliminate any negative values, (2) aver- age the squared values, and (3) take the square root of the average. The RMS amplitude is used in many applications and, fortunately, there are electronic instruments available that directly mea- sure the RMS amplitudes of pure tones and other sounds. Figure 3–14 illustrates two other ways to describe the overall amplitude of pure tones. One way is to take the amplitude change between the positive peak and the negative peak, called peak- to-peak amplitude (Ap-p). Another way is to mea- sure the amplitude from baseline (zero) to one of the peaks, called peak amplitude (Ap). The RMS

amplitude for a pure tone is equal to 0.707 times the peak amplitude; however, this is not the case for more complex sounds.

INTENSITY AND PRESSURE

The overall amplitude of a sound wave is typically quantified and measured in units of sound pres- sure or sound intensity to describe how the sound

FIGURE 3–12. Illustration of pure tones of the same frequency with different amplitudes. notice how the period of the vibration is the same for all three wave- forms and only the height of the waveforms is different.

FIGURE 3–13. Quantication of amplitude by the method of root mean square (RMS). The instantaneous amplitudes across the waveform are squared to re- move the negative numbers, then averaged to nd the mean, and nally the square root of the number is de­ termined to get the total RMS value.

FIGURE 3–14. Illustration of how amplitude can be de- scribed based on its peak (Ap) and peak-to-peak (Ap-p) values. The RMS amplitude is equal to 0.707 × Ap.

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3. PRoPERTIES oF SoUnD 29

SYNOPSIS 3–1

l The back and forth vibration of a sound source sets up alternating areas where the air molecules are pushed together, called condensation, and areas where the air molecules are pulled apart, called rarefaction. The air molecules move only within a small localized area, but the condensation and rarefaction areas are passed to adjacent air molecules, which cause the waveform’s pressure variations to move through the medium. Sound waves travel in air at a speed of about 343 m/s (at 680 F).

l Simple vibrations, called pure tones, sine waves, or sinusoids, are characterized by their physical (acoustic) dimensions of frequency (or period), amplitude, and starting phase.

l The frequency of a sound refers to the number of vibrations that occur in 1 second and has units of hertz (Hz) or kilohertz (kHz); 1 kHz = 1000 Hz. The reciprocal of frequency is the period (T), which is the time it takes to complete one cycle of vibration and has units of seconds (s) or milliseconds (ms) (1 ms = 1/1000 s). If either the frequency or period is known, the other can be calculated (f = 1/T or T = 1/f ).

l The frequency range for human hearing is about 20 to 20,000 Hz. other species may be sensitive to other frequency ranges.

l The starting phase of a sound refers to what position in the cycle a vibration begins its cycle, and is expressed in units of degrees around a circle (0 to 360º). For example, a vibration that begins at its equilibrium and moves toward its area of maximum condensation has a starting phase of 0º; a vibration that begins at the point of maximum rarefaction and moves toward equilibrium has a starting phase of 270º.

l For a single pure tone, humans cannot tell the difference between starting phases. Combining two or more pure tones of the same frequency, but with different starting phases, results in different phase relations at different points in time (instantaneous phase) and produces a complex waveform that is the sum of the two waveforms. Two pure tones of the same frequency that are 180º out of phase will cancel each other out and not produce any sound.

l The amplitude of a sound refers to how far an object moves back and forth and/or the amount of maximum and minimum air pressure created. The larger the movement or pressure variation, the greater the amplitude for any given frequency. A simple way to describe amplitude of a visually displayed waveform is to measure the distance or pressure between the highest point of condensation peak and the lowest point of rarefaction peak, called peak-to- peak amplitude (Ap-p). Measures can also be made between the condensation or rarefaction peak to the equilibrium point, called peak amplitude (Ap). A more practical measure is called root-mean-square (RMS) amplitude, which is the way that instruments measure a sound’s overall amplitude. The RMS method averages the amplitudes across the entire waveform by squaring each value (to remove all negative values), then averaging these squared values, and nally taking the square root to bring it back into scale.

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AUDIOLOGY: SCIENCE TO PRACTICE30

energy is distributed over some area of the prop- agating wave. For any given sound wave, there is a corresponding sound intensity and sound pres- sure. If you know either the intensity or the pres- sure, the other quantity can be derived. Intensity and pressure are related to each other by the fol- lowing formulas:

I = p 2

p = √I

It should be intuitive that the farther away you are from a sound’s source, the softer it will be- come. As the sound gets further away from the sound source (assuming there are no obstruc- tions), it is distributed over a greater spherical area, as illustrated in Figure 3–15.

The decrease in a sound’s intensity with dis- tance is known as the inverse square law. The inverse square law states that the intensity (I) is inversely (decreases) related to the square of the distance between any two points (D), and is ex- pressed by the following formula:

I = 1/D 2, where D = d1 /d 2

For example, if the distance is doubled (D = 2; d1/ d2 = 2/1), the intensity decreases by one-fourth (I = 1/22). Now let’s see how that same sound’s pressure ( p) would change with distance. Be- cause of the previously discussed relation be-

tween intensity and pressure for any given sound wave, the inverse square law for pressure is obtained by substituting p 2 for I in the above formula, which results in the following formula to describe how pressure changes with distance:

p2 = 1/D 2

p = 1/D, where D = d1 /d2

This formula says that the sound pressure is in- versely (decreases) related to the distance be- tween two points (D), rather than the square of the distance that was used for intensity. For exam- ple, if the distance is doubled (D = 2; d1/d2 = 2/1), the pressure decreases by one half ( p = 1/2).

Sound intensity is actually a measure of power that is distributed over an area, and has units of watts/m2 or watts/cm2 depending on the system of measurement being used (MKS or CGS). Sound pressure is a measure of force dis- tributed over an area and has units of dynes/cm2, newton/m2, or micropascals (µPa) depending on the system of measurement being used. For this text, we will only use units of watts/m2 for inten- sity and µPa for pressure. For our purposes, we are most interested in the range of sound intensi- ties or pressures that are audible, that is, from the smallest amount needed to barely hear a sound, up to the largest amount that the ear can toler- ate. Based on accepted standards derived from the lowest average levels (thresholds) obtained from young adults, the lowest average intensity needed to hear a sound, called the reference level for intensity, is 0.000000000001 w/m2 (or 1.0 × 10−12 w/m2). The lowest average pressure needed to hear a sound, called the reference level for pressure, is 20 µPa (or 2.0 × 101 µPa). Remark- ably, these lower levels of audition correspond roughly to a vibration about the size of a hydro- gen molecule (Gelfand, 2009). The highest inten- sity (also called the threshold of pain) that can be tolerated is approximately 100 w/m2 (or 1.0 × 102 w/m2). The highest pressure that can be toler- ated is approximately 200,000,000 µPa (or 2.0 × 108 µPa). As you can see, the upper tolerated limit of sound intensity is 100,000,000,000,000 (or 1014) times greater than the least audible sound (i.e., from 1.0 × 102 w/m2 to 1.0 × 10−12 w/m2). For

D

2 D

3 D

FIGURE 3–15. Illustration of how the energy of a sound is distributed over a larger area as the distance from the sound source is doubled and tripled, and is the basis for the inverse square law.

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3. PRoPERTIES oF SoUnD 31

pressure, the upper tolerated limit is 10,000,000 (or 107) times greater than for the least audible sound (i.e., from 2.0 × 101 µPa to 2.0 × 108 µPa). Recall that intensity and pressure are related by p = √I; therefore, it follows that the range of pressures (107) is equal to the square root of the range of intensities (√1014 = 107).

Table 3–2 summarizes the intensity and pres- sure ranges for the human ear. It may have struck you by now that these amplitude ranges are quite large, and it would be quite cumbersome if you were trying to graph them on a linear scale ( y-axis). Fortunately, linear scales can be transformed into ratio scales, as described in the next section on decibels, and makes working with intensity and pressure ranges much more manageable.

DECIBELS

To avoid linear scales of intensity or pressure ranges that would require working with large numbers or scientific notation, we transform these scales into a more manageable scale called the decibel scale. Any decibel scale is a ratio scale in which a measured value is related to a specified reference value. For example, as was mentioned above, the most intense sound that can be toler- ated is 1014 times greater than the lowest sound intensity that can be heard. If we consider the lowest audible sound intensity as the reference, this least audible sound could be expressed as 1 or 100; then the range of sound intensity can be written as a ratio of 1014/100. The next step

in the transformation to a decibel scale is to take the logarithm3 (or log) of that ratio, the Bel. The log is the mathematical difference in the values of the exponents in the ratio. Stated another way, the log is the power to which 10 must be raised to produce the number defined by the ratio. In our example, the mathematical difference in the expo- nents is 14 (14-0), which means that the range for intensity, in Bels, would be from 0 (least intense) to 14 (most intense). The upper limit would be the same as saying that 10 must be raised to the 14th power (1014). Because the range for Bels only goes from 0 to 14, it is considered too re- strictive to effectively describe the range of audi- ble sounds; thus, to expand the range, the Bel is multiplied by a factor of 10 and is called the deci- bel (dB). The decibel equation can be written as:

dB = 10 log (Xmeas /Xref )

where Xmeas equals the sound that is being mea- sured, and Xref equals the reference sound to which Xmeas is to be compared. It is important to see that when the measured value is the same as

3 A logarithm of a number, in base 10 (log10), is the power to which 10 must be raised to obtain that num- ber. Some simple examples are for powers of 10, in which the exponent is the log; for example, the log of 1000 or 103 = 3. It is also important to remember that the log (1) = 0. Another useful example that you may want to memorize is log (2) = 0.3. Calculators can be used to find logs of other less obvious numbers. Other sources, such as Speaks (2017), should be consulted for a review of logarithms.

