```Website owner:  James Miller
```

[ Home ] [ Up ] [ Info ] [ Mail ]

Natural frequency. Resonance. Harmonics. Overtones. Vibration of taut strings. Nodes and antinodes. Speed of a transverse wave. Wave reflection. Standing waves. Vibrating air columns. Quality of sound.

Def. Natural frequency of a body. The frequency with which a pendulum or an elastic object vibrates when it is displaced from its equilibrium position.

A weight hanging from a spring, a diving board, a bridge or a building will all vibrate with some particular characteristic frequency, called their natural frequency, when displaced. Some bodies have more than one frequency at which they will vibrate. A violin string can vibrate in many ways and has many natural frequencies.

Def. Resonance. In general, when a body which is capable of oscillating is acted on by a periodic series of impulses having a frequency equal to one of the natural frequencies of the body, the body is set into vibration of a relatively large amplitude. This phenomenon is called resonance. The body is said to resonate with the applied impulses.

An example of resonance can be found in the pushing of a playground swing. A swing is a pendulum with a single natural frequency that depends on its length. If one pushes a swing in a series of regularly spaced pushes with a frequency equal to the natural frequency of the swing, the amplitude of the motion can be made quite large. If, however, the frequency of the pushes differs from the natural frequency of the swing or if they occur at irregular intervals, the swing may scarcely move at all.

Amplification of sound through forced vibrations. If we start a tuning fork vibrating and press its stem against a table we find that the sound of the fork becomes much louder. The sound has been amplified by the table. Why? The vibrations of the tuning fork have been impressed on the table and the table has been forced to vibrate at the same rate as the fork even though the natural frequency of the table is presumably different from that of the fork. Since the table has a much larger vibrating area than the tuning fork, the forced vibrations produce a more intense sound. The same thing happens when we place a ticking watch on a table. The sound is greatly amplified. If we stretch a violin string between two clamps and bow it, we find that it does not produce a very intense sound. Stretched across the bridge of a violin, however, the thin wood of the violin is forced to vibrate at the same rate as the string, and one obtains a much more intense sound. The sounding board of a piano acts in the same way to amplify the vibrations of the strings.

Def. Harmonic. A tone whose rate of vibration is an integral multiple of a given primary tone. If f is the lowest frequency of a set of harmonic tones, it is the primary tone and is called the first harmonic. The other harmonics in the series have frequencies such as 2f, 3f, 4f, ..., nf, etc. where n is an integer.

Overtones. The lowest frequency that a vibrating string, air column, or other vibrating body can emit is called the fundamental frequency. The next higher frequency the vibrating body can emit is the first overtone, the next higher frequency is the second overtone, etc. The overtones may or may not be harmonics of the fundamental. The overtones of a bell, for example, are not harmonics because their frequencies are not integral multiples of the fundamental frequency.

I Music production by vibration of taut strings. The music of stringed instruments comes from the vibration of taut strings. Understanding how such stringed instruments work involves understanding the principles underlying transverse wave propagation in taut strings.

It is an observed fact that when a taut string that is fastened at both ends (as in a violin) is caused to vibrate, it will vibrate with a certain natural frequencies. These natural frequencies are determined by the length, density (mass per unit length) and tension of the string. The string can vibrate as a whole and it can also vibrate in segments as shown in Fig. 1. The string can vibrate in one part, two parts, three parts or n parts (n = 1, .. , ∞). In addition, a string may vibrate as a whole and in segments simultaneously. What an observer will see is a blurred envelope of the vibrations.

When a string vibrates as a whole, it produces what is called the fundamental frequency or tone. Let us denote the fundamental frequency by f. This frequency represents the lowest pitch that can be produced by the string.

If a string vibrates in two parts, its frequency of vibration is twice the fundamental frequency i.e. 2f . This is the first overtone and second harmonic.

If a string vibrates in three parts, its frequency of vibration is three times the fundamental frequency i.e. 3f. This is the second overtone and third harmonic.

In general, if a string vibrates in n parts, its frequency of vibration is n times the fundamental frequency i.e. nf. It is the n-th harmonic.

Nodes and antinodes. The points on a vibrating string that do not move are called nodes. The points at which the amplitude is greatest are called antinodes or loops. See Fig. 2.

