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Forced Vibrations Of Simple Systems English Language Essay

Paper Type: Free Essay Subject: English Language
Wordcount: 5371 words Published: 1st Jan 2015

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Mechanical, acoustical, or electrical vibrations are the sources of sound in musical instruments. Some familiar examples are the vibrations of strings violin, guitar, piano, etc, bars or rods xylophone, glockenspiel, chimes, and clarionet reed, membranes (drums, banjo), plates or shells (cymbal, gong, bell), air in a tube (organ pipe, brass and woodwind instruments, marimba resonator), and air in an enclosed container (drum, violin, or guitar body). In most instruments, sound production depends upon the collective behavior of several vibrators, which may be weakly or strongly coupled together. This coupling, along with nonlinear feedback, may cause the instrument as a whole to behave as a complex vibrating system, even though the individual elements are relatively simple vibrators (Hake and Rodwan, 1966).

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In the first seven chapters, we will discuss the physics of mechanical and acoustical oscillators, the way in which they may be coupled together, and the way in which they radiate sound. Since we are not discussing electronic musical instruments, we will not deal with electrical oscillators except as they help us, by analogy, to understand mechanical and acoustical oscillators.

According to Iwamiya, Kosygi and Kitamura (1983) many objects are capable of vibrating or oscillating. Mechanical vibrations require that the object possess two basic properties: a stiffness or spring like quality to provide a restoring force when displaced and inertia, which causes the resulting motion to overshoot the equilibrium position. From an energy standpoint, oscillators have a means for storing potential energy (spring), a means for storing kinetic energy (mass), and a means by which energy is gradually lost (damper). vibratory motion involves the alternating transfer of energy between its kinetic and potential forms. The inertial mass may be either concentrated in one location or distributed throughout the vibrating object. If it is distributed, it is usually the mass per unit length, area, or volume that is important. Vibrations in distributed mass systems may be viewed as standing waves. The restoring forces depend upon the elasticity or the compressibility of some material. Most vibrating bodies obey Hooke’s law; that is, the restoring force is proportional to the displacement from equilibrium, at least for small displacement.

Simple harmonic motion in one dimension:

Moore (1989) has mentioned that the simplest kind of periodic motion is that experienced by a point mass moving along a straight line with an acceleration directed toward a fixed point and proportional to the distance from that point. This is called simple harmonic motion, and it can be described by a sinusoidal function of time, where the amplitude A describes the maximum extent of the motion, and the frequency f tells us how often it repeats.

The period of the motion is given by

That is, each T seconds the motion repeats itself.

Sundberg (1978) has mentioned that a simple example of a system that vibrates with simple harmonic motion is the mass-spring system shown in Fig.1.1. We assume that the amount of stretch x is proportional to the restoring force F (which is true in most springs if they are not stretched too far), and that the mass slides freely without loss of energy. The equation of motion is easily obtained by combining Hooke’s law, F = -Kx, with Newton’s second law, F = ma =. Thus,




The constant K is called the spring constant or stiffness of the spring (expressed in Newton’s per meter). We define a constant so that the equation of motion becomes

This well-known equation has these solutions:


Figure 2.1: Simple mass-spring vibrating system

Source: Cremer, L., Heckl, M., Ungar, E (1988), “Structure-Borne Sound,” 2nd edition, Springer Verlag

Figure 2.2: Relative phase of displacement x, velocity v, and acceleration a of a simple vibrator

Source: Campbell, D. M., and Greated, C (1987), The Musician’s Guide to Acoustics, Dent, London


From which we recognize as the natural angular frequency of the system.

