INTRODUCTION
For any given string a wave on that string has its freqency determined by the boundaries of the string. The power within this topic is this: The boundaries of a physical object can determine the reaction that the object produces to an action, or stimulus, put upon it. Just as the amount of money a person has determines the boundaries of that person's business power (i.e., whether that person can start a huge international telecommunications company or whether that person will run a small convenience store), the boundaries within physical problems determine the range of reaction that a physical system can have. The study of methods of solving the effects that the boundaries within a physical system have on the determination of the output of that the physical system to an input is called the study of boundary value problems.
One of the easiest physical systems from which one can gain a quick idea of the concept of the boundary value problem is that of the vibrating string, like a violin, guitar or piano string.
How Does a Stretched String Produce a Musical Tone?
A guitar, violin or piano can produce a sound when plucked or struck because their strings are elastic and rigidly fastened (bounded) at both ends. The elasticity of a string on these musical instruments can be demonstrated by pressing down on any of the strings with your index finger and noticing that the string bends under the force. The restoring force from the elasticity of the string can be observed by noticing that the string returns to its equilibrium position when the force applied to the string is released. The elasticity of the strings in any of the mentioned musical instruments is assisted by the string supports (the nut and bridge) at the ends of the strings. These can be seen by looking at the string supports on a musical instrument.
Since a string is elastic, when it is plucked or hammered (i.e., forced beyond its equilibrium, or resting, position), the tension in the string will supply restoring forces which supplies the energy for it to vibrate at a particular frequency or tone. The frequency at which a string will vibrate depends on the mass of the string (i.e., the density of the material it is made from and its thickness) and the distance that the string spans between it's end fasteners (the bridge and a fret or the nut of a guitar or violin).
The string supports, or fasteners are particularly interesting
because they tie the strings down at their ends and prevent any movement of the strings at their boundaries. Thus, the strings can only vibrate within the space between its end fasteners . At the end fasteners the string vibration's amplitudes, or displacements, are pulled to zero.
The string end fasteners form boundaries which determine the limits through which the strings can vibrate. In other words, the string end fasteners form the boundaries which actually determine the manner in which the string can vibrate. Understanding how the boundaries of a guitar or violin string determine the frequency at which a string can vibrate opens a student's mind to understanding many other natural occurrences in which boundaries, not just on a string, but, in other circumstances, determine the natural frequencies of vibrations that can occur in nature. There are many other instances of physical phenomena in which the boundaries determine the natural frequencies that are able to exist between the boundaries of a physical system. For example, it known that the boundaries within atoms critically determine particular energy levels that can exist within atoms, and correspondingly, the spectral lines that atoms can liberate when the particular atoms are excited. The mathematical solution of problems of this type involve the same kinds of methods that are used in solving the basic vibrations of a string. This is why a study of the vibrations on a string is important.
We see, then, that the knowledge gained from studying the movement of a vibrating string is directly applicable to a great variety of other problems involving vibrations and waves. So, an important reason for learning the theory of waves is that if you don't know about waves, the advanced work you study will largely be a mystery.
The mathematical solution of problems of this type involves finding a mathematical representation of the vibrations that occur between boundaries. This type of problem is called a boundary value problem. The natural frequencies, or modes (i.e., manners of vibration) at which a stretched string can vibrate are those frequencies that have natural waveforms whose ends have zero value at the boundaries, or end fasteners. Surprisingly, more than one waveform, or frequency, can exist that has zero values at its ends on a bounded string. Each frequency developed on a plucked or hammered string has a corresponding wavelength. These wavelengths correspond to a half-wave (a half of a sine-wave has a value of zero when it begins and when it ends), a full wave, a wave and a half, etc., So, the natural frequencies that at which a stretched string can vibrate are those frequencies that have a wavelength that is equal to one-half the length, L, of the string (L/2), a wavelength equal to the length of the string (L), a wavelength equal to one and one-half times the length of the string [(3/2)L], and so forth. These are called normal modes, or natural vibrating frequencies, of a stretched string.
