Forced Vibrations Of Simple Systems English Language Essay

Forced Vibrations Of Simple Systems English Language Essay 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). 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 mater ial. Most vibrating bodies obey Hookes 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 Hookes law, F = -Kx, with Newtons second law, F = ma =. Thus, and Where = The constant 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 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 Musicians Guide to Acoustics, Dent, London or 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): , 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-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: , and 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 (Obrien, 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 ¬Ã¢â‚¬Å¡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 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 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 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 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 amplitu de 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 Frequency Amplitude Phase 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. 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:- Obrien, 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 instrum ents 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 ÃÅ'Ã…  . 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 à ¢Ã¢â‚¬  Ã¢â‚¬Å" à ¢Ã¢â‚¬  Ã¢â‚¬Å" à ¢Ã¢â‚¬  Ã¢â‚¬Å" à ¢Ã¢â‚¬  Ã¢â‚¬Å" à ¢Ã¢â‚¬  Ã¢â‚¬Å" à ¢Ã¢â‚¬  Ã¢â‚¬Å" à ¢Ã¢â‚¬  Ã¢â‚¬Å" à ¢Ã¢â‚¬  Ã¢â‚¬Å" à ¢Ã¢â‚¬  Ã¢â‚¬Å" à ¢Ã¢â‚¬  Ã¢â‚¬Å" à ¢Ã¢â‚¬  Ã¢â‚¬Å" à ¢Ã¢â‚¬  Ã¢â‚¬Å" 20 21/12 22/12 23/12 24/12 25/12 26/12 27/12 28/12 29/12 210/12 2n/12 S à ¢Ã¢â‚¬  Ã¢â‚¬Å" 212/12 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 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 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 inte nsity 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