The natural frequency fo of our simple oscillator is given by and the amplitude by or by A; is the initial phase of the motion. Differentiation of the displacement x with respect to time gives corresponding expressions for the velocity v and acceleration a (Cardle et al, 2003):




Ochmann (1995) has mentioned that the displacement, velocity, and acceleration are shown in Fig. 1.2. Note that the velocity v leads the displacement by radians (90), and the acceleration leads (or lags) by radians (180). Solutions to second-order differential equations have two arbitrary constants. In Eq. (1.3) they are A and; in Eq. (1.4) they are B and C. Another alternative is to describe the motion in terms of constants x0 and v0, the displacement and velocity when t =0. Setting t =0 in Eq. (1.3) gives and setting t = 0 in Eq. (1.5) gives From these we can obtain expressions for A and in terms of xo and vo:



Alternatively, we could have set t= 0 in Eq. (1.4) and its derivative to obtain B= x0 and C= v0/ from which


2.3 Complex amplitudes

According to Cremer, Heckl and Ungar (1990) another approach to solving linear differential equations is to use exponential functions and complex variables. In this description of the motion, the amplitude and the phase of an oscillating quantity, such as displacement or velocity, are expressed by a complex number; the differential equation of motion is transformed into a linear algebraic equation. The advantages of this formulation will become more apparent when we consider driven oscillators.

This alternate approach is based on the mathematical identity where j =. In these terms,

Where Re stands for the “real part of”. Equation (1.3) can be written as,

Skrodzka and Sek (2000) has mentioned that the quantity is called the complex amplitude of the motion and represents the complex displacement at t=0. The complex displacement is written

The complex velocity and acceleration become

Desmet (2002) has mentioned that each of these complex quantities can be thought of as a rotating vector or phase rotating in the complex plane with angular velocity, as shown in Fig. 1.3. The real time dependence of each quantity can be obtained from the projection on the real axis of the corresponding complex quantities as they rotate with angular velocity

Figure 2.3: Phase representation of the complex displacement, velocity, and acceleration of a linear oscillator

Source: Bangtsson E, Noreland D and Berggren M (2003), Shape optimization of an acoustic horn, Computer Methods in Applied Mechanics and Engineering, 192:1533-1571

2.4 Continuous systems in one dimension

Strings and bars

This section focuses on systems in which these elements are distributed continuously throughout the system rather than appearing as discrete elements. We begin with a system composed of several discrete elements, and then allow the number of elements to grow larger, eventually leading to a continuum (Karjalainen and Valamaki, 1993).

Linear array of oscillators

According to Mickens (1998) the oscillating system with two masses in Fig. 1.20 was shown to have two transverse vibrational modes and two longitudinal modes. In both the longitudinal and transverse pairs, there is a mode of low frequency in which the masses move in the same direction and a mode of higher frequency in which they move in opposite directions. The normal modes of a three-mass oscillator are shown in Fig. 2.1. The masses are constrained to move in a plane, and so there are six normal modes of vibration, three longitudinal and three transverse. Each longitudinal mode will be higher in frequency than the corresponding transverse mode. If the masses were free to move in three dimensions, there would be 3*3 =9 normal modes, three longitudinal and six transverse.

Increasing the number of masses and springs in our linear array increases the number of normal modes. Each new mass adds one longitudinal mode and (provided the masses move in a plane) one transverse mode. The modes of transverse vibration for mass/spring systems with N=1 to 24 masses are shown in Fig. 2.2; note that as the number of masses increases, the system takes on a wavelike appearance. A similar diagram could be drawn for the longitudinal modes.

Figure 2.4: Normal modes of a three-mass oscillator. Transverse mode (a) has the lowest frequency and longitudinal mode (f) the highest

Source: Jaffe, D and Smith, J (1983), “Extension of the Karplus-Strong

plucked string algorithm,” CMJ 7:2, 43-45

Figure 2.5: Modes of transverse vibration for mass/spring systems with different numbers of masses. A system with N masses has N modes

Source: Beranek L (1954), Acoustics. McGraw-Hill, New York

As the number of masses in our linear system increases, we take less and less notice of the individual elements, and our system begins to resemble a vibrating string with mass distributed uniformly along its length. Presumably, we could describe the vibrations of a vibrating string by writing N equations of motion for N equality spaced masses and letting N go to infinity, but it is much simpler to consider the shape of the string as a whole (Bogoliubov, and Mitropolsky, 1961).

