Optional Unit V: Sound
C. Characteristics of Sound

4. Harmonics, Resonance, and Interference

Key Concepts

Every object has a unique natural frequency of vibration.

A periodic force occurring at the same frequency as the natural frequency of vibration of an object may cause the object to vibrate. This is called mechanical resonance.

The force and the object being affected must come into physical contact for mechanical resonance to occur.

Mechanical resonance may cause objects to fail.

An understanding of the causes and effects of mechanical resonance must be considered in the selection and design of materials for specific applications.

In certain applications, particularly in civil and mechanical engineering, an understanding of mechanical resonance is essential to prevent the failure of materials.

The lowest frequency which will produce a standing wave pattern in a one dimensional medium is called the fundamental frequency.

Higher resonant frequencies also produce standing wave patterns. The first overtone of a string fixed at both ends has a nodal point in the centre of the medium, as well as at either end. It has twice the frequency of the fundamental frequency.

Overtones have simple whole number multiples of the fundamental frequency (2f, 3f, 4f, etc.).

Multiples of the fundamental frequency are also called harmonics. The fundamental frequency is the first harmonic, the second harmonic corresponds to the first overtone,the third harmonic is the second overtone, and so on.

Musical instruments produce a series of overtones (or higher harmonics) superimposed on the fundamental frequency. This causes the sounds produced by these different sources to be distinguishable.

Oscilloscope tracings can be used to illustrate differences in sound intensity, frequency, and overtone characteristics.

Oscilloscope patterns can help to reveal differences in sound characteristics between music and noise.

(If an oscilloscope is not available, sketches representing oscilloscope tracings may be used instead.)

The frequency of a vibrating string is determined by its length, its tension, its diameter, and the density of the material.

Frequency (f) varies inversely with the length of the string (L):

Frequency also varies inversely with the diameter of the string (d):

Frequency varies directly with the square root of the tension of the string (T):

Frequency varies inversely with the square root of the density of the string ():

These equations can be combined to give

where f is the fundamental frequency of the string.

Musical string instruments are designed by taking into account the physics of vibrating strings.

Standing wave interference patterns help to explain resonance in air columns.

An air column of a fixed length will resonate at certain specific frequencies.

An adjustable air column can be adjusted to several different lengths to resonate with a specific frequency from the source.

At a resonant length the intensity of sound leaving an air column is at a maximum.

In order for an air column to resonate, an antinode (loop) must be produced at the end from which the sound is escaping.

For a closed air column, the shortest resonant length for a given frequency of sound is ¼ . Additional resonant lengths occur at increases in length of ½ (i.e., at odd integer multiples of /4).

For an open air column the shortest resonant length for a given frequency of sound is ½ . Additional resonant lengths occur at increases in length of ½ (i.e., at integer multiple of /2).

Wind instruments are designed to work based on the physics of air columns.

Resonant boxes, sounding boards, and musical instruments help to amplify sound.

The tines of a tuning fork produce an interference pattern. As the tines move inward, they create a compression on the inside, and a rarefaction in the wake. The reverse effect occurs when the tines move in the opposite direction. Two sets of waves are generated by the tuning fork. Each is out of phase with the other. Regions of destructive interference emanate diagonally from the tines as they vibrate. Hence, changes in sound intensity can be heard when a vibrating tuning fork is rotated near the ear.

The interference pattern produced by two loudspeakers (or from a single speaker and its echo) causes different levels of sound intensities due to constructive and destructive interference. Placement of the speaker(s) is crucial in providing the highest quality of sound.

The reverberation of sound in a room may produce regions where differences in the levels of sound intensity can be a problem.

Two sound sources vibrating at slightly different frequencies (roughly 5 to 10 Hz) produce a series of beats.

What is heard is a sound whose frequency is apparently the average of the two frequencies
(f₁ + f₂) /2which rises and falls in intensity at a frequency equal to the beat frequency

The beat frequency depends on the difference in frequencies between the two sources.

Beat frequency =

Musicians can use their understanding of beats to tune musical instruments.

Learning Outcomes

Students will increase their abilities to:

1. Define the following terms: natural frequency of vibration, mechanical resonance, fundamental frequency, overtones, harmonics, beat frequency.

2. Explain that all objects have a unique natural frequency of vibration.

3. Explain that a periodic force occurring at the same frequency as the natural frequency of vibration of an object may cause the object to begin to vibrate.

4. Explain that the periodic force and the object being affected must come into contact for mechanical resonance to occur.

5. Explain that mechanical resonance may cause objects to undergo failure.

6. Suggest ways in which the failure of an object due to mechanical resonance can be prevented.

7. Transfer an understanding of mechanical resonance to practical examples and common experiences.

8. Explain that mechanical resonance must be taken into account when selecting and designing materials for a specific application.

9. State that the fundamental frequency is the lowest frequency which will produce a standing wave pattern in a one dimensional medium.

