17.5

Transverse Standing Waves

CONCEPTS AT A GLANCE A standing wave is another interference effect that can occur when two waves overlap. Standing waves can arise with transverse waves, such as those on a guitar string, and also with longitudinal sound waves, such as those in a flute. In any case, the principle of linear superposition provides an explanation of the effect, just as it does for diffraction and beats. The Concepts-at-a-Glance chart in Figure 17-17 summarizes the role that this simple but powerful principle has played in this chapter. It is an expansion of the charts in Figures 17-9 and 17-14.

Figure 17-17 CONCEPTS AT A GLANCE This chart, which is an extension of those in Figures 17-9 and 17-14, indicates that transverse and longitudinal standing waves can be explained by the principle of linear superposition. The instrument in the photograph is an alpenhorn, and it uses longitudinal standing waves. (Robert Harding World Imagery/Alamy Images)

Figure 17-18 shows some of the essential features of transverse standing waves. In this figure the left end of each string is vibrated back and forth, while the right end is attached to a wall. Regions of the string move so fast that they appear only as a blur in the photographs. Each of the patterns shown is called a transverse standing wave pattern. Notice that the patterns include special places called nodes and antinodes. The nodes are places that do not vibrate at all, and the antinodes are places where maximum vibration occurs. To the right of each photograph is a drawing that helps us to visualize the motion of the string as it vibrates in a standing wave pattern. These drawings freeze the shape of the string at various times and emphasize the maximum vibration that occurs at an antinode with the aid of a red dot attached to the string.

Figure 17-18 Vibrating a string at certain unique frequencies sets up transverse standing wave patterns, such as the three shown in the photographs on the left. Each drawing on the right shows the various shapes that the string assumes at various times as it vibrates. The red dots attached to the strings focus attention on the maximum vibration that occurs at an antinode. In each of the drawings, one-half of a wave cycle is outlined in red. (Richard Megna/Fundamental Photographs)

Each standing wave pattern is produced at a unique frequency of vibration. These frequencies form a series, the smallest frequency f1 corresponding to the one-loop pattern and the larger frequencies being integer multiples of f1, as Figure 17-18 indicates. Thus, if f1 is 10 Hz, the frequency needed to establish the 2-loop pattern is 2f1 or 20 Hz, while that needed to create the 3-loop pattern is 3f1 or 30 Hz, and so on. The frequencies in this series (f1, 2f1, 3f1, etc.) are called harmonics. The lowest frequency f1 is called the first harmonic, and the higher frequencies are designated as the second harmonic (2f1), the third harmonic (3f1), and so forth. The harmonic number (1st, 2nd, 3rd, etc.) corresponds to the number of loops in the standing wave pattern. The frequencies in this series are also referred to as the fundamental frequency, the first overtone, the second overtone, and so on. Thus, frequencies above the fundamental are overtones (see Figure 17-18).
Standing waves arise because identical waves travel on the string in opposite directions and combine in accord with the principle of linear superposition. A standing wave is said to be standing because it does not travel in one direction or the other, as do the individual waves that produce it. Figure 17-19 shows why there are waves traveling in both directions on the string. At the top of the picture, one-half of a wave cycle (the remainder of the wave is omitted for clarity) is moving toward the wall on the right. When the half-cycle reaches the wall, it causes the string to pull upward on the wall. Consistent with Newton’s action–reaction law, the wall pulls downward on the string, and a downward-pointing half-cycle is sent back toward the left. Thus, the wave reflects from the wall. Upon arriving back at the point of origin, the wave reflects again, this time from the hand vibrating the string. For small vibration amplitudes, the hand is essentially fixed and behaves as the wall does in causing reflections. Repeated reflections at both ends of the string create a multitude of wave cycles traveling in both directions.

Figure 17-19 In reflecting from the wall, a forward-traveling half-cycle becomes a backward-traveling half-cycle that is inverted.

