1 Acoustics and Sound Measurement

We begin our study of audiology by reviewing the nature of sound because, after all, sound is what we hear. The science of sound is called acoustics, which is a branch of physics, and relies on several basic physical principles. Many useful sources are available for students wishing to pursue the areas covered in this chapter in greater detail (e.g., Beranek 1986; Gelfand 2010; Hewitt 1974; Kinsler, Frey, Coppens, & Sanders 1982; Peterson & Gross 1972; Sears, Zemansky, & Young 1982).

Physical Quantities

The basic physical quantities are mass, time, and length (or distance). All other physical quantities are derived by combining these three basic ones, as well as other derived quantities, in a variety of ways. The principal basic and derived quantities are summarized in Table 1.1. These basic quantities are expressed in terms of conventional units that are measurable and repeatable. The unit of mass (M) is the kilogram (kg) or the gram (g); the unit of length (L) is the meter (m) or the centimeter (cm); and the unit of time (t) is the second (s). Mass is not really synonymous with weight even though we express its magnitude in kilograms. The mass of a body is related to its density, but its weight is related to the force of gravity. If two objects are the same size, the one with greater density will weigh more. However, even though an object’s mass would be identical on the earth and the moon, it would weigh less on the moon, where there is less gravity.

When we express mass in kilograms and length in meters, we are using the meter–kilogram-second or MKS system. Expressing mass in grams and length in centimeters constitutes the centimeter-gram-second or cgs system. These two systems also have different derived quantities. For example, the units of force and work are called newtons and joules in the MKS system, and dynes and ergs in the cgs system, respectively. We will emphasize the use of MKS units because this is the internationally accepted standard in the scientific community, known as the Systeme Internationale (SI). Equivalent cgs values will often be given as well because the audiology profession has traditionally worked in cgs units, and the death of old habits is slow and labored. These quantities are summarized with equivalent values in MKS and cgs units in Table 1.1. In addition, the correspondence between scientific notation and conventional numbers, and the meanings of prefixes used to describe the sizes of metric units are shown for convenience and ready reference in Table 1.2 and Table 1.3.

Quantities may be scalars or vectors. A scalar can be fully described by its magnitude (amount or size), but a vector has both direction and magnitude. For example, length is a scalar because an object that is one meter long is always one meter long. However, we are dealing with a vector when we measure the distance between two coins that are one meter apart because their relationship has both magnitude and direction (from point x₁ to point x₂). This quantity is called displacement (d). Derived quantities will be vectors if they have one or more components that are vectors; for example, velocity is a vector because it is derived from displacement, and acceleration is a vector because it involves velocity. We distinguish between scalars and vectors because they are handled differently when calculations are being made.

Velocity Everyone knows that “55 miles per hour” refers to the speed of a car that causes it to travel a distance of 55 miles in a one-hour period of time. This is an example of velocity (v), which is equal to the amount of displacement (d) that occurs over time (t):

Acceleration Driving experience has taught us all that a car increases its speed to get onto a highway, slows down when exiting, and also slows down while making a turn. “Speeding up” and “slowing down” mean that the velocity is changing over time. The change of velocity over time is acceleration (a). Suppose a body is moving between two points. Its velocity at the first point is v₁, and the time at that point is t₁. Similarly, its velocity at the second point is v₂ and the time at that point is t₂. Average acceleration is the difference between the two velocities (v₂ – v₁) divided by the time interval (t₂ – t₁):

Because velocity is the first derivative of displacement, we find that acceleration is the second derivative of displacement:

Elasticity and restoring force It takes some effort (an outside force) to compress or expand a spring; and the compressed or expanded spring will bounce back to its original shape after it is released. Compressing or expanding the spring is an example of deforming an object. The spring bouncing back to its prior shape is an example of elasticity. More formally, we can say that elasticity is the property whereby a deformed object returns to its original form. Notice the distinction between deformation and elasticity. A rubber band and saltwater taffy can both be stretched (deformed), but only the rubber band bounces back. In other words, what makes a rubber band elastic is not that it stretches, but rather that it bounces back. The more readily a deformed object returns to its original form, the more elastic (or stiff) it is.

The unit of power is called the watt (w). One unit of power corresponds to one unit of work divided by one unit of time. Hence, one watt is equal to one joule divided by one second, or 1 w = 1 J/s. Power is also expressed in watts in the cgs system, where work is measured in ergs. Since 1 J = 10⁷ erg, we can also say that 1 w = 107 erg/s.

An analogous phenomenon occurs with sound. It is common knowledge that sound radiates outward in every direction from its source, constituting a sphere that gets bigger and bigger with increasing distance from the source, as illustrated by the concentric circles in Fig. 1.1. Let us imagine that the sound source is a tiny pulsating object (at the center of the concentric circles in the figure), and that it produces a finite amount of power, analogous to the fixed amount of oil in the prior example. Consequently, the sound power will be divided over the ever-expanding surface as distance increases from the source, analogous to the thinning out of the widening oil slick. This notion is represented in the figure by the thinning of the lines at greater distances from the source. Suppose we measure how much power registers on a certain fixed amount of surface area (e.g., a square inch). As a result, a progressively smaller proportion of the original power falls onto a square inch as the distance from the source increases, represented in the figure by the lighter shading of the same-size ovals at increasing distances from the source.