Standing waves

Consider a string of length L fixed at x=0 and x= L. The first condition y (0,t) = 0 requires that A = -C and B = -D in Eq. (2.9), so

Using the sum and difference formulas, sin(xy) = sin x cos y cos x sin y and cos(x

Y = 2A sin kx cos

= 2[A cos

The second condition y (L, t) =0 requires that sin kL =0 or . This restricts to values Thus, the string has normal modes of vibration (O’brien, Cook and Essl, 2001):

These modes are harmonic, because each fn is n times f1= c/2L.

The general solution of a vibrating string with fixed ends can be written as a sum of the normal modes:

and the amplitude of the nth mode is. At any point

Alternatively, the general solution could be written as

Where Cn is the amplitude of the nth mode and is its phase (Keefe and Benade, 1982).

2.5 Energy of a vibrating string

McIntyre et al (1981) has mentioned that when a string vibrates in one of its normal modes, the kinetic and potential energies alternately take on their maximum value, which is equal to the total energy. Thus, the energy of a mode can be calculated by considering either the kinetic or the potential energy. The maximum kinetic energy of a segment vibrating in its nth mode is:

Integrating over the entire length gives

The potential and kinetic energies of each mode have a time average value that is En/2. The total energy of the string can be found by summing up the energy in each normal mode:

Plucked string: time and frequency analyses

According to Laroche and Jot (1992) when a string is excited by bowing, plucking, or striking, the resulting vibration can be considered to be a combination of several modes of vibration. For example, if the string is plucked at its center, the resulting vibration will consist of the fundamental plus the odd-numbered harmonics. Fig. 2.5 illustrates how the modes associated with the odd-numbered harmonics, when each is present in the right proportion; add up at one instant in time to give the initial shape of the center-plucked string. Modes 3,7,11, etc., must be opposite in phase from modes, 1, 5, and 9 in order to give maximum displacement at the center, as shown at the top. Finding the normal mode spectrum of a string given its initial displacement calls for frequency analysis or fourier analysis.

Figure 2.6: Time analysis of the motion of a string plucked at its midpoint through one half cycle. Motion can be thought of as due to two pulses travelling in opposite directions

Source: Gokhshtein, A. Y (1981), ”Role of air¬‚ow modulator in the excitation of sound in wind instruments,” Sov. Phys. Dokl. 25, 954-956

Since all the modes shown in Fig.2.6 have different frequencies of vibration, they quickly get out of phase, and the shape of the string changes rapidly after plucking. The shape of the string at each moment can be obtained by adding the normal modes at that particular time, but it is more difficult to do so because each of the modes will be at a different point in its cycle. The resolution of the string motion into two pulses that propagate in opposite directions on the string, which we might call time analysis, is illustrated in Fig.2.6 if the constituent modes are different, of course. For example, if the string is plucked 1/5 of the distance from one end, the spectrum of mode amplitudes shown in Fig. 2.7 is obtained. Note that the 5th harmonic is missing. Plucking the string ¼ of the distance from the end suppresses the 4th harmonic, etc. (Pavic, 2006).

Roads (1989) have mentioned that a time analysis of the string plucked at 1/5 of its length. A bend racing back and forth within a parallelogram boundary can be viewed as the resultant of two pulses (dashed lines) travelling in opposite directions. Time analysis through one half cycle of the motion of a string plucked one-fifth of the distance from one end. The motion can be thought of as due to two pulses moving in opposite directions (dashed curves). The resultant motion consists of two bends, one moving clockwise and the other counter-clockwise around a parallelogram. The normal force on the end support, as a function of time, is shown at the bottom. Each of these pulses can be described by one term in d’Alembert’s solution [Eq. (2.5)].

