17.3

Diffraction

CONCEPTS AT A GLANCE Section 16.5 discusses the fact that sound is a pressure wave created by a vibrating object, such as a loudspeaker. The previous two sections of this chapter have examined what happens when two sound waves are present simultaneously at the same place; according to the principle of linear superposition, a resultant disturbance is formed from the sum of the individual waves. This principle reveals that overlapping sound waves exhibit interference effects, whereby the sound energy is redistributed within the overlap region. As the Concepts-at-a-Glance chart in Figure 17-9 shows, we will now use the principle of linear superposition to explore another interference effect, that of diffraction.

Figure 17-9 CONCEPTS AT A GLANCE By combining our knowledge of sound waves with the principle of linear superposition, we can understand the interference phenomenon of diffraction. In the illustration a water wave moves to the right and passes through an opening in a breakwater. To the right of the breakwater, the bending of the wave is an example of diffraction.

When a wave encounters an obstacle or the edges of an opening, it bends around them. For instance, a sound wave produced by a stereo system bends around the edges of an open doorway, as Figure 17-10a illustrates. If such bending did not occur, sound could be heard outside the room only at locations directly in front of the doorway, as part b of the drawing suggests. (It is assumed that no sound is transmitted directly through the walls.) The bending of a wave around an obstacle or the edges of an opening is called diffraction. All kinds of waves exhibit diffraction.

Figure 17-10 (a) The bending of a sound wave around the edges of the doorway is an example of diffraction. The source of the sound within the room is not shown. (b) If diffraction did not occur, the sound wave would not bend as it passes through the doorway.

To demonstrate how the bending of waves arises, Figure 17-11 shows an expanded view of Figure 17-10a. When the sound wave reaches the doorway, the air in the doorway is set into longitudinal vibration. In effect, each molecule of the air in the doorway becomes a source of a sound wave in its own right, and, for purposes of illustration, the drawing shows two of the molecules. Each produces a sound wave that expands outward in three dimensions, much like a water wave does in two dimensions when a stone is dropped into a pond. The sound waves generated by all the molecules in the doorway must be added together to obtain the total sound wave at any location outside the room, in accord with the principle of linear superposition. However, even considering only the waves from the two molecules in the picture, it is clear that the expanding wave patterns reach locations off to either side of the doorway. The net effect is a “bending,” or diffraction, of the sound around the edges of the opening. Further insight into the origin of diffraction can be obtained with the aid of Huygens’ principle (see Section 27.5).

Figure 17-11 Each vibrating molecule of the air in the doorway generates a sound wave that expands outward and bends, or diffracts, around the edges of the doorway. Because of interference effects among the sound waves produced by all the molecules, the sound intensity is mostly confined to the region defined by the angle on either side of the doorway.

When the sound waves generated by every molecule in the doorway are added together, it is found that there are places where the intensity is a maximum and places where it is zero, in a fashion similar to that discussed in the previous section. Analysis shows that at a great distance from the doorway the intensity is a maximum directly opposite the center of the opening. As the distance to either side of the center increases, the intensity decreases and reaches zero, then rises again to a maximum, falls again to zero, rises back to a maximum, and so on. Only the maximum at the center is a strong one. The other maxima are weak and become progressively weaker at greater distances from the center. In Figure 17-11 the angle defines the location of the first minimum intensity point on either side of the center. Equation 17.1 gives in terms of the wavelength and the width D of the doorway and assumes that the doorway can be treated like a slit whose height is very large compared to its width:

| (17.1)

Waves also bend around the edges of openings other than single slits. Particularly important is the diffraction of sound by a circular opening, such as that in a loudspeaker. In this case, the angle is related to the wavelength and the diameter D of the opening by

| (17.2)

An important point to remember about Equations 17.1 and 17.2 is that the extent of the diffraction depends on the ratio of the wavelength to the size of the opening. If the ratio is small, then is small and little diffraction occurs. The waves are beamed in the forward direction as they leave an opening, much like the light from a flashlight. Such sound waves are said to have “narrow dispersion.” Since high-frequency sound has a relatively small wavelength, it tends to have a narrow dispersion. On the other hand, for larger values of the ratio , the angle is larger. The waves spread out over a larger region and are said to have a “wide dispersion.” Low-frequency sound, with its relatively large wavelength, typically has a wide dispersion.
In a stereo loudspeaker, a wide dispersion of the sound is desirable. Example 3 illustrates, however, that there are limitations to the dispersion that can be achieved, depending on the loudspeaker design.

A N A L Y Z I N G M U L T I P L E - C O N C E P T P R O B L E M S

| Example 3 | | | Designing a Loudspeaker for Wide Dispersion

A 1500-Hz sound and a 8500-Hz sound each emerges from a loudspeaker through a circular opening that has a diameter of 0.30 m (see Figure 17-12). Assuming that the speed of sound in air is 343 m/s, find the diffraction angle for each sound.

Figure 17-12 Because the dispersion of high frequencies is less than that of low frequencies, you should be directly in front of the speaker to hear both the high and low frequencies equally well.

Reasoning The diffraction angle depends on the ratio of the wavelength of the sound to the diameter D of the opening, according to (Equation 17.2). The wavelength for each sound can be obtained from the given frequencies and the speed of sound.
Knowns and Unknowns The following data are available:

Description | Symbol | Value
Sound frequency | f | 1500 Hz or 8500 Hz
Diameter of speaker opening | D | 0.30 m
Speed of sound | v | 343 m/s
Unknown Variable |  |
Diffraction angle | θ | ?

Modeling the Problem

| Step 1 The Diffraction Angle
| Equation 17.2 indicates that the diffraction angle is related to the ratio of the wavelength of the sound to the diameter D of the opening by . Solving this expression for gives Equation 1 at the right. A value for D is given. To determine a value for , we turn to Step 2.

| (1)

| Step 2 Wavelength
| The wavelength is related to the frequency f and the speed v of the sound according to (Equation 16.1). Solving for gives

which can be substituted into Equation 1 as shown at the right.

Solution Combining the results of each step algebraically, we find that

Applying this result to each of the sound frequencies shows that

The physics of tweeter loudspeakers. Figure 17-12 illustrates these results. With a 0.30-m opening, the dispersion of the higher-frequency sound is limited to only 9.4°. To increase the dispersion, a smaller opening is needed. It is for this reason that loudspeaker designers use a small-diameter speaker called a tweeter to generate the high-frequency sound (see Figure 17-13).

Figure 17-13 Small-diameter speakers, called tweeters, are used to produce high-frequency sound. The small diameter helps to promote a wider dispersion of the sound. (Sony Electronics, Inc./Sony USA)

Problem solving insight When a wave passes through an opening, the extent of diffraction is greater when the ratio is greater, where is the wavelength of the wave and D is the width or diameter of the opening.

Related Homework: Problems 10, 11 and 13
As we have seen, diffraction is an interference effect, one in which some of the wave’s energy is directed into regions that would otherwise not be accessible. Energy, of course, is conserved during this process, because energy is only redistributed during diffraction; no energy is created or destroyed.

Copyright © 2007 John Wiley & Sons, Inc. All rights reserved.