TABLE 3–2. Intensity and Pressure Ranges from the Least Audible (Reference Level) to the Upper Limit that Is Tolerated (Pain Threshold)

Intensity (w/m2)a Pressure (μPa)b

Upper limit (pain) 100 or 1 × 102 200,000,000 or 20 ×107

(or 2.0 × 108)

Lowest audible (reference level)

0.0000000000001 or 1 × 10–12 20 or 20 × 100

(or 2.0 × 101)

aWatts per square meter. bMicroPascals.

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AUDIOLOGY: SCIENCE TO PRACTICE32

the reference value, that is, Xmeas /Xref = 1, the mea- sured value would be equal to 0 dB because the log of 1 (or log 100) is equal to zero. Applying a decibel conversion to describe the range of sound intensity, the upper limit of intensity would be 140 dB greater than the least audible intensity, as determined by the following calculation:

dB intensity (at upper limit)

= 10 log (1014/100) = 10 log (1014) = 10 (14), where log of 1014 equals 14 = 140 dB

The formula for converting to decibels can be ap- plied to anything that can be expressed as a ratio if the reference value is known. For example, we could calculate the decibel difference between a large bag of oranges (Xmeas) as compared with a small bag of oranges (Xref). If there are twice as many oranges in the larger bag than the smaller bag, there would be 3 dB more oranges in the larger bag than in the smaller bag, as shown in the following calculation:

dB in bag of oranges with twice as many as the reference bag:

= 10 log (2/1) = 10 log (2) = 10 (0.3), where log of 2 equals .3 = 3 dB

Decibels of Intensity Level (dB IL)

Let’s look at another example and calculate how many dB some arbitrary measured intensity is above the lowest average intensity needed to hear. In other words, how much more intense is this measured higher intensity level above the lowest possible audible intensity? Recall that the lowest average intensity needed to hear a sound, called the standard reference level for intensity, is 1 × 10−12 w/m2 (or simply 10–12 w/m2). Let’s say we measure a sound intensity to be 1 × 10−6 w/m2. The calculation of dB intensity is:

dB intensity = 10 log (10−6 w/m2/10−12 w/m2)

Whenever the standard reference value for in- tensity (10−12 w/m2) is used or implied in the de- nominator of the decibel formula, this is called decibel intensity level (dB IL). The general equa- tion for dB IL is:

dB IL = 10 log (Imeas w/m2/10−12 w/m2).

When the measured intensity is the same as the reference level for intensity (10−12 w/m2/10−12 w/m2), in decibels this would be the 10 log (1), which would become 0 dB IL. By using a deci- bel scale, we can define the intensity range of human hearing from 0 to 140 dB IL. It is impor- tant to realize that 0 dB IL does not mean the ab- sence of sound; it only means that the measured sound intensity is the same as the standard refer- ence intensity. However, as you will see later, the lowest audible intensity is different depending on frequency, type of earphones, and other spec- ified listening conditions, but all these variations are always referenced to the universally accepted standard reference intensity of 1.0 × 10–12 w/m2.

Now let’s look at some other examples. Sup- pose you make a sound measurement and find it to be 10−5 w/m2 (0.00001 w/m2). How many dB IL is this sound? The calculation is as follows:

dB intensity level (dB IL) for a measured sound = 10−5 w/m2

= 10 log (10−5 w/m2/10−12 w/m2) = 10 log (107) = 10 (7), where log of 107 equals 7 = 70 dB

The decibel can be used to describe the ratio of any two numbers if the proper reference value is specified in the denominator. Suppose you are interested in expressing, in dB, how the inten- sity of one sound compares with the intensity of another sound. For example, assume that one sound has an intensity that is 10,000 times more intense than another sound. In this case, the less intense sound can be considered the reference and written as 1 in the denominator; the ratio of these two sounds would be 10,000/1 (i.e., with- out needing to use the 10−12 w/m2). The dB of the louder sound, as referenced to the softer sound, is calculated as follows:

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3. PRoPERTIES oF SoUnD 33

dB intensity of a sound 10,000 times greater than another sound

= 10 log (10,000/1) = 10 log (104) = 10 (4), where log of 104 equals 4 = 40 dB

As another example, if the intensity of a sound is doubled, how many dB has that sound increased? In this example, the increase in intensity can be thought of as a ratio of 2/1 (the louder sound is twice as intense as the softer sound) and, there- fore, would show an increase of 3 dB. The calcu- lation is as follows:

dB intensity of a sound twice as much as another sound (increase)

= 10 log (2/1) = 10 (0.3), where the log of 2 equals 0.3 = 3 dB

This concept and use of the formula would also apply if one were to decrease the intensity level of a sound. For example, if one sound is half the intensity of another sound, the ratio would be 1/2. The calculation is as follows:

dB intensity of a sound half as much as another sound (decrease)

= 10 log (1/2) = 10 (–0.3), where the log of 2 equals –0.3 = –3 dB

If you find it easier, the above example could be calculated using a ratio of 2/1, but then be sure to indicate that it is negative (representing a de- crease): Notice that in both cases, the answer is 3 dB (either positive or negative).

Decibels of Sound Pressure Level (dB SPL)

In audiology, decibels of sound pressure are commonly used. To derive the decibel scale for sound pressure, it is important to go back to the previous relation between pressure and inten-

sity (I = p 2). This relation requires that pressure squared ( p 2) be substituted for intensity (I ) in the general equation for decibels. The derivation is as follows:

dB sound pressure

= 10 log (p2meas /p2ref) = 10 log (pmeas /pref)2

= 20 log (pmeas /pref), where log (x)2 equals 2 log (x)

Notice that for dB of sound pressure, the log of the pressure ratio is multiplied by 20 instead of by 10 that was used for dB of sound intensity. Using the formula for dB of sound pressure, any pressure ratio can be expressed in decibels. The decibel scale for sound pressure would also range from 0 to 140 dB because the upper limit for pressure is 107 times greater than the lowest pressure. The calculation is expressed as follows:

dB pressure (at upper limit)

= 20 log (107/100) = 20 log (107) = 20 (7), where log of 107 equals 7 = 140 dB

Recall that the standard reference sound pres- sure level is 20 µPa. When a measured pressure is compared to this standard reference pressure, that is, when the specific reference value for sound pressure is used or implied in the denom- inator of the formula for dB sound pressure, this is called decibel sound pressure level (dB SPL). The formula for dB SPL is:

dB SPL = 20 log (Pmeas µPa/20 µPa)

Whenever the measured sound pressure is equal to the standard reference level for sound pressure it would be equal to 0 dB SPL (i.e., 20 log [1] = 0). Using dB SPL, we have now defined the range of human hearing from 0 to 140 dB SPL. It is impor- tant to realize that 0 dB of sound pressure does not mean the absence of sound pressure; it only means that the measured sound pressure is the same as the standard reference sound pressure. A sound that is 0 dB SPL has the same pressure as

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AUDIOLOGY: SCIENCE TO PRACTICE34

the standard reference pressure of 20 µPa. How- ever, as with intensity, the lowest audible pres- sure can depend on frequency, type of earphones, and other specified listening conditions, but these are always referenced to the universally accepted standard reference pressure of 20 µPa.

Let’s look at some other decibel examples us- ing pressure. Suppose you make a sound measure- ment and find it to be 200,000 µPa. What is the dB SPL of this sound? The calculation is as follows:

dB sound pressure level (dB SPL) for a measured sound = 200,000 µPa

= 20 log (200,000 µPa/20 µPa) = 20 log (104) = 20 (4), where log of 104 equals 4 = 80 dB SPL

What if you want to compare one sound to an- other sound? For example, suppose the pressure of one sound is 1000 times more than the pres- sure of another sound. This defines the pressure ratio of these two sounds, that is, 1000/1. In dB pressure, this is expressed using the following equation:

dB pressure of a sound 1000 times greater than another sound

= 20 log (1000/1) = 20 log (103) = 20 (3), where log of 103 equals 3 = 60 dB

How about the situation in which we double the pressure of a sound? How many dB greater is the louder sound? The calculation is as follows:

dB pressure for a sound twice with as much (doubling) as another sound (increase)

= 20 log (2/1) = 20 (0.3), where the log of 2 equals 0.3 = 6 dB

As was discussed above for intensity, this would similarly apply if one were to decrease the pres- sure level of a sound. For example, if one sound is half the pressure of another sound, the ratio

would be 1/2 (softer sound is half the pressure level of the louder sound). The calculation is as follows:

dB pressure of a sound half as much as another sound (decrease)

= 20 log (1/2) = 20 (–0.3), where the log of 2 equals –0.3 = –6 dB

If you find it easier, the above example could be calculated using a ratio of 2/1, but then be sure to indicate that it is negative (representing a de- crease): Notice that in both cases, the answer is 6 dB (either positive or negative).