As each new cycle is formed by the vibrating hand, previous cycles that have reflected from the wall arrive and reflect again from the hand. Unless the timing is right, however, the new and the reflected cycles tend to offset one another, and a standing wave is not formed. Think about pushing someone on a swing and timing your pushes so that they reinforce one another. Such reinforcement in the case of the wave cycles leads to a large-amplitude standing wave. Suppose the string has a length L and its left end is being vibrated at a frequency f1. The time required to create a new wave cycle is the period T of the wave, where T = 1/f1 (Equation 10.5). On the other hand, the time needed for a cycle to travel from the hand to the wall and back, a distance of 2L, is 2L/v, where v is the wave speed. Reinforcement between new and reflected cycles occurs if these two times are equal; that is, if 1/f1 = 2L/v. Thus, a standing wave is formed when the string is vibrated with a frequency of f1 = v/(2L).
Repeated reinforcement between newly created and reflected cycles causes a large-amplitude standing wave to develop on the string, even when the hand itself vibrates with only a small amplitude. Thus, the motion of the string is a resonance effect, analogous to that discussed in Section 10.6 for an object attached to a spring. The frequency f1 at which resonance occurs is sometimes called a natural frequency of the string, similar to the frequency at which an object oscillates on a spring.
There is a difference between the resonance of the string and the resonance of a spring system, however. An object on a spring has only a single natural frequency, whereas the string has a series of natural frequencies. The series arises because a reflected wave cycle need not return to its point of origin in time to reinforce every newly created cycle. Reinforcement can occur, for instance, on every other new cycle, as it does if the string is vibrated at twice the frequency f1, or f2 = 2f1. Likewise, if the vibration frequency is f3 = 3f1, reinforcement occurs on every third new cycle. Similar arguments apply for any frequency fn = nf1, where n is an integer. As a result, the series of natural frequencies that lead to standing waves on a string fixed at both ends is

| (17.3)

It is also possible to obtain Equation 17.3 in another way. In Figure 17-18, one-half of a wave cycle is outlined in red for each of the harmonics, to show that each loop in a standing wave pattern corresponds to one-half a wavelength. Since the two fixed ends of the string are nodes, the length L of the string must contain an integer number n of half-wavelengths: or . Using this result for the wavelength in the relation shows which can be rearranged to give Equation 17.3.

Problem solving insight The distance between two successive nodes (or between two successive antinodes) of a standing wave is equal to one-half of a wavelength.

Standing waves on a string are important in the way many musical instruments produce sound. For instance, a guitar string is stretched between two supports and, when plucked, vibrates according to the series of natural frequencies given by Equation 17.3. The next two examples show how this series of frequencies governs the design of a guitar.

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 4 | | | Playing a Guitar

The heaviest string on an electric guitar has a linear density of 5.28 × 10−3 kg/m and is stretched with a tension of 226 N. This string produces the musical note E when vibrating along its entire length in a standing wave at the fundamental frequency of 164.8 Hz. (a) Find the length L of the string between its two fixed ends (see Figure 17-20a). (b) A guitar player wants the string to vibrate at a fundamental frequency of 2 × 164.8 Hz = 329.6 Hz, as it must if the musical note E is to be sounded one octave higher in pitch. To accomplish this, he presses the string against the proper fret before plucking the string. Find the distance L between the fret and the bridge of the guitar (see Figure 17-20b).

Figure 17-20 These drawings show the standing waves (in blue) that exist on a guitar string under different playing conditions.

Reasoning The series of natural frequencies (including the fundamental) for a string fixed at both ends is given by (Equation 17.3), where n = 1, 2, 3, etc. This equation can be solved for the length L. The speed v at which waves travel can be obtained from the tension and the linear density of the string. The fundamental frequencies that are given correspond to n = 1.
Knowns and Unknowns The given data are summarized as follows:

Description | Symbol | Value | Comment
Explicit Data |  |  |
Linear density of string | m/L | 5.28 × 10−3 kg/m |
Tension in string | F | 226 N |
Natural frequency at which string vibrates | fn | 164.8 Hz or 329.6 Hz | These are fundamental frequencies.
Imlicit Data |  |  |
Integer variable in series of natural frequencies | n | 1 | Fundamental frequencies are given.
Unknown Variable |  |  |
Length | L | ? |