Each of the normal modes described in Eq. (2.13) has two coefficients and Bn whose values depend upon the initial excitation of the string. These coefficients can be determined by Fourier analysis. Multiplying each side of Eq. (2.14) and its time derivative by sin mx/L and integrating from 0 to L gives the following formulae for the Fourier coefficients:

By using these formulae, we can calculate the Fourier coefficients for the string of length L is plucked with amplitude h at one fifth of its length as shown in figure.2.8 time analysis above. The initial conditions are:

y (x,0) = 0

y (x,0) = 5h/L .x, 0 x L/5,

= 5h/4 (1-x/L), L/5 x L.

Using the first condition in first equation gives An=0. Using the second condition in second equation gives



The individual Bn’s become: B1 =0.7444h, B2 =0.3011h, B3 =0.1338h, B4 =0.0465h, B5 =0, B6= -0.0207h, etc. Figure 2.7 shows 20 log for n=0 to 15. Note that Bn=0 for n=5, 10, 15, etc., which is the signature of a string plucked at 1/5 of its length (Shabana, 1990).

Bowed string

Woodhouse (1992) has mentioned that the motion of a bowed string has interested physicists for many years, and much has been written on the subject. As the bow is drawn across the string of a violin, the string appears to vibrate back and forth smoothly between two curved boundaries, much like a string vibrating in its fundamental mode. However, this appearance of simplicity is deceiving. Over a hundred years ago, Helmholtz (1877) showed that the string more nearly forms two straight lines with a sharp bend at the point of intersection. This bend races around the curved path that we see, making one round trip each period of the vibration.

According to Chaigne and Doutaut (1997) to observe the string motion, Helmholtz constructed a vibration microscope, consisting of an eyepiece attached to a tuning fork. This was driven in sinusoidal motion parallel to the string, and the eyepiece was focused on a bright-colored spot on the string. When Helmholtz bowed the string, he saw a Lissajous figure. The figure was stationary when the tuning fork frequency was an integral function of the string frequency. Helmholtz noted that the displacement of the string followed a triangular pattern at whatever point he observed it, as shown in Fig.2.7:

Figure 2.7: Displacement and Velocity of a bowed string at three positions along the length: a) at x = L/4; b) at the center, and c) at x = 3L/4

Source: Smith, J (1986), “Efficient Simulation of the Reed-Bore and Bow-String Mechanisms,” Proc. ICMC, 275-280

The velocity waveform at each point alternates between two values. Other early work on the subject was published by Krigar-Menzel and Raps (1891) and by Nobel laureate C. V. Raman (1918). More recent experiments by Schelleng (1973), McIntyre, et al. (1981). Lawergren (1980), Kondo and Kubata (1983), and by others have verified these early findings and have greatly added to our understanding of bowed strings. An excellent discussion of the bowed string is given by Cremer (1981). The motion of a bowed string is shown in Fig.2.8:

Figure 2.8: Motion of a bowed string. A) Time analysis of the motion showing the shape of the string at eight successive times during the cycle. B) Displacement of the bow (dashed line) and the string at the point of contact (solid line) at successive times. The letters correspond to the letters in (A)

Source: McIntyre, M., Woodhouse, J (1979), “On the Fundamentals of Bowed-String Dynamics,” Acustica 43:2, 93-108

Dobashi, Yamamoto and Nishita (2003) have described that a time analysis in the above figure 2.8 (A) shows the Helmholtz-type motion of the string; as the bow moves ahead at a constant speed, the bend races around a curved path. Fig. 2.8 (B) shows the position of the point of contact at successive times; the letters correspond to the frames in Figure 2.8(A). Note that there is a single bend in the bowed string. Whereas in the plucked string (fig. 2.8), we had a double bend. The action of the bow on the string is often described as a stick and slip action. The bow drags the string along until the bend arrives [from (a) in figure 2.8 (A)] and triggers the slipping action of the string until it is picked up by the bow once again [frame (c)]. From (c) to (i), the string moves at the speed of the bow. The velocity of the bend up and down the string is the usual . The envelope around which the bend races [the dashed curve in Figure 2.8 (A)] is composed of two parabolas with maximum amplitude that is proportional, within limits, to the bow velocity. It also increases as the string is bowed nearer to one end.