Standing waves. If a string is vibrating in n segments, the n segments that are vibrating represent what are called standing waves or stationary waves. Standing waves are a phenomenon that can occur under certain conditions when waves moving back and forth in a medium are in phase and reinforce each other. No lateral motion is seen and they appear to be standing still.

We will now proceed to consider how standing waves are formed on a taut string.

Speed of a transverse wave in a taut string. The speed of a transverse wave in a taut string or wire is given by the formula

where T is the tension in the string and μ is its mass per unit length.

Example. Compute the velocity of a transverse wave in a string under a tension of 15 lb where the string weighs 0.003 lb/ft.

T = 15 lb

Reflection of a transverse wave at the end of a string. Consider an upwardly displaced pulse traveling to the

right on a string whose right end is securely attached to a post as shown in Fig. 3 (a). When the pulse reaches the post, the pulse will reflect, invert, and travel to the left as an inverted downwardly displaced pulse as shown in Fig. 3 (b). What has happened here? When the pulse reaches the post it exerts an upward pull on the post, and because the post is immovable, the upward pull becomes transformed into a downward pulse that is reflected and travels back to the left.

Let us now conceive of tying the right end of the string to a weightless ring that we slip over the post that allows the string to frictionlessly move up and down the post as shown in Fig. 4 (a). Now when the pulse reaches the post, the ring slides up the post as shown in Fig. 4 (b), and the pulse is reflected and returns to the left as an upwardly displaced pulse as shown in Fig. 4 (c).

We thus have two cases for wave reflection:

Case 1. Right end of string fixed. Wave inversion occurs on reflection.

Case 2. Right end of string is free. Wave is reflected with no inversion.

Standing wave formation. The ends of the strings of a stringed musical instrument (such as a violin) are fixed. That means that vibrational waves traveling on the instrument’s strings are inverted when they are reflected at the ends of the string. Riding on the vibrating sting of a musical instrument is not only the original wave produced by the player of the instrument, but all the reflections of the original wave, traveling from end to end on the string. By the superposition principle, the resultant wave is the sum of all the waves on the string. One can thus expect, in general, both constructive and destructive interference. Under certain conditions, however, the waves passing back and forth on the string will be in phase, they will reinforce each other, and standing waves will be formed. Under what conditions will this happen? For waves of certain wavelengths, standing waves will occur. What wavelengths? We first observe that since the string is fixed at both ends, both ends must necessarily be nodes. We next observe that since the nodes are one-half wavelength apart, for both right and left moving waves to be in sync, the length of the string must be some multiple of a half wavelength i.e. the length L of the string must be one of the following:

where λ is the wavelength of the wave. Thus given a string of length L, the equations

give a set of values λi that will result in standing waves.

Solving 1) for λi we get the expression for the possible wavelengths for which standing waves will occur in a taut string of length L:

Since v = fλ, where v is the speed of the wave and f is the frequency, we write the expression for the natural frequencies of a taut string of length L:

Two waves, having the same amplitude and wavelength and meeting the above conditions, traveling in opposite directions on a taut string, will form standing waves. In Fig. 5 are shown two waves, a red wave moving to the right and a blue wave moving to the left, where the above conditions for standing waves are assumed to be met. The wave in black is the sum of the red and blue waves. The figure shows how standing waves are formed. Lines ab and ef represent nodes of the standing wave. Lines cd and gh correspond to antinodes. While the red and blue waves are moving laterally, the black standing wave simply vibrates up and down in the areas of the antinodes, forming vibrating segments.

Equation of a standing wave. The equation of a standing wave can be formed by adding the displacements of two waves of equal amplitude, period, and wavelength which are traveling in opposite directions. The equation of the wave traveling in the positive x direction is

The equation of the wave traveling in the negative x direction is

Employing the expression for the cosine of the sum and difference of two angles

and combining terms we get

This is the equation of a cosine wave whose amplitude (the expression in brackets) varies with time. It is the equation of the standing wave.

How a guitar player produces a tone. How does a guitar player produce a particular tone? By placing a finger on a string at a chosen point, he can force a node at that point and suppress all harmonics that do not share that node.

II Music production by vibrating air columns. Many musical instruments produce music through a vibrating air column inside a hollow metal tube. In some cases the tube is bent to conserve space. In some musical instruments both ends are open. If both ends of the tube are open the instrument is said to contain an open-end air column. Many musical instruments are of this type. In other instruments one end is closed. Such instruments are said to contain a closed-end air column.