Modeling the Problem

| Step 1 Natural Frequencies
| According to Equation 17.3, the natural frequencies for a string fixed at both ends are given by where n takes on the integer values 1, 2, 3, etc., v is the speed of the waves on the string, and L is the length between the two fixed ends. Solving this expression for L gives Equation 1 at the right. In this equation, only the speed v is unknown. We will obtain a value for it in Step 2.

| (1)

| Step 2 Speed of the Waves on the String
| The speed v of the waves traveling on the string is given by Equation 16.2 as

where F is the tension and m/L is the linear density, both of which are given. The substitution of this expression for the speed into Equation 1 is shown at the right.

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

It is given that the two natural frequencies at which the string vibrates are fundamental frequencies. Thus, n = 1 and fn = f1. We now determine the desired lengths in parts (a) and (b).

The length in part (b) is one-half that in part (a) because the fundamental frequency in part (b) is twice that in part (a).
Related Homework: Problems 23, 26 and 28

Need more practice?

Interactive LearningWare17.2
Electric guitars often come equipped with a “whammy bar,” which is a lever that allows the performer to adjust the tension in the strings. In this way, the performer can “dive bomb.” To dive bomb means to produce a tone that begins with a high frequency and ends with a low frequency and to go through all the frequencies in between. Suppose that a performer dive bombs the frequency in half. By what factor has the whammy bar reduced the tension in the strings?

| Conceptual Example 5 | | | The Frets on a Guitar

Figure 17-21 shows the frets on the neck of a guitar. They allow the player to produce a complete sequence of musical notes using a single string. Starting with the fret at the top of the neck, each successive fret shows where the player should press to get the next note in the sequence. Musicians call the sequence the chromatic scale, and every thirteenth note in it corresponds to one octave, or a doubling of the sound frequency. The spacing between the frets is greatest at the top of the neck and decreases with each additional fret further on down. The spacing eventually becomes smaller than the width of a finger, limiting the number of frets that can be used. Why does the fret spacing decrease going down the neck?

Figure 17-21 The spacing between the frets on the neck of a guitar decreases going down the neck toward the bridge.

The physics of the frets on a guitar.

Reasoning and Solution Our reasoning is based on Equation 17.3, with . The value of n is 1 because a string vibrates mainly at its fundamental frequency when plucked, as mentioned in Example 4. This equation indicates that the fundamental frequency f1 is inversely proportional to the length L between a given fret and the bridge of the guitar. Thus, in Example 4 we found that the E-string had a length of 0.628 m, corresponding to a frequency of f1 = 164.8 Hz. We also found that the length between the bridge and the fret that must be pressed to double this frequency to 329.6 Hz is one-half of 0.628 m, or 0.314 m. To understand why the spacing between frets decreases going down the neck, consider the fret that must be pressed to double the frequency again, from 329.6 Hz to 659.2 Hz. The length between the bridge and this fret would be one-half of 0.314 m, or 0.157 m. The three lengths of 0.628 m, 0.314 m, and 0.157 m are shown in Figure 17-21. The distances D1 and D2 are also shown and give the spacings between the starting fret at the top, the fret for the first doubling, and the fret for the second doubling. Clearly, D1 is greater than D2, and the frets near the top have more space between them than those further down.
Related Homework: Problem 32

Check Your Understanding 3

A standing wave that corresponds to the fourth harmonic is set up on a string that is fixed at both ends, (a) How many loops are in this standing wave? (b) How many nodes (excluding the nodes at the ends of the string) does this standing wave have? (c) Is there a node or an antinode at the midpoint of the string? (d) If the frequency of this standing wave is 440 Hz, what is the frequency of the lowest-frequency standing wave that could be set up on this string? (The answers are given at the end of the book.)
Background: A transverse standing wave is the featured player here. A vibrating string can have a series of natural frequencies, and the fourth harmonic is one of them.
For similar questions (including calculational counterparts), consult Self-Assessment Test 17.2, which is described at the end of 17.6.

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