Forced Vibrations Of Simple Systems English Language Essay

Forced Vib symmetryns Of Simple Systems English Language EssayMechanical, acoustic, or electrical quiverings argon the sources of sound in tuneful instruments. Some familiar examples are the shakinesss of sucks violin, guitar, piano, etc, bars or rods xyloph hotshot, glockenspiel, chimes, and clari angiotensin-converting enzymet reed, membranes (drums, banjo), plates or shells (cymbal, gong, bell), air in a tobacco pipe (organ pipe, validation and woodwind instruments, marimba resonator), and air in an enclosed container (drum, violin, or guitar embody). In most instruments, sound production numbers upon the collective behavior of several vibrators, which may be weakly or potently coupled together. This coupling, along with non elongate feedback, may ca intention the instrument as a whole to be make water as a multifactorial vibrating agreement, even though the individual elements are relatively simple vibrators (Hake and Rodwan, 1966).In the first seven chapters, we pr ovide 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 recognize with electrical oscillators except as they serve well us, by analogy, to understand mechanical and acoustical oscillators.According to Iwamiya, Kosygi and Kitamura (1983) m all objects are loose of vibrating or oscillating. Mechanical vibrations ask that the object possess two basic properties a stiffness or wince like quality to provide a restoring force when displaced and inertia, which drivings the resulting exploit to blast the equilibrium position. From an skill standpoint, oscillators urgency a content for storing potential goose egg (spring), a means for storing kinetic energy ( people), and a means by which energy is gradually lost (damper). vibratory motion involves the alternating transfer of energy amongst its kinetic and p otential forms. The inertial softwood may be every concentrated in one location or distributed throughout the vibrating object. If it is distributed, it is usually the mass per building block length, area, or volume that is important. Vibrations in distributed mass agreements may be viewed as standing waves. The restoring forces depend upon the elasticity or the compressibility of some material. Most vibrating bodies obey Hookes law that is, the restoring force is proportional to the displacement from equilibrium, at least(prenominal) for small displacement.Simple harmonic motion in one dimensionMoore (1989) has mentioned that the simplest kind of detailic motion is that experienced by a point mass moving along a straight line with an quickening directed toward a fixed point and proportional to the distance from that point. This is called simple harmonic motion, and it female genital organ be described by a sinusoidal function of time, where the bountifulness A describes th e maximum extent of the motion, and the oftenness f tells us how often it repeats.The period of the motion is crackn byThat is, each T guerillas the motion repeats itself.Sundberg (1978) has mentioned that a simple example of a constitution 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 Hookes law, F = -Kx, with Newtons second law, F = ma =. Thus,andWhere=The never-ending K is called the spring constant or stiffness of the spring (expressed in Newtons per meter). We define a constant so that the equation of motion becomesThis well-known equation has these solutions)Figure 2.1 Simple mass-spring vibrating systemSource Cremer, L., Heckl, M., Ungar, E (1988), Structure-Borne Sound, 2nd edi tion, Springer VerlagFigure 2.2 Relative phase of displacement x, velocity v, and acceleration a of a simple vibratorSource Campbell, D. M., and Greated, C (1987), The Musicians Guide to Acoustics, Dent, LondonorFrom which we recognize as the pictorial angulate absolute frequence 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),And.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- gear up differential equations concord two discretional 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 tick offting t = 0 in Eq. (1.5) gives From these we throne obtain expressions for A and in terms of xo and vo,andAlternatively, 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 amplitudesAccording to Cremer, Heckl and Ungar (1990) another approach to solving elongate 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 need 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) to oshie be compose 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 writtenThe complex velocity and acceleration becomeDesmet (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 velocityFigure 2.3 form representation of the complex displacement, velocity, and acceleration of a linear oscillatorSource Bangtsson E, Noreland D and Berggren M (2003), create optimization of an acoustic horn, Computer Methods in Applied Mechanics and Engineering, 1921533-15712.4 Continuous systems in one dimensionStrings and barsThis 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 accordingly allow the number of elements to grow larger, eventually leading to a continuum (Karjalainen and Valamaki, 1993).Linear array of oscillatorsAccording to Mickens (1998) the oscillating system with two masses in Fig. 1.20 was shown to have two crosswise vibrational modes and two longitudinal modes. In two 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 turnaround 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. severally 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 last(a) frequency and longitudinal mode (f) the highestSource Jaffe, D and Smith, J (1983), Extension of the Karplus-Strong pull draw off algorithm, CMJ 72, 43-45Figure 2.5 Modes of transverse vibration for mass/spring systems with different numbers of masses. A system with N masses has N modesSource Beranek L (1954), Acoustics. McGraw-Hill, New YorkAs t he 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 pass 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 fancy the wreak of the string as a whole (Bogoliubov, and Mitropolsky, 1961).Standing wavesConsider 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), soUsing the sum and difference formulas, sin(xy) = sin x romaine lettuce y cos x sin y and cos(xY = 2A sin kx cos= 2A cosThe second condition y (L, t) =0 requires that sin kL =0 or . This restricts to values Thus, the string has normal modes of vibration (Obrien, Cook and Essl, 2001)These modes are harmonic, because each fn is n times f1= c/2L.The habitual solution of a vi brating string with fixed ends can be written as a sum of the normal modesand the amplitude of the nth mode is. At any pointAlternatively, the general solution could be written asWhere Cn is the amplitude of the nth mode