Notice that if the pressure of a sound is doubled, it increases by 6 dB, whereas if the intensity of a sound is doubled, it increases by 3 dB. However, it is important to realize that for a specific sound, the intensity and pressure must vary together (I = p 2), that is, one cannot dou- ble the sound’s pressure and at the same time double that sound’s intensity. For example, if we double the intensity of a sound, the decibel level increases by 3 dB and the pressure also increases by 3 dB because the sound’s pressure would in- crease by the square root of two. On the other hand, if we double the pressure of a sound, the decibel level increases by 6 dB and the intensity of that sound also increases by 6 dB because the sound’s intensity is squared. These comparisons can be illustrated by the following calculations (keeping in mind that I = p 2); (a) the pressure of a sound is doubled, and (b) the pressure of a sound is increased by a factor of 10:

(a) dB pressure increase dB intensity increase

= 20 log (2/1) = 10 log (22/1) = 20 log (2) = 10 log (4) = 20 (0.3) = 10 (0.6) = 6 dB = 6 dB

(b) dB pressure increase dB intensity increase

= 20 log (10/1) = 10 log (102/1) = 20 log (10) = 10 log (100) = 20 (1) = 10 (2) = 20 dB = 20 dB

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3. PRoPERTIES oF SoUnD 35

Table 3–3 summarizes how the ranges of pres- sure and intensity for human hearing are related and how the linear scales are transformed into their respective decibel (ratio) scales. The deci- bel is defined as 10 times the log of an intensity ratio and 20 times the log of a pressure ratio. The decibel range between the least audible sound and the upper limit (threshold of pain) is 140 dB for either intensity or pressure. However, a ten- fold increase in intensity results in a 10 dB in- crease, whereas a tenfold increase in pressure results in a 20 dB increase because of the rela- tion between intensity and pressure ( p = √I ). For more information on and practice with decibels, see Audiology Workbook (Kramer & Small, 2019) or the textbook by Speaks (2017).

Combining Levels from Different Sound Sources

One thing to keep in mind is that decibels cannot be simply added or subtracted, that is, adding a sound of 40 dB to another sound of 40 dB does not equal 80 dB; the decibels must be converted back to intensity before being combined. Let’s look at what happens to the level of a sound when you combine two or more sound sources, each producing the same or different levels. The most important thing to keep in mind is that when combining sounds, you should work with the intensity levels of the sounds that combine (not the pressures); therefore, the standard deci- bel formula for intensity, 10 log (Imeas /10–12 w/m2) must be used. However, as you learned earlier “a dB is a dB,” and the combined level you obtain for dB IL would be the same in dB SPL. Since we more often measure sounds in dB SPL, the fol- lowing examples will have the levels in dB SPL; however, it is the actual intensity levels (not in dB) that must be added together. In the case where all the sound sources have equal output levels, you can treat this like examples discussed earlier in which the combined level can be calcu- lated by taking the log of the number of sources (added to the level of one source). In other words, if there are three sources with equal lev- els, the combined output (in dB) is calculated as follows:

Combined (dB SPL or IL) = x dB from 1 source + 10 log (3/1)

Example: You have three fans, each with an out- put level of 72 dB SPL. What is the combined level of the three fans? The solution is as follows:

dB combined = 72 + 10 log (3/1)

= 72 + 10 (0.48), where log of 3 = 0.48 = 72 + 4.8 = 76.8 dB SPL (or dB IL)

It is a bit more difficult when combining sound sources with different output levels. To do this, you must: (1) calculate the intensity level (not in dB) of each source; (2) add them together to get the combined numerator of the intensity ratio (Imeas), and; (3) calculate the dB level using the formula for intensity level. The tricky part is cal- culating the intensity levels of the sounds, akin to finding the antilog, i.e., that is, what is the numerator of the intensity ratio for a given dB level. The general formula for calculating the Imeas for each source when the dB level is specified is as follows:

x dB (given) = 10 log (Imeas w/m2/10–12 w/m2); then solve for Imeas.

Example: You have two radios, one with an out- put of 80 dB SPL and the other with an output of 70 dB SPL. What is the combined level of the two radios? The solution is as follows:

1) Radio 1: 80 dB SPL = 10 log (Imeas/10–12) or 8.0 = log (Imeas/10-12)

Imeas = 1 × 10–4; determined so that the addi- tion of exponents would = 108

2) Radio 2: 70 dB SPL = 10 log (Imeas/10–12) or 7.0 = log (Imeas/10–12)

Imeas = 1 × 10–5 ; determined so that the addi- tion of exponents would = 107

3) Convert Radio 2 so it has the same exponent as Radio 1 (10–4):

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36

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1 ×

1 0

-1 2

1 :1

1 0

0 0 .0

0 2 0 ×

1 0

0 √

1 :1

√ 1

0 0

0 .0

0

a d B I L =

1 0 l o g (

I m e a s/

I r e f)

b d B S

P L =

2 0 l o g (

P m

e a s/

P re

fe re

n ce )

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3. PRoPERTIES oF SoUnD 37

Radio 2: 1 x 10–5 = .1 x 10–4 (multiply expo- nent x 10; therefore, divide 1 by 10)

4) Combine the intensities from each radio:

(1 × 10–4) + (.1 × 10–4) = 1.1 × 10–4

5) Calculate the dB level from these combined levels:

= 10 log (1.1 × 10–4/10–12) = 10 log (1.1 × 108) = 10 log (1.1) + log (108) = 10 (0.04 + 8) = 80.4 dB SPL (or IL)

AUDIBILITY BY FREQUENCY

As mentioned earlier, the human ear is respon- sive to frequencies from 20 to 20,000 Hz; how- ever, the ear is not equally sensitive across the frequency range. The relation between normal thresholds (in dB SPL) and frequency is referred to as the threshold of audibility curve. Fig- ure  3–16  shows an example of a threshold of audibility curve from 100 to 10,000 Hz. As you can see, humans are most sensitive to frequen- cies between 500 and 2000 Hz, and it takes slightly higher dB SPLs to reach threshold in the lower and higher frequencies. Also shown in Fig- ure 3–16 is an estimate of the upper limit for hearing, called the threshold of pain. Notice that the upper limit does not vary much as a function of frequency, probably because it involves the threshold of feeling within the tympanic mem- brane, which is relatively constant across fre- quency (Durrant & Lovrinic, 1995; Kent & Read, 2002). The area between the lower threshold curve and the upper pain limit curve defi nes the useable range for human hearing. However, most listeners fi nd sounds above 100 dB SPL uncom- fortably loud. Many loud music venues may have sound levels on the order of 110 to 120 dB (and may damage hearing!).

WAVELENGTH

An additional acoustic parameter is the distance that pure tones travel in one cycle, the wave- length. The symbol for wavelength is the Greek symbol λ (lambda). The wavelength has a unit of length (e.g., feet, meters) rather than unit of time, which is used to defi ne a pure tone’s period. Wavelength is a measure of the distance between one point on the waveform to the same point on the next cycle, most easily seen as the dis- tance between adjacent points of condensation (or rarefaction) in Figure 3–1. You should be able to surmise that higher frequencies have shorter wavelengths than lower frequencies because the cycles are closer together. Since the speed of sound is determined by the properties of the air, and is the same for all frequencies, wavelength can be defi ned by the following equation:

Threshold of Pain

100 1,000 10,000 Frequency (Hz)

S ou

nd P

re ss

ur e

(d B

S P

L)

Threshold of Hearing

Sound not audible to human ear

140

120

100

80

60

40

20

0

-20

FIGURE 3–16. Thresholds and upper range of hearing as a function of frequency in humans. The variation in thresholds (in dB SPL) as a function of frequency is called the threshold of audibility curve. In this example, the closed circles represent the threshold reference levels based on American national Standards Institute [ANSI] (2010). Higher dB SPLs are needed in the low and high frequencies to reach threshold than in the middle frequencies. The threshold of pain (140 dB SPL) is also shown; although most listeners do not tolerate levels above 100 dB SPL.

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AUDIOLOGY: SCIENCE TO PRACTICE38

SYNOPSIS 3–2

l Intensity (w/m2) and pressure (µPa) are related to each other by the equation p = √I (or I = p2). In audiology, pressure is usually used to quantify the level (amplitude) of sounds. The minimum mean sound pressure for audibility, called the reference level for pressure, is 20 µPa. The mean upper limit of pressure that can be tolerated is 20 × 107 µPa. For intensity, the reference level is 1012 w/m2 and the mean upper limit is 102 w/m2. The range of hearing from the lowest to the highest is 107 for pressure and 1014 for intensity (notice the relation 107 = √1014). In either case, the range on a linear scale is too cumbersome to be very useful, so the ranges are converted to decibel (dB) scales.

l Decibel scales are based on a logarithmic (log) scale. The log of a number (x) is dened as the power to which 10 must be raised to be equal to the number (x). Stated another way, the exponent of a number is the log of that number. Most calculators can easily calculate the log of any number. Some simple examples to keep in mind are:

log (1014) = 14 log (107) = 7 log (102) = 2 log (1) = 0 log (2) = 0.3 log (4) = log (2) + log (2) = 0.6

l A decibel (dB) is dened as 10 times the log of the ratio of two numbers [10 log (Xmeas/Xref)]. This formula is directly applicable to the ratio scale for intensity level [10 log (Imeas/Iref)]; however, because I = p 2, the conversion to decibels for sound pressure follows basic rules of logs and becomes dened as 20 times the log of the ratio of two pressures [20 log (Pmeas/Pref)]. When the reference (denominator) for the ratio is 20 μPa it is called dB sound pressure level (dB SPL). When the reference for the ratio is 10–12 w/m2, it is called dB intensity level (dB IL).

l When the measured sound pressure or sound intensity is equal to its respective reference level (giving a ratio of 1/1), it would be equal to 0 dB because the log of 1 = 0. The range of hearing in decibels is 140 dB for either pressure or intensity as calculated for dB pressure = 20 log (107) or dB intensity = 10 log (1014).

l Sound pressure increases by 6 dB when the pressure is doubled [20 log (2/1)]. Sound intensity increases by 3 dB when the intensity is doubled [10 log 2/1]. A sound that is 100 times greater in pressure than another sound would be 40 dB greater; a sound that is 100 times greater in intensity than another sound would be 20 dB greater. However, since you cannot simultaneously double the intensity and double the pressure of the same sound (recall that I = p2), the number of decibel change would be the same for pressure and intensity.

l The inverse square law denes how the intensity or pressure of a sound changes with distance. As the distance increases, the sound energy spreads out in a spherical form, and the decrease in the level of the sound can be described for intensity as I = 1/D 2 or for pressure as p = 1/D, where D is the ratio of the distance between two sounds (d1/d2). For example, if one doubles the distance, the intensity would decrease by 1/4 (or –6 dB) and the pressure would decrease by 1/2 (or –6 dB).

