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Theory of Waves

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

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.

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.

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

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

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

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

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

A

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

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.

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.