How to Survive a Zombie Apocalypse

How to survive a zombie apocalypse Zombies, do they exist? Can there be actual zombies roaming around on the streets, ready to make anyone and everyone who comes in sight a zombie? Well, Hollywood thinks so. The reason I'm basing this theory of ‘can zombies be real' or the fact that zombies can one day roam amongst us (eventually destroying mankind), is because they are the biggest influence in our lives. Don't believe me? Then I guess the long list of zombie movies might help you change your perspective.Their ideology of assuring us into believing that the concept of ‘zombies' or ‘zombie apocalypse is real' is quite believable. And movies like Resident Evil, Dawn of the Dead, and 28 Days Later are perfect examples of such an event. Each of these movies has groups of people who are trying to get to a safe place, away from all these zombies. But there isn't any guide or rules that they can follow while they're on this road to freedom. It's just one step at a time (an d some of them get eaten in the process as well). So, there should be a guide which tells them how to survive if such an apocalypse ever occurred.In order to get infected by a zombie, you need to be bitten, or bite one of them, although I have no idea why you would ever want to. The saliva and blood of a zombie have the potential to infect a person and eventually turn them, so keep your mouth shut if you're the stupid one using a chainsaw. To be honest I prefer a good old fashioned flamethrower to burn those zombies where they stand. How about an idea; perhaps someone can read this article (a movie director) and magically keeps a few copies of it in the movie for the actors to find. Alright, alright, that was not a good joke.So let's move past that and get a bit serious. But before we do get to the specific rules to keep yourself safe from a zombie apocalypse, I'd like to divert your attention to yet another movie, Zombieland (by the way, I think it's one of the best zombie movies o f all time), It's funny, intelligently made, and the cast is small, but perfect. If you have seen the movie, then you'll find some of those rules implemented here as well. And if you didn't get the opportunity to watch this classic movie, then may I suggest, please do so. And with that small note, we will proceed towards our golden rules to survive a zombie apocalypse.What is a zombie? There are multiple definitions of the word zombie. The dictionary defines zombie as â€Å"an animated corpse that feeds on living flesh. † Other sources define zombie as â€Å"One who moves or acts as if in a daze† or â€Å"a member of Congress† or â€Å"a Voodoo snake god. † While snake gods are interesting, this how-to article will deal with only the first and second definitions of zombie. While Chavs were once thought to be zombies it has been recently proven that it is not the case. It would be an insult to zombies if they were even contemplated to be in the same catego ry as Chavs.Such contemplators were subsequently eaten. Types of zombies There are twelve types of zombies. They will be listed with a little fact about them: 1. Horde zombies are dangerous to combat effectively. They require much more potent weaponry to approach with confidence, but are easier to flee and avoid. 2. Grue Zombies, grues who have (somehow) died and became zombies. There is really no way to kill grue zombies unless you have Chuck Norris or can successfully pull off the shoop da whoop. 3. Camper zombies, zombies who hide and wait for prey, are the other common type.Campers hide in darkened corridors, side rooms, and even in the drywall. They prefer to wait for their prey to come near and then burst out and seize the victim. 4. Fast Zombies are some of your most dreaded enemies when facing legions of the undead. Fast Zombies can vary in many ways, but the main item is that their entire physique (i. e. rotting of fat, skin, veins, and organs. ) is morphed to allow them to strike fast. The majority of fast zombies have been found to have been overzealous gym instructors and annoying co-workers who won't let go of the â€Å"Can do attitude† even post-mortem. 5.Elvis Impersonator Zombies These are usually the rarest, if not the most entertaining variety of undead you may encounter. They can be easily identified by their signature hairdo, glittery suits, and trademark lurching walk which they plagiarized from Elvis. This isn’t really even a walk, since they don’t actually get anywhere. 6. When you know a Smart Zombie is around, the best thing you can do is avoid him at all costs. Smart Zombies have the ability to set up and carry out plans. 7. Sewer Zombies like to spend all their time in sewers and in drain areas, they are afraid of light so taking a flashlight is a good idea.A better idea is to just avoid sewers. 8. Lady Zombies are exactly what they sound like. Every necrophilia’s nightmare, they are girl zombies. But surp risingly, they're not that different from the live ones, they still moan and groan about the men never taking them anywhere. 9. The Zombie†¦.. Chuck Norris!!! , this is the most dangerous zombie ever. Some of the zombie abilities he is theorized to possess include infecting people by burping on them; complete cellular regeneration (means he can't be killed. Ever. ) 10. Animal Zombies Depend on your luck. A zombie snail is most common but they're totally harmless.Other kind of animal zombies do exist. Some of these are â€Å"dog† zombies. If a zombie gets to be this big, then it's lethal. However, since much of its muscle has degraded by then, it won't be able to catch you. (THANK GOD!!! ) If a CHEETAH zombie attacks you, even without its degraded muscle, then you're screwed. 11. Peanut zombies' A. K. A. The Terrors of the junkyard, these vermicious brutes wander the junkyards, in poorly built helies, or controlling guns. Avoid these brutes; they have a leader, Project #2 95. He has two dual glocks, and is not afraid to use it. beware the ultimate power. 