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3. PRoPERTIES oF SoUnD 39

λ = c/f, where c is the speed of sound and f is the frequency.

This equation shows that as frequency gets higher, the wavelength gets shorter. For example, a 2000 Hz pure tone, traveling in air (c = 343 m/s), has a wavelength of 0.17 m (or about 0.56 feet), whereas a 250 Hz sinusoid traveling in the same air would have a wavelength of 1.32 m (or about 4.5 feet). The same pure tones traveling in water would have wavelengths that are approximately four times longer because the speed of sound in water is about four times faster than in air. Con- versely, if you know the wavelength of a sound, the frequency can be calculated by the equation:

f = c/λ.

The wavelength, to some extent, determines how a sound is affected as it encounters objects in its path. In a simple sense, the longer a sound’s wavelength is relative to the size of the object en- countered, the less likely the object will have an effect on the sound. However, if the wavelength is short (as for higher frequencies) relative to the size of an object, then the object will tend to block (and reflect) the sound. You may have no- ticed that it is much easier to hear drums over a greater distance than the higher frequency band instruments, like a flute; this is partly due to the higher frequencies being blocked by objects along the way, whereas the lower frequencies more easily go around the objects. As we will also see later in this chapter, part of our abil- ity to localize sounds is related to the different amplitudes that occur between the two ears for higher frequencies, which tend to be blocked by the head because of their shorter wavelengths.

COMPLEX SOUNDS

As mentioned earlier, most sounds we listen to are complex sounds, which means that they are the result of combining two or more individual pure tones. Any complex vibration can be cre- ated or described by knowing the frequencies (or periods), amplitudes, and starting phases of the individual pure-tone components. The num- ber of pure tones, along with their relative ampli- tudes and starting phases, will determine the type of sound we hear. A spectrum (plural = spectra) is a way to describe a complex vibration by plotting a graph that shows the amplitudes as a function of frequency, called a frequency spectrum, or the starting phases as a function of frequency, called a phase spectrum. Figure 3–17 shows an example of a complex vibration that is composed of two different pure tones with amplitudes and phases shown in the corresponding spectra. The ampli- tude spectra of complex periodic vibrations, as shown on the right side of Figure 3–17, show ver- tical lines at the discrete frequencies that make up the vibration, and this type of spectrum is called a line spectrum.

Vibrations are generally classified as peri- odic or aperiodic. A periodic vibration is one in which the vibratory pattern repeats at regular in- tervals. A pure tone (sinusoid) is an example of a simple periodic vibration. However, when two or more pure tones are combined into a nonsinusoi- dal pattern they may also be considered periodic if the wave pattern repeats itself as a function of time. These nonsinusoidal periodic vibrations are called complex periodic vibrations (or complex periodic tones). Complex periodic vibrations typ- ically have a tonal or buzzing quality. The low- est frequency component in a complex periodic

SYNOPSIS 3–2 (continued )

l When combining output levels from more than one source (whether specied in dB IL or dB SPL), it is the intensity levels (w/m2) that are combined; therefore, the formula 10 log (Imeas/10–12 w/m2) must be used. If sounds are unequal, rst calculate Imeas for each sound, then combine the Imeas from each sound (converting exponents to be the same), then calculate the dB of the combined Imeas relative to 10–12 w/m2.

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AUDIOLOGY: SCIENCE TO PRACTICE40

vibration is called the fundamental frequency (f0). Integer multiples of the fundamental frequency are called harmonics, such as 1f0, 2f0, 3f0, and so forth. Generally, complex periodic vibrations occur when harmonically related pure tones are combined. For example, combining 100 Hz, 200 Hz, 300 Hz, and 400 Hz will produce the com- plex periodic waveform shown in Figure 3–18. If additional sequential harmonics were added to those shown in Figure 3–18, the resulting wave- form would smooth out the smaller bumps and, with enough harmonics, would produce what is called a sawtooth waveform. A sawtooth wave- form is a complex periodic waveform that has more of a buzzing sound quality rather than a tonal quality. You can see in Figure 3–18 that the longest period of this complex periodic wave- form is the same as the lowest frequency compo- nent (100 Hz). The fundamental frequency usu- ally determines the primary pitch of the sound, but the other components can also be heard and will contribute to the perception/quality of the complex periodic vibration. Adding different combinations of pure tones and using different amplitudes or phases can affect the overall shape of complex periodic waveforms.

On the other hand, aperiodic vibrations are those in which the pattern of vibration does not regularly repeat itself over time; in other words, there is no periodicity in the wave pattern. The waveform shown in Figure 3–19 is an example of an aperiodic vibration. Aperiodic vibrations are generally called noise. Noise is produced by combining many pure tones with random start- ing phases. When there are an infi nite number of frequencies with random phases and equal amplitudes over the entire frequency range it is called white noise (analogous to white light). The spectrum shown on the right side of Figure 3–19 is a horizontal line, rather than discrete vertical bars, to indicate that there are infi nite frequencies present over the indicated range, and this type of spectrum is called a continuous spectrum.

Aperiodic noise-type vibrations are encoun- tered frequently in our environment, including many speech sounds (e.g., /s/, /sh/, /f/, /th/), as well as sounds produced by things such as run- ning water, rustling leaves, or engines. In addi- tion, many sounds we listen to have components that give it a tonal (periodic) quality as well as a noise (aperiodic) quality, such as the speech sounds (/v/, /z/, /j/) or the different pitches as- sociated with the buzzing of different types of

FIGURE 3–17. Example of a complex periodic wave- form composed of two frequencies (left), which are described by their corresponding amplitude spectrum (top right) and starting phase spectrum (bottom right). These types of spectra are called line spectra.

Time (ms) 0 5 10 15 20

A m

pl itu

de (

ar bi

tr ar

y)

-8 -6 -4 -2 0 2 4 6 8

A m

pl itu

de (

ar bi

tr ar

y)

0 1 2 3 4 5 6 7 8

Frequency (Hz)

P ha

se

0

90

180

270

360

100 100 200 00 300 00 400

Frequency Hz

100 100 200 00 300 00 400

Frequency Hz

FIGURE 3–18. Example of a complex periodic wave- form with a fundamental frequency of 100 Hz and its harmonics, 200 Hz, 300 Hz, and 400 Hz. With addi- tional sequential harmonics added, the resulting wave- form would be a sawtooth waveform.

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3. PRoPERTIES oF SoUnD 41

motors. In some of these latter cases, the under- lying periodicity or the frequency range of ape- riodic combinations will determine the overall perceptions. If we know the frequencies (or pe- riods), amplitudes, and starting phases of all the individual components of a complex periodic or aperiodic vibration, we can construct the pre- dictable vibration pattern that would result from their combination. Instruments are available that can perform a fast Fourier transform (FFT) on complex vibrations to determine the frequencies, amplitudes, and starting phases of the individual components. Figure 3–20 shows some additional examples of complex vibrations with their corre- sponding amplitude spectra.

RESONANCE

The frequencies to which objects vibrate most easily are called resonant (or resonance) fre- quencies. You are undoubtedly familiar with this concept when you think about musical instru- ments, such that strings of different length vi- brate best at certain frequencies, or the smaller violin’s sounding board emphasizes higher fre- quencies compared with the much larger bass, which emphasizes the lower frequencies. In the case of a guitar string that is attached at both ends, when plucked, there are waves that move toward the ends of the strings and are then re- flected back. This interaction of the two waves (incident and reflected) results in places where the displacements cancel each other, called nodes, and places where they combine with each other, called antinodes. For the guitar string, there are nodes at both ends (where the string cannot move) and an antinode at the center of the string that produces the string’s primary musical note. The pattern of vibration between the nodes at the ends of the string and the antinode in the middle of the string is half of a cycle, as shown in Figure 3–21A. These patterns of displacement as a function of distance along the string are related to the frequency’s wavelength (λ). The longest wavelength determines the string’s primary reso- nant frequency (a.k.a. first mode or fundamental frequency) and is equal to half of a wavelength (λ/2). For a given length of string, the fundamen- tal frequency can be calculated as:

f0 = c/2L, where c = speed of sound; L = length of string.