When a guitar string is plucked, impulses of energy are forced to travel in opposite directions along the string until they reach the string fasteners, or boundaries. These impulses are reflected from the string fasteners, causing one wave to go along the string to the right, and, another wave to go along the string to the left. The consequence is that two oppositely moving waves develop, both traveling at the same velocity, and, traveling through each other in opposite directions. From a study of identical waves traveling in opposite directions and traveling through each other it is known that a combination of such waves produces a new, resultant wave, called a standing wave. This new wave has a waveform which does not move horizontally, but, has a maximum excursion, or amplitude, that goes up and down in a periodic fashion. Another name for a standing wave is a stationary wave.
Perhaps one of the greatest mathematical achievements (because it provided one of the first practical, or empirical, views of the connection between mathematics and real-life problems) was the discovery of the sine-curve, a curve which followed, or represented, the form of a vibrating object like a string. The sine-curve has also been found to exactly have a form that represents an object moving around a circle. So, by putting the two things equal to the same thing, analysts were able to make a direct comparison, or isomorphism, between the waveform, or shape of a vibration along a string and an object moving around a circle. Of course, when you think about it, an object moving around in a circle is really vibrating back-and-forth. So, a circular motion is really equivalent to the motion of a vibration.
Probably, one of the greatest discoveries that has come out of the concept of the sine-curve representation of an object moving around a circle was this: The sides of a triangle drawn from a point on the circle will have a unique ratio whose value will be critically dependent on the specific location of the original point on the circle. For each different location of a point on a circle, the ratio of the triangle sides will have a different value. This unique relationship between a point on a circle and the ratio of the lengths of the vertical and horizontal projections has permitted the development of an entirely new subject of mathematical representation called trigonometry.
THEORY OF WAVES 2.0 The Boundary Value Problem
When a guitar string is picked (or plucked), it vibrates, emitting a sound that has a pitch, or frequency, dependent on the gauge (thickness of the string), the material composition of the string, or mass/unit length, :, of the string, the tension on the string, T, and the length, d, of the string. For a given string mass and tension how does the length of the string determine the particular pitch that the string will produce when plucked? In other words, why does a guitar or violin string vibrate at one particular tone, or frequency, when it is plucked?
The answer is that, for a given gauge and tension of a particular string, it is the boundaries of the string that determine the frequency at which the string vibrates. Guitar and violin strings are supported by a bridge at their bottoms and a nut at their tops. These string supports hold the strings firmly and prevent them from moving vertically (in the y direction). This means that when a string is picked, the disturbance will travel along the length of the string (with a velocity given by Eq. 1.1) until it hits the solid nut, usually made of bone, or the bridge, usually made of metal. The disturbance will be reflected between the nut and the bridge because the waves propagated on the string have an amplitude, or vertical component, and no vertical component of a wave can exist at the boundaries, or ends, of the string.
Since the boundaries of a string are separated by a distance, d, the only waves that can exist on the string are the waves which have null points, or zero amplitude, at its boundaries. This means a minimum frequency wave, f1, called a fundamental frequency, and multiples of the fundamental frequency, called harmonics can exist on the string. The second harmonic, f2, is a sine wave that is twice the frequency of the fundamental. The third harmonic, f3, is three times the frequency of the fundamental. Higher harmonics (frequencies), f4, f5, f6, etc., can also be sustained on the string. Only those complete waves that can fit, perfectly (have zero amplitude at these boundaries) between the string boundaries can exist. For any wave that would have
The bridge and nut are fasteners at the ends of a guitar or violin string. These form boundaries at which y = 0, which means the string cannot vibrate vertically at these boundaries.
some amplitude at the boundary would immediately be pulled to zero by the boundary and be forced to release it's energy, through reflection, to the principal waves that have minima at the boundaries.
The mathematical procedure for finding these principal frequency values (i.e., the fundamental and the harmonics of the string) is called a boundary value problem, or eigenvalue problem (from the German word, eigenwerte, meaning characteristic value). These naturally occurring frequencies are called canonicalfrequencies because they form the basis, or source of the combined, or superimposed, vibrations, or tones, that one hears from a vibrating string. The word canonical merely means the regular, or standard, waves that are produced which form the model from which the gross super impositions (combinations) of sounds from a plucked string are obtained. The word canonical is derived from the word canon meaning a fundamental or regular law, or principle, of a religion.