12.Ninja/Samurai/Genghis Khan Zombies Once feared and respected warriors of Japan; the zombie infection has made them some of the most terrifying zombies to invade your country. All have high level martial arts skills and requires extreme weapon tactics to wipe out; if using hand-to-hand combat, the chances are you will not survive. Certain firearms to have Everybody needs a weapon so here are the top ten: 1. Chainsaw- Most of all Resident Evil or horror films and countless Hollywood scenes show the awesome power of the chainsaw. As cool as it is though, it ranks very low on the practical zombie-killing weapon list. . Lawn Mower- All the same problems as Chainsaw, with even greater problems. You know how hard one of those things are to lift, let alone to a position that the blades can even do their work. Also, it is not made to be lifted up into the air, so even if you can lift the mower, it will not stay up there for long. It might of looked cool in Brain Dead, it will never work. 3. Slingshot- It wouldn't be the wisest of choices. Using it against a zombie will only alert it to your presence, stupid. Unless you throw a bomb with it. 4. Edged Weapons- Imagine using a sword to kill a crowd of zombies nd you stab some zombie through the skull in the brain. Now imagine you turn to the zombie behind you to do the same, only to realize that it's stuck on the first zombie that you stabbed and you can't seem to get it back†¦Ã¢â‚¬ ¦. not a pretty picture, is it? So use one without a serrated edge. Unless you are a skilled fencer, this is an inadvisable choice. 5. Rubber Ducky- †¦. not sure if that is the best idea, but just remember it's your funeral JOINING THE LEGIONS OF THE UNDEAD. So if you REALLY want to use it and see what happens, by all means go ahead (dumbass).If you are dumb smart enough to do this, make sure you have a friend tape it and stick it up on YouTube, yo u will get more hits than Achmed the Dead Terrorist, easy. 6. Mop If there are no other blunt weapons available, you can always run to that janitorial closet(provided you didn't stuff zombies in there earlier) and take out a Long, Wooden, MOP! Only to be used against 5-6 zombies, otherwise the long and heavy mop may sort of, backfire on you. 7. Weed Whacker- Insert into the mouth and let the fun begin (not usable for groups, will become the zombie's â€Å"fun†)! 8.Large Minigun: will kill a lot of zombies, and is fun! Though whoever is using it will die just as his buddies got to relative safety. 9. Shotgun: The second best weapon for zombies, there buck shots will blow off the heads of several zombies, the person who is carrying a shotgun will survive most zombie outbreaks. 10. Pistol: short range, small clip, small bullet, only use on small groups, otherwise you are dead (or undead). Guide on Surviving a Zombie Apocalypse If we happen to be caught up in the world-ending zom bie plague, then it wouldn't hurt for us all to have a certain ‘how-to' list with us, right?There can be tons of rules to survive a zombie apocalypse as every one of us has a different way to tackling situations. And since I've never been in this type of position before (thank God), I will be listing rules that I feel are ideal. So here are my top rules on how to survive a zombie apocalypse. 1. Run 2. Run even faster 3. Run faster than that guy next to you 4. Trip the guy next to you 5. Don't let the spazzy woman try and get her dog back 6. Shoot people randomly 7. Drive to Alaska (Zombies will freeze into corpsicles, delicious AND nutritious! , but get there fast, the roads will be too congested and if you're too slow†¦ It'll only be a tasty flesh bottleneck 8. Go out to sea (zombies can't swim, but it’s fun to see them try. Warning:Peanut Zombies know how to swim, so run) 9. Sacrifice Ms. Barbra, the old lady across the street (no one liked her anyways. ) 10. Don 't fall asleep in the open 11. If surrounded, just distract them with a classy dance (Warning: May cause: Zombification, Death, Death, and more Death. ) 12. Always find the nearest gun and ammo Shop, and always trade at least a 10,12, or 20 gauge shotguns, one hit kills 13.Notice that we said trade. Not even think about break into the shop, the shop keeper is always good with gun and you won't stand a chance. Hell, he's most likely the boss in that area with several goons, each carries gun bigger than yours. 14. Don't, for even one second think that you are safe. 15. You need 1 shot in their heads to kill them. So keep a few shotguns with you. 16. If you're in the house, board up the windows and doors properly. 17. Fill as many containers as you possibly can with water. 18. Keep food supplies and other essentials. 19.Along with guns, gather anything that can smash skulls; keep them handy. 20. Find a safe place for you, your friends, and family members. 21. Before sunset, find a secu red location to sleep. 22. Always keep your travel bags light; less weight to carry means more easily to flee. 23. Do not harbor people who have been attacked or bitten by zombies. 24. Always wear comfortable clothing. Avoid movie-like wardrobe. 25. Wear as many layers as you can reasonably get away with. Have a trusted friend try to bite or claw their way through your wardrobe to test fabric strengths ahead of time. 6. We do want to survive so ladies no high heels, and fellas no saggin pants. 27. Have a first aid kit with you in the house and in your getaway car. 28. While traveling, DO NOT go in the woods or lonely areas. 29. If you're trying to stop the zombies from entering in the building, please don't try to hold the door. 30. Don't go anywhere alone. Follow the buddy system. But if you're the only human left, then I guess you're on your own. 31. Learn the zombie dance from Thriller. Because what if the zombies dance? And what is the difference between surviving and not dancin g with them? 2. Guys we know you want to impress your girl, but please follow rule 33. If you want to impress her just keep her safe and stay alive yourself. 33. And our final golden rule: DON'T BE A HERO AND GO OUT IN SEARCH OF ZOMBIES TO KILL. Zombies are slow who drag their feet as they walk. As they've lost control over their brain or the brain doesn't function properly, they are not very intelligent. But you are. So take necessary precautions before you go out there and try really hard to make it. Living in an undead world can be tough. So perhaps these rules can make existing less scary.