We will consider the principles underlying music production in a pipe organ.

Reflection of sound waves in a cylindrical pipe. There are two types of organ pipes: closed pipes and open pipes. See Fig. 6. In the figure, the left end of the pipes (where the source of the vibrations is located) is always regarded as open. In the closed tube, the left end is open and the right end is closed; in the open tube, both ends are open. The length of the air column is the length indicated in the figure.

In the case of both open and closed tubes, sound waves are reflected at both ends. One knows that sound waves are reflected when they strike a solid object (as with an echo) so he would expect them to be reflected at a closed end. However, at an open end one might expect the sound wave to just pass out through the end. In fact, however, whatever the explanation, they are reflected at an open end. They are not reflected totally, for if they were, one would hear no sound. However, about 99% of the sound energy is reflected back.

RULE: Sound is reflected from an open end of a tube with phase inversion. It is reflected from a closed end of a tube without inversion.

Standing wave formation. Standing waves are formed in tubes in a way similar to the way they are formed on taut strings. A tube of a particular length will have its own set of natural frequencies. It will have its fundamental frequency and a set of overtones, or higher harmonics. The considerations that determine the fundamental frequency and overtones are similar to those for taut strings.

The waves travel back and forth from end to end of the tube in the same way waves on a taut string (fastened at both ends) travel back and forth. Standing waves are formed when the right conditions are met. Under the right conditions, the waves passing back and forth in the tube will be in phase, they will reinforce each other, and standing waves will be formed. Under what conditions will this happen? For waves of certain wavelengths, standing waves will occur. What wavelengths?

RULE: For a standing wave to occur, a node must exist at the closed end of a tube and an antinode must exist at the open end of a tube. Thus an open tube must have an antinode at both ends and a closed tube must have a node on the closed end and an antinode at the open end. Keeping these requirements in mind, see Fig. 7 and 8 for the possible configurations that will yield standing waves.

We consider open and closed tubes separately:

Open tube. In open tubes all harmonics are possible. If we denote the first harmonic (fundamental tone) by f1, then the harmonics are

f2 = 2 f1

f3 = 3 f1

f4 = 4 f1

f5 = 5 f1

........

1. Production of fundamental frequency (first harmonic). This case is represented by Fig. 7(a). If λ1 and f1 are the wavelength and frequency of the fundamental tone:

where v is the velocity of sound.

2. Production of first overtone (second harmonic). This case is represented by Fig. 7(b). If λ2 and f2 are the wavelength and frequency of the first overtone and second harmonic:

3. Production of second overtone (third harmonic). This case is represented by Fig. 7(c). If λ3 and f3 are the wavelength and frequency of the second overtone and third harmonic:

Closed tube. In closed tubes only odd harmonics are possible. If we denote the first harmonic (fundamental tone) by f1, then the harmonics are

f3 = 3 f1

f5 = 5 f1

f7 = 7 f1

........

1. Production of fundamental frequency (first harmonic). This case is represented by Fig. 8(a). If λ1 and f1 are the wavelength and frequency of the fundamental tone:

2. Production of first overtone (third harmonic). This case is represented by Fig. 8(b). If λ3 and f3 are the wavelength and frequency of the first overtone and third harmonic:

3. Production of second overtone (fifth harmonic). This case is represented by Fig. 8(c). If λ5 and f5 are the wavelength and frequency of the second overtone and fifth harmonic

________________________________________________________________

Quality of sound. A bell that has been rung or a string that has been plucked will vibrate with many frequencies at the same time. It is rare for a body to vibrate with only a single frequency. A carefully made tuning fork struck lightly on a rubber block may vibrate with a single frequency, but in the case of musical instruments, the fundamental and many frequencies are usually present at the same time. Different instruments playing the same note at the same intensity sound differently. Each instrument has its own characteristic sound. This difference is a property of sounds called quality. The difference in the sound from different instruments is related to the number of overtones present and their relative intensities.

Overtones added to a fundamental give a sound a much fuller and richer quality. The quality of a sound depends, in general, on the number of overtones present and their relative intensities.

References

Dull, Metcalfe, Brooks. Modern Physics

Sears, Zemansky. University Physics

Semat, Katz. Physics