The string analogy and other vibrating objects have additional modes (e.g., harmonics) of vibra- tions that create other possible nodes and an- tinodes, corresponding to f2, f3, f4, and so on, as illustrated in Figure 3–21B and C. When a reso- nating source has a fundamental frequency that is equal to one-half of a wavelength, it is called a half-wave resonator, and generates a specific fundamental frequency based on its character- istics, and also generates harmonics at integer multiples of the fundamental frequency.

FIGURE 3–19. Example of an aperiodic noise vibration (left) and its corresponding amplitude spectra (right). These types of spectra are called continuous spectra.

Pure tone

Square Wave

White Noise

Click

Tone Burst

Waveform Spectrum

Time (ms) Frequency (kHz) 1 2 3 4 5 6 7 8 9 100 1 2 3 4

+5 0

-5

10 5 0

+5 0

-5

10 5 0

+5 0

-5

10 5 0

+5 0

-5

10 5 0

+5 0

-5

10 5 0

FIGURE 3–20. Examples of some continuous and transient signals with their corresponding amplitude spectra.

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AUDIOLOGY: SCIENCE TO PRACTICE42

Resonance also occurs in tubes of different dimensions whereby the air molecules within the tube interact and produce regions of nodes and antinodes within the tube depending on the length of the tube. For example, blowing across a small tube produces a higher pitched tone com- pared with a lower pitched tone from a longer tube. The relationship of nodes and antinodes will depend also on whether the tube is open on both ends or only on one end, as illustrated in Figure 3–22. For the same-length tube, being open on both ends produces a higher-pitch tone than one open on only one end because of the different relationships of nodes to antinodes as illustrated in Figure 3–22A. As with the string ex- ample, a tube open at both ends involves half of a wavelength, and the resonant frequency can also be obtained by the formula f0 = c/2L. This would also be a half-wave resonator with a fundamen- tal frequency and harmonics at integer multiples of the fundamental frequency. Let’s now look at

a tube that is open only on one end (thinking ahead to our vocal tract or ear canal that acts much like a tube open at one end). As shown in Figure 3–22B, there must be a node at the closed end and an antinode at the open end. For a tube that is open only on one end, a one-quarter wave- length can fit within the tube (between nodes and antinodes). This type of resonator is called a quarter-wave resonator, and generates a specific fundamental (f0), but the harmonics are only at odd multiples of the fundamental frequency. The resonant frequency of a quarter-wave resonator is calculated as:

f0 = c/4L, where c = speed of sound; L = length of tube.

Node

Node

Antinode

Antinode Node

Antinode

Antinode Antinode

Antinode Antinode

Antinode

A B

Node

Node

Node

FIGURE 3–22. A and B. Examples of modes of vibra- tion that would occur in tubes. A. The wave patterns for a tube open on both ends, called a half-wave reso- nator. A half-wave resonator can produce harmonics at integer multiples of the fundamental mode. B. The wave patterns in a tube open only on one end, called a quarter-wave resonator. A quarter-wave resonator can produce harmonics at odd multiples of the fundamental mode.

Node

Node Node

Node

Node

Node Node

Node Node

Antinode

Antinode Antinode

Antinode Antinode Antinode

A B C

FIGURE 3–21. A–C. Examples of modes of vibration that would occur in a string that is attached at both ends showing the rst mode or fundamental frequency (A), the second mode or second harmonic (B), and the third mode or third harmonic (C). Additional modes can occur at integer multiples of the rst mode. At the nodes, the displacement is zero and creates a standing wave.

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3. PRoPERTIES oF SoUnD 43

ACOUSTICS OF SPEECH

The primary means of producing speech is through air-flow supplied from the lungs, which passes through the vocal structures of the larynx and the oral and nasal cavities. The positioning and movements of the articulators (tongue, lips, velum, and jaws) alter the shape of the vocal path that results in specific resonance patterns of frequencies, amplitudes, and timing. The speech sounds are propagated into the environment, re- ceived, and ideally perceived by a listener. For example, what acoustic parameters are neces- sary for the listener to decide that they heard the sound /d/ in the word day? Or how does one perceive the word day differently than the words die or bay? Also, in today’s world, human speech can be synthesized and digitally generated by a computer, largely based on what is known about the meaningful acoustic characteristics of the speech. Although the acoustic parameters of speech are important, keep in mind that com- munication involves much more than the simple production, recognition, and perception of the acoustic properties of speech.

Speech sounds can be classified into differ- ent types. On the most basic level, there are vow-

els, which are mostly complex periodic sounds, and consonants, which can be either complex pe- riodic (with vocal fold vibration) or aperiodic vi- brations (without vocal fold vibration). Table 3–4 lists the labels used to describe different types of sounds. When the vocal folds vibrate, there is a complex periodicity to the sound, called a voiced sound. When the vocal folds do not vibrate, the sound is aperiodic and called a voiceless sound. Vowels are voiced, but consonants can be either voiced or unvoiced. The normal average intensity level of ongoing connected speech is about 65 to 75 dB SPL (Killion & Mueller, 2010; Thibodeau, 2007), and this “volume” is primarily carried by the vowels. Consonants have less energy than the vowels during connected speech, and contrib- ute the most to word intelligibility. Figure 3–23 shows how conversational-level speech sounds are distributed across the frequency and inten- sity scales. This general distribution of speech sounds is often referred to as the speech banana due to its general outline encompassing ranges of the vowels and consonants during ongoing speech. As you can see in Figure 3–23, there is as much as a 30 dB SPL difference between the loudest vowel ( /u/) and the softest consonant ( /th/). Notice also that the vowels tend to be

TABLE 3–4. Labels Used to Describe Different Types of Sounds Related to the Place of Articulation

Manner of Articulation Voicinga Bilabial Labio-dental Lingua-dental Alveolar Palatal Velar Glottal

Plosives (Stops) – +

p(pea) b(bee)

t(tea) d(d id)

k(kit) g(go)

Fricatives – +

f(f in) v(v ine)

θ(thin) ð(the)

s(so) z(zoo)

ʃ(she) ʒ(luge)

h(he)

Affricates – +

tʃ(chin) dʒ( jot)

nasals – + m(me) n(no) ŋ(bang)

Liquids – +

l(let) r(red)

Glides – + w(we)

ʍ(whet) j(yet)

aSome consonants are produced without voicing (–), and some are produced with voicing (+).

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AUDIOLOGY: SCIENCE TO PRACTICE44

lower in frequency, whereas many of the aperi- odic noise-like consonants, called fricatives, are higher in frequency.

In the following sections, only the very ba- sics of the acoustic properties of the different types of speech sounds are given, primarily their frequency components; however, keep in mind that amplitude and timing variations are also im- portant acoustic properties of speech. In Chap- ter 8 of this textbook, you will learn about the clinical speech tests that are used to assess how well a patient is able to recognize words and sen- tences, and how a patient’s speech recognition is altered by various disorders of the auditory system. For a more in-depth understanding of speech production, speech acoustics, and speech perception, the interested reader is referred to other sources, such as (Kent & Read, 2002; Ra- fael, Borden, & Harris, 2007).

Spectrogram

Speech is composed of basic periodic and ape- riodic vibrations that occur in relatively short time periods, and is interspersed with short si- lent periods between sounds. The frequency, am-

plitude, and time characteristics of speech can be analyzed with some basic equipment. One of the most important pieces of equipment used to analyze speech sounds is a spectrograph, which measures the spectra of speech sounds, words, or sentences (in a relatively short time window) recorded through a microphone. A spectrogram is the graphical output from a spectrograph for a specifi c speech utterance. A spectrogram dis- plays frequency (along the y-axis) as a function of time (along the x-axis). The amplitudes of the different frequencies are also represented in a spectrogram by the relative darkness of the fre- quency bands, that is, the more intense frequen- cies are seen as darker bands. Figure 3–24 shows examples of spectrograms for some vowels and consonants. As you can see in Figure 3–24A, the

Frequency (Hz)

100 1000 10000

dB S

ou nd

P re

ss ur

e Le

ve l

0

10

20

30

40

50

60

70

80

90

100

110

120

Frequency (Hz) 100 1000 10000

dB S

ou nd

P re

ss ur

e Le

ve l (

dB S

P L)

0 10 20 30 40 50 60 70 80 90

100 110 120

Vowels F1

Vowels F1

1st peak voiced fricatives and nasals Voiceless

fricatives

2nd peak voiced fricatives and

nasals

Vowels F2

FIGURE 3–23. General distribution of speech sounds during normal conversational level of connected speech. The overall level of conversational speech is about 65 dB SPL. The outer most outlined area is called the speech banana.

F re

qu en

cy

(k H

z)

5 4 3 2 1

A m

pl itu

de

A m

pl itu

de

5 4 3 2 1 F

re qu

en cy

(k

H z)

i a u

a f a a v a

5 4 3 2 1 F

re qu

en cy

(k

H z)

Time

A m

pl itu

de

a s a a z a

A

B

FIGURE 3–24. A and B. Spectrographic recordings for three different vowels in isolation (A), and some voiceless and voiced fricatives (B).