2.6 Vibration of air columns:

According to Moore and Glasberg (1990) the familiar phenomenon of the sound obtained by blowing across the open and of a key shows that vibrations can be set up in an air column. An air column of definite length has a definite natural period of vibrations. When a vibrating tuning fork is held over a tall glass is pured gradually, so as to vary the length of the air column, a length can be obtained which will resound loudly to the note of the tuning fork. Hence it is the air column is the same as that of the tuning fork.

A vibration has three important characteristics namely




2.6.1 Frequency:-

Frequency is defined as the number of vibration in one second. The unit is Hertz. It is normally denoted as HZ. Thus a sound of 1000 HZ means 1000 vibrations in one second. A frequency of 1000 HZ can also be denoted as 1 KHZ. If the frequency range of audio equipment is mentioned as 50 HZ to 3 HZ it means that audio equipment will function within the frequency range between 50HZ and 3000 HZ.

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2.6.2 Amplitude:-

Amplitude is defined as the maximum displacement experienced by a particle in figure will help to understand amplitude. Let us consider two vibrating bodies having the same frequency but different amplitudes. The body vibrating with more amplitude will be louder than the body vibrating with less amplitude. The following figure represents two vibrating bodies having the same frequency but different amplitudes (Takala and Hahn, 1992).

2.6.3 Phase:-

Phase is defined as the stage to which a particle has reached in its vibration. Initial phase means the initial stage from which the vibration starts. The following will help to understand the concept of phase. From the source travels in the form of waves before reaching the ear sound cannot travel in vacuum. Sound needs medium for its travel. The medium may be a solid or liquid or gas (Brown and Vaughn, 1993).

Support a glass tube open at both ends in a vertical position, with its lower and dipping into water contained in a wider cylinder. Hold over the upper end of the tube a vibrating tuning form. Adjust the reinforcement of the sound is obtained. Adjust the distance of the air column till we get actually the resonance or sympathetic note. Repeat the adjustments and take the average of the results from the observation. It will be found from the repeated experiments, that the longer the air column is produced when the tuning fork becomes identical.

Vibration of air column in a tube open at both ends:-

O’brien, Shen and Gatchalian (2002) have described that if they think of an air column in a tube open both ends, and try to imagine the ways in which it can vibrate; we shall readily appreciate that the ends will always be antinodes, since here the air is free to move. Between the antinodes there must be at least one node, and the ends, the moving air is either moving towards the center from both ends or away from the centre at both ends. Thus the simplest kind of vibration has a node at the centre and antinodes at the two ends. This can be mathematically expressed as follows:

Wave length of the simplest kind of vibration is four times the distance from node to antinode – 2L where L is the length of the pipe.

Vibration of air column in a tube closed at one end:

The distance from node to antinode in this case is L, the whole length of the pipe, the wavelength is therefore = 4L.

2.7 Resonance-sympathetic vibration

Sloan, Kautz and Synder (2002) have described that everybody which is capable of vibration has natural frequency of its own. When a body is made to vibrate at its neutral frequency, it will vibrate with maximum amplitude. Resonance is a phenomenon in which a body at rest is made to vibrate by the vibrations of another body whose frequency is equal to that of the natural frequency of the first. Resonance can also be called sympathetic vibrations. The following experiment will help to understand resonance:

Consider two stretched stings A and B on a sonometer. With the help of a standard tuning form we can adjust their vibrating lengths [length between the bridges] to have the same frequency. Thus we can place a few paper riders on string B and pluck string A to make it vibrate. The string B will start vibrate and paper riders on it will flutter vigorously and sometimes A can be stopped simply by touching it. Still the string B will continue to vibrate. The vibration in the string B is due to resonance and it can be called as sympathetic vibration. If instead of the fundamental frequency one of the harmonics of string B is equal to the vibrating frequency of string A then the string B will start vibrating at that harmonics frequency. But in the case of harmonics the amplitude of vibration will be less. In Tambura when the sarani is sounded the anusarani also, vibrates thus helping to produce a louder volume of sound. The sarani here makes the anusarani to vibrate. In all musical instruments the material, the shape of the body and enclosed volume of air make use of resonance to bring out increased volume and desired upper partials of harmonics.

2.8 Intonations

Spiegel and Watson (1984) have described that during the course of the history of music, several of music intervals were proposed aiming at a high degree of maturing consonance and dissonance played important role in the evolution of musical scales. Just intonation is the result of standardizing perfect intervals. Just Intonation is limited to one single-key and aims at making the intervals as accordant as possible with both one another and with the harmonics of the keynote and with the closely related tones. The frequency ratio of the musical notes in just Intonation is given below.

Indian note Western note Frequency ratio

r C 1

K2 D 9/8

f2 E 5/4

M1 F 4/3

P G 3/2

D2 A 5/3

N2 B 15/8

S C 2

Ward (1970) has mentioned that most of the frequency ratios are expressed is terms of comparatively small numbers. Constant harmonics are present when frequency ratios are expressed in terms of small numbers. The interval in frequency ratio are:

Between Madhya sthyai C[Sa] and Tara sthayi c[sa] is 2 [1*2=2].

Between Madhya sthyai C[Sa] and Madhya sthayi G[pa] is 3/2 [1*3/2=3/2].

Between Madhya sthayi D[Ri] and Madhya sthayi E[Ga] is 10/9 [9/8*10/9=5/4]

Between Madhya sthyai E[Ga] and Madhya sthayi F[Ma] is 16/15-[5/4*16/15=4/3].

Between Madhya sthyai F[Ma] and Madhya sthayi G[Pa] is 9/8-[4/3*9/8=3/2].

Between Madhya sthyai G[Pa] and Madhya sthayi A[Dha] is 10/9[3/2*10/9=5/3].

Between Madhya sthyai A[Dha] and Madhya sthayi B[Ni] is 9/8-[5/3*9/8=5/8].

Between Madhya sthyai Sa[C] and Ri2[D] there is a svarasthanam [CH]. Hence the interval between Sa[C] and Ri2[D] and Ga2[E] is known as a tone. But there is no svarasthanam [semitone] between Ga2[E] and Ma1[F]. Hence the interval between Ga[E] and Ma1[F] is known as a semitone. Between Pa[G] and Dha[A] we have a tone. Between mathya styayi Ni2[B] and Tara sthyai C[Sa] we have a semitone.

In just Intonation we find that tones are not all equal. But the semitones are equal. In just Intonation the modulation of key of musical notes will be difficult for example, if the keynote is changed from Sa[C] to Pa[G] then the frequency of etatusruthi Dhairatam [A] will change from 1.687, time the frequency of Sa[c]. A musical instrument tuned in just intonation to play sankarabaranam ragam cannot be used to play kalyani ragam. Hence the modulation of key of musical notes will be difficult in just Intonation (Doutaut , Matignon, and Chaigne, 1998).

Equal temperature

Lehr (1997) has described that the above mentioned problem in just Intonation can be solved in the Equal Temperament scale. In Equal temperament all the 12 music intervals in a sthayi [octave] are equal. The frequency ratios of semitones in Equal temperament scale was first calculated by the French Mathematician Mersenne and was published in ‘Harmonic Universelle’ in the year 1636. But it was not put into use till the latter half of seventeenth century. All keyboard instruments are tuned of Equal Temperature scale. Abraham pandithar strongly advocated Equal Temperament scale and in his famous music treatise ‘karunamitha sagaram’ he tried to prove that the Equal Temperament scale was in practice in ancient Tamil music.