Sample Topics for Comparison Contrast Essays

In high school and college literature classes, one common type of writing assignment is the comparison and contrast essay. Identifying points of similarity and difference in two or more literary works encourages close reading and stimulates careful thought. To be effective, a comparison-contrast essay needs to be focused on particular methods, characters, and themes. These ten sample topics demonstrate different ways of achieving that focus in a critical essay. Short Fiction: The Cask of Amontillado and The Fall of the House of UsherAlthough The Cask of Amontillado and The Fall of the House of Usher rely on two notably different types of narrator (the first a mad murderer with a long memory, the second an outside observer who serves as the readers surrogate), both of these stories by Edgar Allan Poe rely on similar devices to create their effects of suspense and horror. Compare and contrast the story-telling methods employed in the two tales, with particular attention to point of view, setting, and diction.Short Fiction: Everyday Use and A Worn PathDiscuss how details of character, language, setting, and symbolism in the stories Everyday Use by Alice Walker and A Worn Path by Eudora Welty serve to characterize the mother (Mrs. Johnson) and the grandmother (Phoenix Jackson), noting points of similarity and difference between the two women.Short Fiction: The Lottery and The Summer PeopleAlthough the same fundamental conflict of tradition vers us change underlies both The Lottery and The Summer People, these two stories by Shirley Jackson offer some notably different observations about human weaknesses and fears. Compare and contrast the two stories, with particular attention to the ways Jackson dramatizes different themes in each. Be sure to include some discussion of the importance of setting, point of view, and character in each story.Poetry: To the Virgins and To His Coy MistressThe Latin phrase carpe diem is popularly translated as seize the day. Compare and contrast these two well-known poems written in the carpe diem tradition: Robert Herricks To the Virgins and Andrew Marvells To His Coy Mistress. Focus on the argumentative strategies and specific figurative devices (for example, simile, metaphor, hyperbole, and personification) employed by each speaker.Poetry: Poem for My Fathers Ghost, Steady as Any Ship My Father, and Nikki RosaA daughter investigates her feelings for her father (and, in the process, reveals so mething about herself) in each of these poems: Mary Olivers Poem for My Fathers Ghost, Doretta Cornells Steady as Any Ship My Father, and Nikki Giovannis Nikki Rosa. Analyze, compare, and contrast these three poems, noting how certain poetic devices (such as diction, repetition, metaphor, and simile) serve in each case to characterize the relationship (however ambivalent) between a daughter and her father.Drama: King Oedipus and Willy LomanDifferent as the two plays are, both Oedipus Rex by Sophocles and Death of a Salesman by Arthur Miller concern a characters efforts to discover some kind of truth about himself by examining events from the past. Analyze, compare, and contrast the difficult investigative and psychological journeys taken by King Oedipus and Willy Loman. Consider the extent to which each character accepts difficult truths--and also resists accepting them. Which character, do you think, is ultimately more successful in his journey of discovery--and why?Drama: Queen Jo casta, Linda Loman, and Amanda WingfieldCarefully examine, compare, and contrast the characterizations of any two of the following women: Jocasta in Oedipus Rex, Linda Loman in Death of a Salesman, and Amanda Wingfield in The Glass Menagerie by Tennessee Williams. Consider each womans relationship with the leading male character(s), and explain why you think each character is primarily active or passive (or both), supportive or destructive (or both), perceptive or self-deceived (or both). Such qualities are not mutually exclusive, of course, and may overlap. Be careful not to reduce these characters to simple-minded stereotypes; explore their complex natures.Drama: Foils in Oedipus Rex, Death of a Salesman, and The Glass MenagerieA foil is a character whose main function is to illuminate the qualities of another character (often the protagonist) through comparison and contrast. First, identify at least one foil character in each of the following works: Oedipus Rex, Death of a Salesm an, and The Glass Menagerie. Next, explain why and how each of these characters may be viewed as a foil, and (most importantly) discuss how the foil character serves to illuminate certain qualities of another character.Drama: Conflicting Responsibilities in Oedipus Rex, Death of a Salesman, and The Glass MenagerieThe three plays Oedipus Rex, Death of a Salesman, and The Glass Menagerie all deal with the theme of conflicting responsibilities--toward self, family, society, and the gods. Like most of us, King Oedipus, Willy Loman, and Tom Wingfield at times try to avoid fulfilling certain responsibilities; at other times, they may appear confused as to what their most important responsibilities should be. By the end of each play, this confusion may or may not be resolved. Discuss how the theme of conflicting responsibilities is dramatized and resolved (if it is resolved) in any two of the three plays, pointing out similarities and differences along the way.Drama and Short Fiction: Trif les and The ChrysanthemumsIn Susan Glaspells play Trifles and John Steinbecks short story The Chrysanthemums, discuss how setting (i.e., the stage set of the play, the fictional setting of the story) and symbolism contribute to our understanding of the conflicts experienced by the character of the wife in each work (Minnie and Elisa, respectively). Unify your essay by identifying points of similarity and difference in these two characters.

Essay about Politicians Of The Gilded Age - 870 Words

Politicians during this time period worried more about ensuring their own financial success, securing votes by any means, granting jobs or favors in return for votes, and remaining popular. They were not concerned with social issues, but supported or crushed these issues in accordance with the decision that would benefit them personally. If politicians were judged to be good personally, they were automatically viewed as good politically. Changes were made for personal benefit, not the good of the community. Read political ideologies were not central to this time period. Use specific people mentioned in Chapter 19 to validate or invalidate this statement. *** As stated by Henry Adams, the Gilded Age which occurred through†¦show more content†¦Conkling followed nothing of this sort. In fact, in more than two decades in Congress, he never drafted one bill. Instead, Conkling distributed very profitable jobs at the New York customhouse and spent most of his career as senator by rewarding his party who stayed faithful with government jobs. Conklings actions show the changes made only for personal benefit of this time period because instead of during what was needed for the nation, he responded with actions that would help out himself with more votes and by helping out only his friends. The second of the two senators was a senator named James G. Blaine. Blaine was a Republican senator from Maine during this same time period. Despite his corrupt actions, Blaine was probably the most popular Republican politician of the era. Charming, intelligent, witty, and able, he served twice as secretary of state and was a serious contender for the presidency in every election from 1876 to 1892. Blaine was a corrupt politician though. After being paid off by favors to railroads in return for money, Blaine lied to the public denying that it ever happened. 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