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3. PRoPERTIES oF SoUnD 45

vowels show three or four darker frequency re- gions (bands). These darker frequency bands are called formants, beginning with the first formant (F1), the next higher F2, and so forth. Formants vary depending on the resonance properties as- sociated with the positions of the articulators, and are similar to those harmonics seen with the example of the quarter-wave resonator. Vowels behave more like the complex periodic vibra- tions that we saw in the preceding section. Other speech sounds may be aperiodic, like the /s/ shown in Figure 3–24B, which does not have the discrete frequency bands like those seen with vowels, but instead shows a wider range of fre- quencies as expected with a noise-type sound. The following sections give a brief overview of the acoustics properties for the various vowel- like speech sounds and the various noise-like speech sounds. Although not covered in this text, keep in mind that there are also corresponding amplitude variations associated with connected speech, as well as important temporal factors such as duration of sounds and silent intervals.

Vowels

Vowels carry most of the audible energy in speech, and generally have lower frequencies and higher intensities than consonants. Vowels are complex periodic vibrations (voiced) that re- sult from the vibration of the vocal folds. The frequency of vocal fold vibration is the funda- mental frequency (f0) which gives the sound its perceived pitch, and can vary depending on the vowel as well as the size of the larynx (which relates to males generally having a lower sound- ing voice). As described earlier, vowels can be characterized by their F1 and F2. In general, the F1 varies inversely with the height of the tongue, and F2 varies with the forward/backward posi- tion of the tongue. For example, /i/ is produced with the tongue in its highest position and most forward in the mouth, whereas /a/ is produced with the tongue in its lowest position and as far back as possible. Lip rounding is done for some back and center vowels and its effect is to ex- tend the vocal tract and thus lower all formant frequencies. Keep in mind that the formant fre-

quencies are not precise numbers, but are best considered as elliptical regions that vary depend- ing on the speaker due to variations in size of vocal tract, articulators, and dialect. Figure 3–25 shows how the different vowels can be sepa- rated into their F1 and F2 elliptical areas, and be relatively distinct from each other.

During speech, vowels can also vary by their duration. For example, some vowels are longer in duration, like those in open syllables (e.g., “see” “so”), whereas others are shorter in dura- tion like those in closed syllables (e.g., “sit” “sat”). Additionally, vowels that are produced in con- text with other consonants also have a dynamic shifting of their formant frequencies, called for- mant transitions, where there may be a rising or falling frequency transition depending on the preceding and/or target consonant. Combining vowel sounds, called semivowels or diphthongs, are characterized by formant transitions, and the shifts in F2 are the most distinguishing charac- teristic used to identify different semivowels and diphthongs.

Consonants

Consonants are considered the sounds that con- tribute most toward intelligibility as they precede and/or follow vowels to define words or parts of words. The acoustic properties of consonants are

F1 (kHz)

1.0

2.0

4.0

0 0.4 0.8 1 2

F 2

( k H

z )

heard

h e e d

h id

h e a d

ha d

hu d

w h o ̕d

h o o d ho

d

ha w ed

i æ

ɔ

ɑЗ

u

ˆ

.

FIGURE 3–25. Distribution of F1 and F2 formants for English vowels for a variety of speakers. Source: From Kent and Read, 2002, p. 170, with permission of the authors.

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AUDIOLOGY: SCIENCE TO PRACTICE46

a bit more complicated than vowels, and one is not able to give as general a description as we were able to do with the formant structure of vowels. Consonants are divided into several dif- ferent types. Some consonants are voiced, some have a noise quality, some only have a period of silence, and some may involve the nasal cavity. Only a brief description of these characteristics is given here, and only some selective examples are shown on a spectrogram. With some prac- tice, you may be able to recognize the patterns of vowels and consonants in the complete sentence “The sunlight strikes raindrops in the air,” shown in Figure 3–26.

Stops (or plosives) are produced by a brief period in which the airflow is blocked. The blockages can occur at the lips, alveolar area, or velum, and are referred to as bilabial, alveolar, or velar stop. Stops can be voiced or unvoiced depending on if the vocal folds are set into vibra- tion. During the closure, air pressure is built up and when the stop is opened, there is a burst of air flow. Fricatives are noise-like sounds that are produced by passing air through the oral cavity in which the articulators are positioned in a way to create turbulence in the airflow. The different fricatives are produced by the location (place) of constriction from the most forward point of the oral cavity at the lips to the rearmost position at the glottal area. Fricatives can be voiced (e.g., /v/ ) or unvoiced (e.g., /f/ ) at the same place of articu- lation. Affricates are created by the transition of

a stop into a fricative. With nasals, the velopha- ryngeal port is opened so that sound energy can pass through both the nasal and oral tracts or through only the nasal tract. The formants of the nasals depend on the length of the cavity from the uvula to the nostrils, and are voiced with the vibration of the vocal folds.

FILTERING

Filtering is a means by which certain frequencies are excluded and certain frequencies are allowed to pass. Filtering can be used to generate a sound that is composed of a specified range of frequen- cies by filtering out some portion of a wider range of frequencies. For example, one can start with white noise and then filter out some of the fre- quencies so that a more restricted range of fre- quencies is passed.

Figure 3–27 shows the spectra for different types of commonly used filters. The band of fre- quencies that is passed is represented under the curve, and those outside the curve are the fre- quencies that are filtered out. The point where frequencies begin to be filtered out is called the cutoff frequency and is usually defined at the point that is 3 dB less than the peak (called 3 dB down-points or half-power point). The extent to which the frequencies are excluded is deter- mined by the slope of the curve, called atten- uation rate or rejection rate, which is usually

F re

qu en

cy (k

H z)

A

m pl

itu de

5 4 3 2 1

ð s ʌ n l ᾱĪ t s t r ᾱĪ ks rēĪn drα p s I n ð i Ԑ r

FIGURE 3–26. Sample waveform and spectrogram of the sentence, “The sunlight strikes raindrops in the air.”

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3. PRoPERTIES oF SoUnD 47

specified in dB per octave (dB/octave). An octave means a doubling or halving of the frequency. As an example (see Figure 3–27A), a high-pass filter passes frequencies higher than the cutoff frequency and rejects frequencies lower than the cutoff frequency at a specified dB/octave slope toward the lower frequencies. In this example, the filter would be called a high-pass filter with a 2000 Hz low frequency cutoff and a rejection rate of 10 dB/octave. The other examples in Fig- ure 3–27 include a low-pass filter that passes all the frequencies lower than the cutoff frequency and rejects frequencies higher than the cutoff frequency at a specified dB/octave slope directed toward the higher frequencies. A band-pass filter passes a band of frequencies as defined by the high and low cutoff frequencies and rejects fre-

quencies above and below the cutoff at the spec- ified rejection rates toward the higher and lower frequencies. The band-reject filter (also known as a notched filter) specifies a range of frequen- cies in the middle of a wider range of noise that is rejected, and the frequencies on both sides of the specified band-reject area are passed.

A special type of band-pass filtered noise is called narrowband noise, where there is a rela- tively restricted range of frequencies. These fre- quencies are often described by the width of the curve as measured across the 3 dB down points from the center frequency. A commonly used nar- rowband noise is called a one-third octave nar- rowband noise, which means the filter is one third of an octave wide at the 3 dB down points. Fig- ure 3–28 shows the spectra for some one-third

Frequency (kHz)

A tte

nu at

io n

(d B

)

-50

-40

-30

-20

-10

0

.25 1 2 4 8.50 16

Frequency (kHz)

A tte

nu at

io n

(d B

)

-50

-40

-30

-20

-10

0

.25 1 2 4 8.50 16 Frequency (kHz)

A tte

nu at

io n

(d B

)

-50

-40

-30

-20

-10

0

.25 1 2 4 8.50 16

Frequency (kHz)

A tte

nu at

io n

(d B

) -50

-40

-30

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-10

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.25 1 2 4 8.50 16

10 dB/octave

3 dB down

LF cut-off

HF cut-off

High Pass

Low Pass

Band Pass

LF cut-off HF cut-off

Band Reject

LF cut-off HF cut-off

30 dB/octave

A

B

C

D

FIGURE 3–27. A–D. Examples of different types of lters. The region under the curves shows those frequencies that are heard (passed). The places where the lter begins to reject frequencies are called cutoff frequencies, which are at the 3 dB down points. The lter’s rate of frequency rejection is indicated by the dB/octave. A. High­pass lter with 2000 Hz cutoff with 10 dB/octave rejection rate. B. Low­pass lter with 2000 Hz cutoff with 10 dB/octave rejec- tion rate. C. Band­pass lter with 1000 Hz low frequency cutoff and 4000 Hz high frequency cutoff, with 10 dB/octave rejection rates. D. Band­reject lter with 1000 and 4000 Hz cutoff points, with 30 dB/ octave rejection rates.

