17.6

Longitudinal Standing Waves

Standing wave patterns can also be formed from longitudinal waves. For example, when sound reflects from a wall, the forward- and backward-going waves can produce a standing wave. Figure 17-22 illustrates the vibrational motion in a longitudinal standing wave on a Slinky. As in a transverse standing wave, there are nodes and anti-nodes. At the nodes the Slinky coils do not vibrate at all; that is, they have no displacement. At the antinodes the coils vibrate with maximum amplitude and, thus, have a maximum displacement. The red dots in Figure 17-22 indicate the lack of vibration at a node and the maximum vibration at an antinode. The vibration occurs along the line of travel of the individual waves, as is to be expected for longitudinal waves. In a standing wave of sound, the molecules or atoms of the medium behave as the red dots do.

Figure 17-22 A longitudinal standing wave on a Slinky showing the displacement nodes (N) and antinodes (A).

Musical instruments in the wind family depend on longitudinal standing waves in producing sound. Since wind instruments (trumpet, flute, clarinet, pipe organ, etc.) are modified tubes or columns of air, it is useful to examine the standing waves that can be set up in such tubes. Figure 17-23 shows two cylindrical columns of air that are open at both ends. Sound waves, originating from a tuning fork, travel up and down within each tube, since they reflect from the ends of the tubes, even though the ends are open. If the frequency f of the tuning fork matches one of the natural frequencies of the air column, the downward- and upward-traveling waves combine to form a standing wave, and the sound of the tuning fork becomes markedly louder. To emphasize the longitudinal nature of the standing wave patterns, the left side of each pair of drawings in Figure 17-23 replaces the air in the tubes with Slinkies, on which the nodes and antinodes are indicated with red dots. As an additional aid in visualizing the standing waves, the right side of each pair of drawings shows blurred blue patterns within each tube. These patterns symbolize the amplitude of the vibrating air molecules at various locations. Wherever the pattern is widest, the amplitude of vibration is greatest (a displacement antinode), and wherever the pattern is narrowest there is no vibration (a displacement node).

Figure 17-23 A pictorial representation of longitudinal standing waves on a Slinky (left side of each pair) and in a tube of air (right side of each pair) that is open at both ends (A, antinode; N, node).

To determine the natural frequencies of the air columns in Figure 17-23, notice that there is a displacement antinode at each end of the open tube because the air molecules there are free to move.* As in a transverse standing wave, the distance between two successive antinodes is one-half of a wavelength, so the length L of the tube must be an integer number n of half-wavelengths: or . Using this wavelength in the relation shows that the natural frequencies of the tube are

| (17.4)

At these frequencies, large-amplitude standing waves develop within the tube due to resonance. Examples 6 and 7 and Examples 6 and 7 illustrate how Equation 17.4 is involved when a flute is played.

| Example 6 | | | Playing a Flute

When all the holes are closed on one type of flute, the lowest note it can sound is a middle C (fundamental frequency = 261.6 Hz). The air temperature is 293 K, and the speed of sound is 343 m/s. Assuming the flute is a cylindrical tube open at both ends, determine the distance L in Figure 17-24—that is, the distance from the mouthpiece to the end of the tube. (This distance is approximate, since the antinode does not occur exactly at the mouthpiece.)

Figure 17-24 The length L of a flute between the mouthpiece and the end of the instrument determines the fundamental frequency of the lowest playable note.

The physics of a flute.

Reasoning For a tube open at both ends, the series of natural frequencies (including the fundamental) is given by (Equation 17.4), where n takes on the integer values 1, 2, 3, etc. To obtain a value for L, we can solve this equation, since the given fundamental frequency corresponds to n = 1 and the speed v of sound is known.
Solution Solving Equation 17.4 for the length L, we obtain

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 7 | | | Tuning a Flute

A flautist is playing the flute discussed in Example 6, but now the temperature is 305 K instead of 293 K. Therefore, with the length calculated in Example 6, the note middle C does not have the proper fundamental frequency of 261.6 Hz. In other words, the flute is out of tune. To adjust the tuning, the flautist can alter the flute’s length by changing the extent to which the head joint (see Figure 17-24) is inserted into the main stem of the instrument. To what length must the flute be adjusted to play the middle C at its proper frequency?
Reasoning As in Example 6, we will make use of the series of natural frequencies (including the fundamental) represented by (Equation 17.4), for a tube open at both ends. In this expression, n takes on the integer values 1, 2, 3, etc. and has the value of n = 1 for the given fundamental frequency. We can again solve Equation 17.4 for L, but now must deal with the fact that no value is given for the speed v of sound at the temperature of 305 K. To obtain the necessary value, we will assume that air behaves as an ideal gas and utilize the value given for v at 293 K to calculate a value at 305 K.
Knowns and Unknowns The given data are summarized as follows:

Description | Symbol | Value | Comment
Explicit Data |  |  |
Natural frqquency for middle C | fn | 261.6 Hz | This is a fundamental frequency.
Speed of sound at 293 K | v293 | 343 m/s |
Temperature at which flute is played | T305 | 305 K |
Imlicit Data |  |  |
Integer variable in series of natural frequencies | n | 1 | The fundamental frequency is given.
Unknown Variable |  |  |
Length | L | ? |

Modeling the Problem

| Step 1 Natural Frequencies
| For a tube open at both ends, thenatural frequencies are given by , where n takes on theinteger values 1, 2, 3, etc., v305 is the speed of sound at 305 K, and L is the length between the two open ends. Solving this expression for L gives Equation 1 at the right. Since the speed v305 is not given, we proceed to Step 2 in order to evaluate it.

| (1)

| Step 2 The Speed of Sound
| We assume that air behaves as an ideal gas. For an ideal gas, the speed v of sound is given by Equation 16.5 as , where is the ratio of specific heat capacities at constant pressure and constant volume, k is Boltzmann’s constant, T is the Kelvin temperature, and m is the average mass of the molecules and atoms of which the air is composed. Applying this equation at the temperatures of 293 and 305 K, we have

Dividing these two expressions gives

In this result, the speed v293 and the two temperatures are known, so that we may solve for the unknown speed v305:

This expression can be substituted into Equation 1, as shown at the right.

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

Since the given frequency is the fundamental frequency, it follows that in this result n = 1 and fn = f1. The length to which the flute must be adjusted is

Comparing this result with that in Example 6, we see that to play in tune at the higher temperature, a flautist must lengthen the flute by 0.013 m.
Related Homework: Problem 42

Standing waves can also exist in a tube with only one end open, as the patterns in Figure 17-25 indicate. Note the difference between these patterns and those in Figure 17-23. Here the standing waves have a displacement antinode at the open end and a displacement node at the closed end, where the air molecules are not free to move. Since the distance between a node and an adjacent antinode is one-fourth of a wavelength, the length L of the tube must be an odd number of quarter-wavelengths: and for the two standing wave patterns in Figure 17-25. In general, then, , where n is any odd integer (n = 1, 3, 5, …). From this result it follows that , and the natural frequencies fn can be obtained from the relation ;

| (17.5)

A tube open at only one end can develop standing waves only at the odd harmonic frequencies f1, f3, f5, etc. In contrast, a tube open at both ends can develop standing waves at all harmonic frequencies f1, f2, f3, etc. Moreover, the fundamental frequency f1 of a tube open at only one end (Equation 17.5) is one-half that of a tube open at both ends (Equation 17.4). In other words, a tube open only at one end needs to be only one-half as long as a tube open at both ends in order to produce the same fundamental frequency.

Figure 17-25 A pictorial representation of the longitudinal standing waves on a Slinky (left side of each pair) and in a tube of air (right side of each pair) that is open only at one end (A, antinode; N, node).

Energy is also conserved when a standing wave is produced, either on a string or in a tube of air. The energy of the standing wave is the sum of the energies of the individual waves that comprise the standing wave. Once again, interference redistributes the energy of the individual waves to create locations of greatest energy (displacement antinodes) and locations of no energy (displacement nodes).

Check Your Understanding 4

A cylindrical bottle, partially filled with water, is open at the top. When you blow across the top of the bottle a standing wave is set up inside it. Is there a node or an antinode (a) at the top of the bottle and (b) at the surface of the water? (c) If the standing wave is vibrating at its fundamental frequency, what is the distance between the top of the bottle and the surface of the water? Express your answer in terms of the wavelength of the standing wave, (d) If you take a sip, is the fundamental frequency of the standing wave raised, lowered, or does it remain the same? (The answers are given at the end of the book.)
Background: The properties of a longitudinal standing wave are featured here. They depend on the length of the “tube” and whether the tube is open at both ends or open at only one end.
For similar questions (including calculational counterparts), consult Self-Assessment Test 17.2, which is described next.

Test your understanding of the material in Sections 17.5 and 17.6:

· | Transverse Standing Waves
· | Longitudinal Standing Waves

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