The examples just described reveal that a new quantity, called intensity (I), develops when power is distributed over area. Specifically, intensity is equal to power per unit area, or power divided by area, or

Intensity decreases with increasing distance from a sound source according to a rule called the inverse square law. It states that the amount of intensity drops by 1 over the square of the change in distance. Two examples are illustrated in Fig. 1.2. Frame a shows that when the distance from a loudspeaker is doubled from 5 m to 10 m, the amount of intensity at 10 m will be one quarter of what it was at 5 m (because 1/2² = 1/4). Similarly, frame b shows that tripling the distance from 5 m to 15 m causes the intensity to fall to one ninth of its value at the closer point because 1/3² = 1/9.

An important relationship to be aware of is that power is equal to pressure squared,

P = p²

A vibrating tuning fork is illustrated in Fig. 1.3. The initial force that was applied by striking the tuning fork is represented by the green arrow in frame 1. The progression of the drawings represents the motion of the prongs at selected points in time after the fork has been activated. The two prongs vibrate as mirror images of each other, so that we can describe what is happening in terms of just one prong. The insert highlights the motion of the right prong. Here the center position is where the prong would be at rest. When the fork is struck the prong is forced inward as shown by arrow a. After reaching the leftmost position it bounces back (arrow b), accelerating along the way. The rapidly moving prong overshoots the center and continues rightward (arrow c). It slows down along the way until it stops for an instant at the extreme right, where it reverses direction again and starts moving toward the left (arrow d) at an ever-increasing speed. It overshoots the center again, and as before, the prong now follows arrow a, slowing down until it stops momentarily at the extreme left. Here it reverses direction again and repeats the same process over and over again. One complete round trip (or replication) of an oscillating motion is called a cycle. The number of cycles that occur in one second is called frequency.

This form of motion occurs when a force is applied to an object having the properties of inertia and elasticity. Due to its elasticity, the deformation of the fork caused by the applied force is opposed by a restoring force. In the figure the initial leftward force is opposed by a restoring force in the opposite direction, that is, toward the right. The rightward restoring force increases as the prong is pushed progressively toward the left. As a result, the movement of the prong slows down and eventually stops. Under the influence of its elasticity the prong now reverses direction and starts moving rightward. As the restoring force brings the prong back toward the center, we must also consider its mass. Because the prong has mass, inertia causes it to accelerate as it moves back toward its center resting position. In fact, the prong is moving at its maximum speed as it passes through the resting position. The force of inertia causes the prong to overshoot the center and continue moving rightward. The deformation process begins again once the prong overshoots its resting position. As a result, opposing elastic restoring forces start building up again, now in the leftward direction. Just as before, the increasing (leftward) restoring force eventually overcomes the rightward inertial force, thereby stopping the prong’s displacement at the rightmost point, and causing a reversal in the direction of its movement. Hence, the same course of events is repeated again, this time in the leftward direction; then rightward, then leftward, etc., over and over again. This kind of vibration is called simple harmonic motion (SHM) because the oscillations repeat themselves at the same rate over and over again.

A diagram summarizing the concepts just described is shown in Fig. 1.4. The curve in the figure represents the tuning fork’s motion. The amount of displacement of the tuning fork prong around its resting (or center) position is represented by distance above and below the horizontal line. These events are occurring over time, which is represented by horizontal distance (from left to right). The initial displacement of the prong due to the original applied force is represented by the dotted segment of the curve. Inertial forces due to the prong’s mass and elastic restoring forces due to the elasticity of the prong are represented by labeled arrows. Damping of the oscillations due to friction is shown by the decline in the displacement of the curve as time goes on. The curve in this diagram is an example of a waveform, which is a graph that shows displacement (or another measure of magnitude) as a function of time.

Sound Waves

Tuning fork vibrations produce sound because the oscillations of the prongs are transmitted to the surrounding air particles. When the tuning fork prong moves to the right, it displaces air molecules to its right in the same direction. These molecules are thus displaced to the right of their own resting positions. Displacing air molecules toward the right pushes them closer to the air particles to their right. The pressure that exists among air molecules that are not being disturbed by a driving force (like the tuning fork) is known as ambient or atmospheric pressure. We can say that the rightward motion of the tuning fork prong exerts a force on the air molecules that pushes them together relative to their undisturbed, resting situation. In other words, forcing the air molecules together causes an increase in air pressure relative to the ambient pressure that existed among the undisturbed molecules. This state of positive air pressure is called compression. The amount of compression increases as the prong continues displacing the air molecules rightward. A maximum amount of positive pressure occurs when the prong and air molecules reach their greatest rightward displacement.