and is its phase (Keefe and Benade, 1982).2.5 Energy of a vibrating stringMcIntyre 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 compound over the entire length givesThe potential and kinetic energies of each mode have a time honest value that is En/2. The total energy of the string can be found by summing up the energy in each normal modePlucked string time and frequency analysesAccording to Laroche and Jot (1992) when a string is excited by radic aling, 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 fashion of the center-plucked string. Modes 3,7,11, etc., moldiness 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. doubt can be thought of as due to two pulses travelling in opposite directionsSource Gokhshtein, A. Y (1981), Role of airow modulator in the pro vocation of sound in wind instruments, Sov. Phys. Dokl. 25, 954-956Since 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 crabby time, but it is more catchy 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 of the distance from the end suppresses the 4th harmonic, etc. (Pavic, 2006).Roads (1989) have mentioned that a time analysis of the string pluc ked at 1/5 of its length. A bend racing back and forth within a parallelogram boundary can be viewed as the serial 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 dAlemberts 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 coefficientsBy 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 preceding(prenominal). The initial conditions arey (x,0) = 0y (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 Bns 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 stringWoodhouse (1992) has mentioned that the motion of a arced 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 betwixt two cut boundar ies, 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 almost 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 instigate 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 lift. 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 ascertained it, as shown in Fig.2.7Figure 2.7 Displace ment 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/4Source Smith, J (1986), Efficient Simulation of the Reed-Bore and Bow-String Mechanisms, Proc. ICMC, 275-280The velocity waveform at each point alternates amidst 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.2.8Figure 2.8 Motion of a bowed string. A) Time analysis of the motion showing the shape of the string at eight straight 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 earn correspond to the letters in (A)Source McIntyre, M., Woodhouse, J (1979), On the Fundamentals of Bowed-String Dynamics, Acustica 432, 93-108Dobashi, 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 triplex 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 be nd 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 columnsAccording to Moore and Glasberg (1990) the familiar phenomenon of the sound obtained by blowing across the heart-to-heart 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 namelyFrequencyAmplitudePhase2.6.1 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 frequency 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.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 unde rstand the concept of phase. From the source travels in the form of waves before reaching the ear sound cannot travel in vacuum. Sound needs specialty for its travel. The medium may be a solid or liquid or gas (Brown and Vaughn, 1993).Support a glass tubing open at some(prenominal) ends in a vertical position, with its begin and dipping into water contained in a wider cylinder. Hold over the upper end of the thermionic valve-shaped structure a vibrating tuning form. array 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-Obrien, Shen and Gatchalian (2002) have described that if they think of an air column in a tube open both ends, and try to hypothecate the ways in which it can vibrate we shall readily appreciate that the ends will unendingly 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 rove 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 endThe distance from node to antinode in this consequence is L, the whole length of the pipe, the wavelength is therefore = 4L.2.7 Resonance-sympathetic vibrationSloan, 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 fre quency, 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 resonanceConsider 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 fewer 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 o f 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 chantingsSpiegel 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 ratior C 1K2 D 9/8f2 E 5/4M1 F 4/3P G 3/2D2 A 5/3N2 B 15/8S C 2Ward (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 areBetween Madhya sthyai CSa and Tara sthayi csa is 2 1*2=2.Between Madhya sthyai CSa and Madhya sthayi Gpa is 3/2 1*3/2=3/2.Between Madhya sthayi DRi and Madhya sthayi EGa is 10/9 9/8*10/9=5/4Between Madhya sthyai EGa and Madhya sthayi FMa is 16/15-5/4*16/15=4/3.Between Madhya sthyai FMa and Madhya sthayi GPa is 9/8-4/3*9/8=3/2.Between Madhya sthyai GPa and Madhya sthayi ADha is 10/93/2*10/9=5/3.Between Madhya sthyai ADha and Madhya sthayi BNi is 9/8-5/3*9/8=5/8.Between Madhya sthyai SaC and Ri2D there is a svarasthanam CH. Hence the interval betwee n SaC and Ri2D and Ga2E is known as a tone. But there is no svarasthanam semitone between Ga2E and Ma1F. Hence the interval between GaE and Ma1F is known as a semitone. Between PaG and DhaA we have a tone. Between mathya styayi Ni2B and Tara sthyai CSa 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 SaC to PaG then the frequency of etatusruthi Dhairatam A will change from 1.687, time the frequency of Sac. 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 temperatureLehr (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 mus ic 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 TemperamentMadhya sthayi Sac frequency ratio=1=2 .Tara sthayi Sai 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/12S212/12All semitones are equal is Equal Temperament scale. Each represents the sa me 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 kind of 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 pointIwamiya 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 Natures 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 warmth 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 lift 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 followsTsingos 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 tubeC, 16 feetC 8 ,.c 4,,c 2,,c I footc 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. ExamplesTrumpet (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