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AUDIOLOGY: SCIENCE TO PRACTICE48

SYNOPSIS 3–3

l Humans are capable of hearing sounds from 20 to 20,000 Hz; however, we are most sensitive to frequencies in the 500 to 2000 Hz range, and it takes slightly greater amounts of sound pressure (or intensity) for lower and higher frequencies to be just audible (threshold). The variation threshold as a function of frequency is referred to as the threshold of audibility curve.

l The wavelength of a sound (λ) describes how far a pure tone travels in one cycle. The wavelength can be calculated by the equation λ = c/f, where c is the speed of sound and f is the frequency. Conversely, if you know the wavelength, the frequency can be calculated from f = c/λ.

l Complex sounds are much more typical in our environment than pure tones; however, any complex sound is the result of some combination of pure tones with specic amplitudes and starting phases. The individual sinusoidal components of complex sounds can be determined using equipment that can perform a fast Fourier transform (FFT).

l A spectrum describes a sound’s amplitude or starting phase (along the y-axis) as a function of frequency (along the x-axis). A line spectrum is used when discrete (and usually limited) frequencies contribute to the complex vibration. A continuous spectrum is used when a range of frequencies (all inclusive) contributes to the complex vibration.

l A periodic vibration repeats itself at regular time intervals. Complex vibrations can be periodic if the combined waveform repeats itself over time; periodic complex vibrations occur when the combined pure tones are harmonically related as integer multiples of the lowest frequency, called the fundamental

Frequency (kHz)

A tte

nu at

io n

(d B

)

-15

-12

-9

-6

-3

0

.25 1 2 4 8.50 16

Frequency (kHz)

A tte

nu at

io n

(d B

)

-15

-12

-9

-6

-3

0

.25 1 2 4 8.50 16

1 kHz NB Noise (1/3 octave wide)

A

12 dB/octave

B 1 kHz NB Noise (1/3 octave wide) .5 1 2 4

NB Noises (CF in kHz)

FIGURE 3–28. A and B. Spectra for some one-third octave narrowband noises. A. The width of the lter is described by how wide the spectrum is at the 3 dB down points from the center frequency, with the corresponding dB/octave rejection rates. B. Some narrowband noises, centered at 0.5, 1, 2, and 4 kHz that are commonly used in audiology.

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3. PRoPERTIES oF SoUnD 49

octave narrowband noises. Narrowband noises are used frequently as noise maskers during basic hearing tests. Filtering can also be used when analyzing or measuring a complex wave pattern to exclude frequencies that you are not interested in analyzing or measuring, and instead focuses only on those frequencies that are of in- terest. This type of filtering is used in many of

the physiological measures from the auditory system.

PSYCHOACOUSTICS

The study of how humans perceive the acous- tic properties of sound is called psychoacoustics.

SYNOPSIS 3–3 (continued )

frequency (f0). A sawtooth waveform results when all harmonics (1f0 + 2f0 + 3f0 + 4f0 . . .) are included and a square wave results when only the odd harmonics (1f0 + 3f0 + 5f0 + 7f0 . . .) are included.

l An aperiodic complex vibration (often called noise) does not repeat itself at regular time intervals and typically results from a range of pure tones with random starting phases. White noise (analogous to white light) has an innite number of frequencies present with random phases.

l Most vibrating objects (except for a pure tone), including air molecules in tubes, have a fundamental frequency and additional harmonics. The vocal tract and ear canal are similar to a tube that is open at one end, called a quarter-wave resonator. A quarter-wave resonator produces vibrations at a fundamental frequency and at odd integer harmonics. The resonance frequencies are dependent on the wavelengths associated with the length of the tube.

l Speech has specic acoustic properties generated by airow passing through the vocal fold, oral cavity, and nasal cavity, resulting in complex periodic and aperiodic acoustic waveforms. The articulators modify the airow depending on the targeted speech sound.

l Vowels are more intense than consonants in connected speech and produce the perception of voice loudness. The long-term average level of conversational speech is about 65 dB SPL.

l Vowels are complex periodic vibrations with a fundamental frequency (giving rise to the pitch of the voice), and additional bands of energy called formants. The F1 and F2 formants are most important for vowel differentiation.

l Consonants can be periodic if accompanied by vocal fold vibration (voicing) or aperiodic if entirely noise-like (unvoiced). There are many types of consonants depending on the manner and place of articulation. Consonants are generally less intense than vowels and contribute most to intelligibility when in connected speech.

l Filtering is a way to exclude certain frequencies and allow other frequencies to pass through. Filters are used to shape the spectra of noise stimuli and/or to focus on a specic frequency range that is to be analyzed. Common types of lters include high­pass, low­pass, band­pass, and band­reject as dened by their cutoff frequencies and rejection rates. narrowband noises, used extensively in hearing testing, are ltered noises that are typically one­third of an octave wide.

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AUDIOLOGY: SCIENCE TO PRACTICE50

The following sections will present basic infor- mation on how frequency and intensity translate into our perception of loudness and pitch; how our thresholds change as the duration of a sound is shortened, called temporal integration; and, fi- nally, how we use acoustic information to deter- mine where sounds are coming from, called local- ization. The introductory material covered in this chapter only touches the surface of this fascinat- ing area and only covers how normal hearing hu- mans perceive simple sounds. For more advanced coverage of this topic for a variety of simple and complex sounds, the interested reader is referred to other texts (Durrant & Lovrinic, 1995; Gelfand, 2009; Moore, 2013; Zwicker & Fastl, 2013).

Loudness

Loudness is generally considered the psycholog- ical correlate of intensity. Most of us have a gen- eral idea that soft and loud sounds are related to low and high intensities (or pressures) of sounds. As discussed earlier, the ear responds to a wide range of intensities (or pressures); the minimum level is perceived as threshold and the upper limit is perceived as being uncomfortably loud. As we learned from the threshold of audibility curve, it takes a different amount of intensity to reach threshold for different frequencies. So, an obvious question is what intensities are needed to maintain equal loudness across frequencies? To answer this question, a loudness-matching procedure is typically used in which the dB SPL for different frequencies are adjusted until they sound equally loud to a 1000 Hz reference tone (Fletcher & Munson, 1933; Robinson & Dadson, 1956). The scaling unit used to compare loudness across frequencies is called a phon. Figure 3–29 shows a series of phon curves. The phon equates loudness across frequencies (also called equal loudness contours). The level of each phon curve is defined as the dB SPL of a given level of a 1000 Hz tone. For example, a loudness level of 30 phons is equal to a 1000 Hz tone presented at 30 dB SPL, and a loudness level of 60 phons is equal to a 1000 Hz tone presented at 60 dB SPL. Different phon curves are established for differ-

ent levels of the 1000 Hz reference tone. The dB SPLs needed for other frequencies to sound equally loud to the 1000 Hz tone, at a specified phon level, are measured to establish the corre- sponding phon curve across the frequency range. In other words, all frequencies at the given phon level (along the same phon curve) are judged to be equally loud even though their actual SPLs are different. As you can see from Figure 3–29, the general shape of the phon curves follows the threshold of audibility curve; however, they tend to flatten out across frequency as the phon level increases, especially in the lower frequen- cies. This means that at higher sound levels it does not take as much increase in dB SPL in the lower frequencies to sound equally loud to the mid frequencies.

The phon scale does not tell us much about how our perception of loudness is related to the continuum of sound pressures. Another question of interest is how does a scale of loudness re- late to a range of sound pressures? For example, if the dB SPL of a sound is doubled, does the

FIGURE 3–29. Phon scales of loudness (or equal loudness contours). A phon is dened as the loudness associated with a 1000 Hz reference tone. Each line represents frequencies that are perceived as equally loud for the given phon level. Source: From Yost, 2013, p. 190. Copyright 2013 by Koninklijke Brill.

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3. PRoPERTIES oF SoUnD 51

perceived loudness also double? Or how much increase in dB SPL is needed to achieve a dou- bling of the perceived loudness? The relationship of loudness to pressure (or intensity) is gener- ally determined by using magnitude estimation or scaling method (Stevens, 1956). This type of loudness scale uses sones as the unit of loudness, where one sone is defined as the loudness of a 1000 Hz tone at 40 dB SPL. In these types of ex- periments, the subject is asked to adjust the dB SPL of the tone until the loudness is judged to be half, double, triple, and so on, of the loudness of the reference value. In other words, how many dB SPL corresponds to 2 sones or 0.5 sones? The data are typically presented for a 1000 Hz tone; however, similar data can be obtained for differ- ent frequencies by making the reference value equal to the loudness in phons for the frequency being measured. Figure 3–30 shows the relation- ship that occurs between the loudness in sones and the dB SPL of a 1000 Hz tone (or loudness level in phons). As you can see, the sone scale for loudness (on a log-log plot) shows a rela- tively straight line above 30 dB SPL for a 1000 Hz tone. In this region, a doubling of loudness cor- responds to a 10 dB increase in sound pressure and approximates a power function, where the slope of the line is the exponent of the power function; in this case, loudness = pressure6. For

levels between threshold and 30 dB SPL, the loudness function is much steeper and does not fit the simple power function. Notice also that the entire range of sound pressures (107) is com- pressed into a range of only around 100 sones.

Pitch

Pitch is generally considered the psychological correlate of frequency. Most of us have a sense that low and high-pitch sounds are related to low and high frequencies. Although we can detect a wide range of frequencies and can attribute a general perception of pitch to these frequencies, the question of interest here is how does a scale of pitch relate to a scale of frequency, that is, if we double the frequency of a sound, does the pitch also double? In other words, how much of a frequency increase is needed for a doubling of the perceived pitch? As you will see, there is not a one-to-one correspondence between pitch and frequency. The relationship of pitch to frequency was first described by Stevens and Volkmann (1940). The pitch scale is typically presented in units called mels. The mel scale assigns a stan- dard reference value of 1000 mels to the pitch as- sociated with 1000 Hz. The subject then adjusts the frequency of a tone until the pitch is judged to be half, double, triple, and so on, of the pitch of the 1000 Hz reference tone. In other words, what frequency best corresponds to 500, 2000, or 3000 mels? Figure 3–31 shows the relationship that occurs between pitch (linear scale) and fre- quency (logarithmic scale). As you can see, the mel scale for pitch does not show a one-to-one relation to frequency. For example, a doubling of frequency from 1000 to 2000 Hz corresponds to a 1.5 increase in pitch (from 1000 to 1500 mels). Conversely, a doubling of pitch from 1000 to 2000 mels corresponds to about a threefold in- crease in frequency (from 1000 to 3000 Hz). No- tice also that the entire range of frequencies (20 to 20,000 Hz) is compressed into a range of only 3500 mels, and that the data follow a curvilinear function in a semi-log plot across the frequency range.