The tuning fork prong then reverses direction, overshoots its resting position, and proceeds to its leftmost position. The compressed air molecules reverse direction along with the prong. This occurs because air is an elastic medium, so the particles compressed to the right develop a leftward restoring force. Small as they are, air particles do have mass. Therefore, inertia causes the rebounding air particles to overshoot their resting positions and to continue toward their extreme leftward positions. As the particles move leftward, the amount of compression decreases and is momentarily zero as they pass through their resting positions. As they continue to move to the left of their resting positions, the particles are now becoming increasingly farther from the molecules to their right (compared with when they are in their resting positions). We now say that the air particles are rarefied compared with their resting states, so that the air pressure is now below atmospheric pressure. This state of lower than ambient pressure is called rarefaction. When the air particles reach the leftmost position they are maximally rarefied, which means that the pressure is maximally negative. At this point, the restoring force instigates a rightward movement of the air molecules. This movement is enhanced by the push of the tuning fork prongs that have also reversed direction. The air molecules now accelerate rightward (so that the amount of rarefaction decreases), overshoot their resting positions, and continue to the right, and so on. The tuning fork vibrations have now been transmitted to the surrounding particles, which are now also oscillating in simple harmonic motion. Sounds that are associated with simple harmonic motion are called pure tones.

Let us consider one of the air molecules that has already been set into harmonic motion by the tuning fork. This air particle now vibrates to-and-fro in the same direction that was originally imposed by the vibrating prong. When this particle moves toward its right it will cause a similar displacement of the particle that is located there. The subsequent leftward motion is also transmitted to the next particle, etc. Thus, the oscillations of one air particle are transmitted to the molecule next to it. The second particle is therefore set into oscillation, which in turn initiates oscillation of the next one, and so forth down the line. In other words, each particle vibrates back and forth around its own resting point, and causes successive molecules to vibrate back and forth around their own resting points, as shown schematically in Fig. 1.5. Notice that each molecule vibrates “in place” around its own average position; it is the vibratory pattern that is transmitted through the air.

This propagation of vibratory motion from particle to particle constitutes the sound wave. This wave appears as alternating compressions and rarefactions radiating from the sound source in all directions, as already suggested in Fig. 1.1. The transmission of particle motion along with the resulting variations in air pressure with distance from the source are represented in Fig. 1.6. Most people are more familiar with the kinds of waves that develop on the surface of a pond when a pebble is dropped into the water. These are transverse waves because the particles are moving at right angles to the direction that the wave is propagating. That is, the water particles oscillate up and down (vertically) even though the wave moves out horizontally from the spot where the pebble hit the water. This principle can be demonstrated by floating a cork in a pool, and then dropping a pebble in the water to start a wave. The floating cork reflects the motions of the water particles. The wave will move out horizontally, but the floating cork bobs up and down (vertically) at right angles to the wave. In contrast, sound waves are longitudinal waves because each air particle oscillates in the same direction in which the wave is propagating (Fig. 1.6). Although sound waves are longitudinal, it is more convenient to draw them with a transverse representation, as in Fig. 1.6. In such a diagram the vertical dimension represents some measure of the size of the signal (e.g., displacement, pressure, etc.), and left to right distance represents time (or distance). For example, the waveform in Fig. 1.6 shows the amount of positive pressure (compression) above the baseline, negative pressure (rarefaction) below the baseline, and distance horizontally going from left to right.

The Sinusoidal Function

Simple harmonic motion is also known as sinusoidal motion, and has a waveform that is called a sinusoidal wave or a sine wave. Let us see why. Fig. 1.7 shows one cycle of a sine wave in the center, surrounded by circles labeled to correspond to points on the wave. Each circle shows a horizontal line corresponding to the horizontal baseline on the sine wave, as well as a radius line (r) that will move around the circle at a fixed speed, much like a clock hand but in a counterclockwise direction.

Point a on the waveform in the center of the figure can be viewed as the “starting point” of the cycle. The displacement here is zero because this point is on the horizontal line. The radius appears as shown in circle b when it reaches 45° of rotation, which corresponds to point b on the sine wave. The angle between the radius and the horizontal is called the phase angle (θ) and is a handy way to tell location going around the circle and on the sine wave. In other words, we consider one cycle (one “round trip” of oscillation) to be the same as going around a circle one time. Just as a circle has 360°, we also consider one cycle to have 360°. Since 45/360 = ⅛, a phase angle (θ) of 45° is the same as one eighth of the way around a circle or one eighth of the way into a sine wave. Returning to the circle, the vertical displacement from the horizontal to the point where r intersects the circle is represented by a vertical line labeled d. This vertical line corresponds to the displacement of point b on the sine wave, where the displacement of the air particle is represented by the height of the point above the baseline. Notice that we now have a right triangle in the circle, where r is the hypotenuse, θ is an angle, and d is the leg opposite that angle. Recall from high school math that the sine of an angle equals the length of the opposite leg over the length of the hypotenuse. Here, sin θ = d/r. If we conveniently assume that the length of r is 1, then displacement d becomes the sine of angle θ, which happens to be 0.707. In other words, displacement is determined by the sine of the phase angle, and displacement at any point on the sine wave corresponds to the sine of θ. This is why it is called a sine wave.

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