10. State that the first overtone has twice the frequency of the fundamental frequency.

11. Explain that a closed pipe does not exhibit first (or any odd number) overtone, but only even numbered overtones (or odd harmonics).

12. State that overtones have whole number multiples of the fundamental frequency.

13. Identify different variables which will affect the frequency produced by a vibrating string.

14. Apply mathematical relationships governing the frequency of vibrating strings in problem solving.

15. Recognize that musical instruments are designed based on important underlying physical principles.

16. Suggest how an understanding of standing wave interference patterns can help to explain resonance in air columns.

17. State that an air column will resonate if certain specific frequencies of sound pass through it.

18. Recognize that resonance will occur when an antinode (loop) is allowed to form at the end of the air column from which sound is escaping.

19. Explain that an adjustable air column can be made to resonate at several different lengths for a given frequency of sound.

20. Solve problems relating to the resonance of sound in open and closed air columns.

21. Explain why a vibrating tuning fork produces an interference pattern.

22. Explain why the placement of one or more speakers in a room affects the quality of sound produced.

23. Explain that beats are produced when two sound sources vibrate at slightly different frequencies.

24. Explain that the beat frequency depends on the difference in the frequencies of the two vibrating sources.

25. Apply an understanding of beat frequency in problem solving.

Teaching Suggestions, Activities and Demonstrations

1. A very useful apparatus to have in the lab is a pair of sympathetic tuning forks attached to resonance boxes. Clamping the tines on one of the tuning forks changes the pitch slightly and causes noticeable beats to occur. The resonance boxes can also be used to demonstrate that if one tuning fork is struck with the open ends of the resonance boxes facing the other, then the other tuning fork will begin to vibrate, since it has the same resonant frequency.

   An inexpensive alternative to sympathetic tuning forks is to use two tuning forks that have the same frequency. Use elastic bands on the tines of one of the forks to lower its pitch and produce beats. The two tuning forks can be placed with their stems touching the sounding box of a stringed instrument, such as a guitar. This amplifies the sound. As an interesting side effect, some of the strings on the guitar may begin to resonate.

2. Carefully place a hollow tube over one of the vibrating prongs of a tuning fork. Rotate the tuning fork. Notice that no evidence of interference is heard. This illustrates that the interference pattern produced by a tuning fork only results from the combined effect of the waves travelling from both of the prongs.

3. Observe a standing wave pattern produced in a one dimensional medium, record an illustration of the pattern, determine the number of wavelengths exhibited by the pattern, label a node and an antinode, and identify which of the harmonics the pattern represents.

4. Give an example of a situation involving the failure of an object due to mechanical resonance.

   Show students the documentary film describing the Tacoma Narrows bridge collapse. It illustrates how mechanical resonance in structures can cause them to fail.

5. A relatively inexpensive sonometer can be made by anchoring one end of a wire to a table. Run the wire over a table clamp pulley and attach weights to the free end. Changing the amount of weight on the free end will change the tension of the wire, affecting the pitch. If a spoke tensiometer is available, it can be used to determine the tension in the wire. The actual tension can be compared with the expected tension based on the mechanics of the pulley arrangement.

6. Explain why the same frequencies produced by various different musical instruments all have sounds which can be distinguished from one another.

7. Have students bring a variety of string instruments to class. They can examine the instruments to see ways in which the physics of sound is being applied. In particular, some of the factors which affect the frequency of a string can be illustrated very nicely.

8. Perform an activity to determine the effect of (at least one of) length, tension, density, or diameter on the frequency of a vibrating string.

9. Determine the ways in which the frequency, intensity, and overtone characteristics of different sounds are depicted in oscilloscope tracings.

10. Examine oscilloscope patterns to determine important distinguishing features between noise and music.

11. Experimentally determine at what wavelengths adjustable open or closed air columns resonate for a given frequency of sound passing through them.

12. Have students predict what might occur if an understanding of mechanical resonance were not taken into consideration when selecting and designing materials for some specific application.

13. Perform an activity to investigate the beats produced by two sound sources vibrating at slightly different frequencies.

14. Perform an activity which illustrates one way in which musical instruments can be tuned.

15. Experimentally determine what frequencies of sound could be used to cause fixed length open or closed air columns to resonate.

16. A Canadian inventor designed a new type of sports whistle which produces a shrill sound without a spinning ball in the whistle housing. As an interesting activity, students can bring in a variety of different types of whistles and examine their design and sound characteristics.

17. Perform an activity to investigate the interference pattern produced by a vibrating tuning fork or some other objects.

18. Sketch the standing wave patterns produced in a one dimensional medium from the fundamental frequency and at least one of the overtones.

19. Investigate the hearing range of different animals. Using an audio frequency generator set at frequencies above the range of human hearing, determine if different types of animals respond to the sound.

    Use a dog whistle to show how a dog can respond to sound frequencies above the normal range of human hearing.