Pitch also changes with intensity, as shown in Figure 3–32, for a variety of frequencies (Stevens,

dB Sound Pressure Level

0 10 20 30 40 50 60 70 80 90 100

S on

es

0.01

0.10

1.00

10.00

100.00

1000 Hz

FIGURE 3–30. Sone scale of loudness for 1000 Hz. one sone equals the loudness associated with a 1000 Hz tone at 40 dB SPL. A doubling of loudness occurs for tenfold increases in stimulus level.

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AUDIOLOGY: SCIENCE TO PRACTICE52

1935). These pitch versus intensity data are called equal pitch contours. In general, increas- ing intensity results in an increased pitch for the higher frequencies and a decreased pitch for the lower frequencies (Durrant & Lovrinic, 1995; Gu- lick, Gescheider, & Frisina, 1989). It should be pointed out that these increases and decreases in pitch with intensity are relatively small and not generally noticeable except under controlled laboratory conditions (Cohen, 1961).

Temporal Integration

Temporal integration describes how the thresh- old of audibility for a sound changes with the du- ration of the sound. In general, as the duration of a sound is shortened to less than 200 ms, the level of the sound must be increased in order for the sound to be audible (Durrant & Lovrinic, 1995; Watson & Gengel, 1969). Figure 3–33 shows how the threshold changes as a function of duration. As you can see, there is about a 10 dB increase in level (threshold shift) for a tenfold decrease in duration. In other words, as the duration is shortened from 200 to 20 ms, the sound level must be increased by 10 dB to remain audible. Similarly, as the duration is shortened from 20 to 2 ms, an additional 10 dB increase in the level of the sound is needed to remain audible. It should

also be pointed out that for tones less than 10 ms in duration, the quality of the sound also changes significantly, such that the tonality is lost and it is perceived as a brief click (transient) type sound. These brief transients also spread their energy across a wider frequency range called spectral splatter. You can also see from Figure 3–33 that for sounds with durations greater than 200 ms, the threshold remains constant.

Localization

Localization refers to the ability to determine the direction from which a sound is coming. In gen-

150 (220)

(400)

(1000) 500 (700)

300

1K 2K (1600) 3K (2000)

4K (2300)

5K (2500)

8K (2900)

30 40 50 60 70 80 90 100 110 120

2 %

SOUND LEVEL (dB)

Δ F

f o

r C

N S

T P

IT C

H

( %

F )

FIGURE 3–32. Equal pitch contours. Each line repre- sents equal pitch as stimulus level increases. For lower frequencies, as stimulus level is increased, the fre- quency tends to decrease. For higher frequencies, as stimulus level is increased, the frequency tends to in- crease. Source: From Durrant and Lovrinic, 1995, p. 279. Copyright 1995 by Williams and Wilkins.

Frequency (Hz) 100 1000 10000

M el

s

0

1000

2000

3000

4000

FIGURE 3–31. Mel scale of pitch. The pitch associated with 1000 Hz is dened as 1000 mels. The range of audible frequencies (20 to 20,000 Hz) is compressed into about 3500 mels.

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3. PRoPERTIES oF SoUnD 53

eral, our ability to localize is greatly dependent on the use of both ears, called binaural hearing. Most animals have a keen sense of localization, and many animals are able to move their pin- nae to help them localize. Humans, on the other hand, do not move their auricles as a means of localizing. Instead, humans rely (unconsciously) on different arrival times and/or intensities of sounds at the two ears to determine from where sounds are coming. These mechanisms are re- ferred to as interaural time differences or inter- aural intensity differences. These two mecha- nisms can explain localization of simple sounds that come from one side of the head. For local- ization, clearly two ears are better than one, and a typical complaint of people with hearing loss in one ear is that they have some difficulty local- izing sounds.

Figure 3–34 shows how interaural time and interaural intensity mechanisms operate when listening with two ears to a sound presented to one side of the head. As shown in Figure 3–34B, higher frequencies (>1500 Hz) have shorter wavelengths as compared to the size of the head, and these frequencies tend to be blocked by the head (Fedderson, Sandel, Teas, & Jeffress, 1957). In this case, the sound at the ear farther away

from the sound source is less audible, called head shadow. For lower frequencies, shown in Figure 3-34C, the wavelength is larger than the size of the head and wraps around the head due to diffraction and does not create interau- ral intensity differences; however, it does take slightly more time for the sound to get to the ear farther away from the sound source. These interaural time differences appear to be impor- tant for lower frequency localization as long as

2 20 200 2000

0

10

20

30

40

Duration (ms)

10 fold

10 dB

dB s

hi ft

re la

tiv e

to s

te ad

y st

at e

FIGURE 3–33. Temporal integration function. This shows how the threshold for a pure tone changes as a function of duration. As the duration is shortened by a factor of 10 (e.g., from 200 to 20 ms), the sound level must be increased by 10 dB to remain audible.

>

A

B >

>

C

FIGURE 3–34. A–C. Illustrations showing factors important for sound localization in the horizontal plane. A. Interaural time differences between the two ears; varies as a function of azimuth. B. Interaural intensity differences between the two ears resulting from sound shadow area that occurs at the ear farther away from the sound source when wavelength is short, which may explain some localization at high frequencies. C. Sound shadow does not occur when wavelength is at least as long as the width of the head.

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AUDIOLOGY: SCIENCE TO PRACTICE54

the wavelength is larger than the distance be- tween the two ears; however, this simple model has been shown to have some discrepancies in more recent studies (Kuhn, 1977) and these are beyond the scope of this textbook. As the direc- tion of the sound source moves more toward the center, localization based on interaural time and intensity differences becomes more difficult. In those situations, and in vertically directed local- ization, spectral differences in the sound from

front to back, due to the shape of the auricle, may help with localization at higher frequen- cies because the wavelengths are smaller than the auricle (Durrant & Lovrinic, 1995). In the real world, humans are able to move their heads and/or use visual cues to help them identify the source of a sound; and while listeners with uni- lateral hearing loss have some localization dif- ficulty, they are able to localize to some degree by using these other cues.

SYNOPSIS 3–4

l Psychoacoustics is the study of how we perceive the different acoustic properties of sound; loudness is the perceptual correlate of intensity and pitch is the perceptual correlate of frequency.

l The phon scale equates loudness across frequencies (also called equal loudness contours), as matched to the dB SPL of a 1000 Hz tone: For example, a 40 dB phon curve equates the loudness across frequencies to a 1000 Hz tone at 40 dB SPL.

l The sone scale describes how perceived loudness of a tone relates to changes in its change in dB SPL. One sone equals the loudness of a 40 dB SPL 1000 Hz tone (also 40 phons). For 1000 Hz above 30 dB SPL, an increase of 10 dB SPL results in a doubling in loudness, and is described as a power function. The range of audible SPLs is compressed into a range of only about 100 sones.

l The mel scale describes how changes in pitch correspond to changes in frequency. The mel scale is based on the pitch of 1000 Hz = 1000 mels. The mel scale shows less of an increase in perceived pitch for a corresponding increase in frequency. The entire range of audible frequencies is compressed into only about 3500 mels.

l Equal pitch contours describe how pitch of a specic tone changes as a function of intensity. Although the effects are quite small, increasing the intensity of high frequencies causes the pitch to increase; increasing the intensity of low frequencies causes the pitch to decrease.

l Temporal integration relates a tone’s threshold to changes in the tone’s duration. There is about a 10 dB increase in threshold for each tenfold decrease in duration below 200 ms (e.g., 200 to 20 ms). For durations longer than 200 ms, threshold remains constant (complete integration).

l Localization of a sound’s source is easiest with binaural hearing. Localizing the source of a sound from the sides is facilitated by the interaural intensity differences for higher frequencies (due to head shadow effect), and interaural time differences for lower frequencies. Localizing from front/back and in the vertical plane are facilitated by spectral differences due to the auricle.

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3. PRoPERTIES oF SoUnD 55

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Durrant, J. D., & Lovrinic, J. H. (1995). Bases of Hear- ing Science (3rd ed.). Baltimore, MD: Williams & Wilkins.

Fedderson, W. E., Sandel, T. T., Teas, D. C., & Jeffress, L. A. (1957). Localization of high-frequency tones. Journal of the Acoustical Society of America, 29, 988–991.

Fletcher, H., & Munson, W. A. (1933). Loudness, its definition, measurement, and calculation. Journal of the Acoustical Society of America, 5, 82–105.

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Robinson, D. W., & Dadson, R. S. (1956). A re- determination of the equal loudness relations for pure tones. British Journal of Applied Physiology, 7, 166–181.

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