2.1 Stationary Waves
A string under tension (from being stretched) has two independent directions, +x and -x. When a string is plucked and released, disturbances will go in both the +x and the -x directions. But, each of these is soon reflected at one of the boundaries. This produces reflected waves going in opposite directions to the original incident impulses. Both waves have an amplitude envelope form, or waveform, which varies as a sine-curve varies. So, in effect the string has a moving incident-wave combining with an oppositely moving reflected-wave. The net result, (See Fig 2.2) is that the instantaneous values of the heights, or amplitudes, of the two waves, add, as one wave pushes through the other. The is the same as one 5-lb bag of salt thrown on top of another 5-lb bag of salt being equal to a 10-lb bag of salt. Because the waves are moving and their amplitudes constantly changing, the eventual result is a totally new wave, different from either of the original waves, and, having an entirely different character from the original two waves. The production of a totally different, new wave, from the instantaneous addition of the corresponding amplitudes of two or more original waves is called superposition of waves.
A resultant wave, C, formed from the combination of the incident wave, A, and the reflected wave, B. Wave C is a sine wave but is not a moving sine wave. Instead it is a standing wave, also called a stationary wave. It is called a stationary wave because it has waveform envelope whose amplitude goes up-and-down, but, whose ends stay fastened (do not move) and remain at a value of zero. When you look at the edge of a standing wave it looks like the top of the surface of a balloon when the air is being let out of it. The amplitude decreases from a maximum down to the x axis. Then, the wave expands like a balloon filling with air. The total wave form doesn't move horizontally the way a traveling wave does. When it's center amplitude reaches zero, the process repeats itself again; the waveform starts to expand in the negative direction to a maximum and, then, decreases to zero again, etc. For, a stationary wave generated from high-speed waves, of course, the vibrating amplitude occurs so fast that high-speed photography would be needed to observe the wave envelope variations. When a fastened string is pulled the forces along the string cause one pull from the right and one pull from the left.
When the string is released, the internal string forces generate two impulses traveling away from the point of disturbance.
One impulse travels along the string to the right and the other travels along the string to the left.
The impulses are reflected at the boundaries (with a 180o phase reversal) and pass through each other without interfering with one another (This is called the principal of orthogonality).
Because the two impulses are both traveling at the same velocity, their combination, upon passing through each other, produces a resultant form, or envelope, that is a standing wave.
The generation of a standing wave occurs when two oppositely traveling identical waves, moving with the same velocity, pass through each other.
An incident sine wave, A, combined with a reflected sine-wave, B, produces a stationary wave, C.
How to Determine the Frequencies at Which a String Will Vibrate - ANALYSIS OF THE FIRST NATURAL WAVELENGTH Limited by boundary conditions put on a string by it's end supports, the only waves that can exist on a string are those that have amplitudes that are nulls, or zeros, at the boundaries, or binding points of the string. Expressed mathematically, the only stationary waves that can exist on a string of length, L, are those that have wavelengths that can exactly fit between the two fastened string ends.
Because of this boundary condition, the first vibration that can naturally exist on a fastened string is a vibration that has a wavelength whose distance traverses half the length of the string. For this vibration, the wavelength, 81, of the completed sine wave that can fit between the string fastening points A and B would be the length of two half sine waves, or,
81 = 2L
Other wavelengths, one representing one half of the above wavelength and others representing multiples of the above wavelength, can exist upon a vibrating string. In all cases the only wavelengths that can exist on a vibrating string are those that have a zero value at the string boundaries. Theoretically, an infinite number of frequencies can exist on a vibrating string, each discrete, or separate, frequency being the frequency corresponding to an allowed wavelength on the string.
Conclusion
The above shows how the boundaries, or supports, of a string affect the response of the string to an input (impulse, or plucking). Although the string is a very important thing to study of its own right it turns out that almost everything that you can uncover regarding the mathematical relationships existing on the vibrating string holds for conditions existing within the atom, radio antennas, wave-guides, and a great variety of mathematics, like Fourier series, Fourier integrals, orthogonal functions, etc., ordinarily studied in college and graduate school. So, the sooner that a student becomes aware of the importance of learning the facts behind the mathematics of the vibrating string the sooner she/he will be able to get on board with really understanding the mathematical concepts of advanced high school, technical school and college courses. What is Algebra?