A simple mathematical exercise will help to under the basis of Equal Temperament scale.

Equal Temperament

Madhya sthayi Sa[c] frequency ratio=1=2 ÌŠ.

Tara sthayi Sa[i] frequency ratio = 2=212/12=2.

Frequency ratios of 12 svarasthanams are given below.

S R1 R2 G1 G2 M1 M2 P D1 D2 N1 N2

↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓

20 21/12 22/12 23/12 24/12 25/12 26/12 27/12 28/12 29/12 210/12 2n/12




All semitones are equal is Equal Temperament scale. Each represents the same frequency ratio 1.05877. The great advantage in Equal Temperament scale is that music can be played equal well in all keys. This means that any of the 12 semitones can be used as ‘Sa’ in a music instrument tuned to Equal Temperament scale. There is no need to change tuning every time the Raga is changed. Since keyboard instruments are pre-tuned instruments they follow Equal Temperament.

2.9 Production and transmission of sound:-

According to Boulanger (2000) the term sound is related to quite definite and specific sensation caused by the stimulation of the mechanism of the ear. The external cause of the sensation is also related to sound. Anybody in vibration is an external cause of the sensation. A veena [after plucking] or violin [after blowing] in a state of vibration is an external cause of the sensation. A body in a state of vibration becomes a source of sound. A vibration is a periodic to and fro motion about a fixed point

Iwamiya and Fujiwara (1985) have mentioned that the pitch of a musical sound produced on a wind instrument depends on the rate or frequency of the vibrations which cause the sound. In obedience to Nature’s law, the column of air in a tube can be made to vibrate only at certain rates, therefore, a tube of any particular length can be made to produce only certain sounds and no others as long as the length of the tube is un-altered. Whatever the length of the tube, these various sounds always bear the same relationship one to the other, but the actual pitch of die series will depend on the length of the tube. The player on a wind instrument, by varying the intensity of the air-stream which he injects into the mouthpiece, can produce at will all or some of the various sounds which that particular length of tube is capable of sounding; thus, by compressing the air-stream with his lips he increases the rate of vibration and produces higher sounds, and by decompressing or slackening the intensity of the air-stream he lowers the rate of vibration and produces lower pitched sounds. In this way the fundamental, or lowest note which a tube is capable of sounding, can be raised becoming higher and higher by intervals which become smaller and smaller as they ascend. These sounds are usually called harmonics or upper partials, and it is convenient to refer to them by number, counting the fundamental as No. t, the octave harmonic as No. 2, and so on. The series of sounds available on a tube approximately 8 feet in length is as follows:

Tsingos et al (2001) has mentioned that a longer tube would produce a corresponding series of sounds proportionately lower in pitch according to its length, and on a shorter tube the same series would be proportionately higher. The entire series available on any tube is an octave lower than that of a tube half its length, or an octave higher than that of a tube double its length ; thus, the approximate lengths of tube required to sound the various notes C are as follows :

Fundamental Length of tube

C, 16 feet

C 8 ,.

c 4,,

c’ 2,,

c” I foot

c”’ 1/2,,

Shonle and Horen (1980) has mentioned that the addition of about 6 inches to a 4-foot tube, of a foot to an 8-foot tube, or of 2 feet to a i6-foot tube, will give the series a tone lower (in B flat), and a proportionate shortening of the C tubes will raise the series a tone (D) ; on the same basis, tubes which give any F as the fundamental of a series must be about midway in length between those which give the C above and the C below as fundamental. Examples:

Trumpet (modern) in C-length about 4 feet

,, in F ,, ,, 6 ,,

,, (old) in C ,, ,, 8 ,,

Horn in F ,, ,, 12 ,,

,, ,, C ,, ,, 16 ,,

It will be noticed that the two lower octaves of the harmonic series are ve


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