(Learn This Subject And Get A Jump Start On All Your College And Graduate School Math Courses)
What Caused People to Become Interested In the Study of Waves?
The Formation of Waves
A LOOK INTO THE FORMATION OF WATER WAVES
The Theory of Oscillation
Max Planck (1858-1947)
A. H. Compton (1892-1962)
A LOOK INTO THE FORMATION OF WATER WAVES
INTRODUCTION TO APPLIED MATHEMATICS
1.0 Why Do We Study Waves?
Mechanical vibrations such as waves observed on the surface of water have been looked upon as an oddity since time immemorial, since the way that the waves are actually formed has been a mystery to most people. Today, however, from work done by past researchers, there is a vast practical knowledge accumulated concerning mechanical vibrations, also known as wave phenomena. When students, technicians and engineers realize that by knowing the reasons behind the formation and motion of waves they are opening the door to a genuine understanding of mathematical ideas, they are often lifted with inspiration. For, there are few experiences that can compare with the personal satisfaction received from achieving understanding of a difficult subject such as mathematics.
Suddenly, difficult mathematical ideas become highly apparent and obvious to them. Suddenly, difficulties in understanding mathematical ideas such as the theory of trigonometry, the physical interpretation of the derivative, the purpose of differential and partial differential equations, and the meanings behind advanced notions like Fourier series and orthogonal functions, disappear. For, one of the best ways to interpret a large variety of mathematical concepts is to associate them with the subject from which they were originally derived, namely, the theory of waves.
In addition, when they discover that the mathematical knowledge attained from the study of wave theory develops their understanding of the operation of a vast number of technological concepts involving electrical vibrations such as alternating current theory, antenna theory, transmission line theory, cellular and laser communications, and atomic theory they often want to learn as much as they can about wave theory, so that they may be more employable to an employer.
Because a study of waves - how they are formed, and how they travel - forms the basis of so many of the subjects that must be known by a student in a mathematics, science or engineering curriculum, the ambitious student will, likewise, want to learn as much as he/she can about wave theory in order to develop the background required for an understanding of these subjects so as to be qualified and, consequently, be accepted for a job opening. So, the purpose of this chapter is to start the development of your understanding of wave theory starting with seemingly simple, but relevant, examples of waves on water and strings and gradually moving into a pictorial exposition of wave concepts using trigonometric identities and, ultimately, into an advanced description of waves using higher mathematics.
1.1 What Caused People to Become Interested In the Study of Waves?
A stone thrown into a pool of water will generate a circular expansion of waves moving away from the location of the disturbance. In this instance, a vertical motion (the falling stone) produces a horizontal motion (a traveling wave). The traveling (or propagating) wave has mechanical energy. One of the main reasons that people want to learn about waves is because generating a wave brings into being a means for quickly sending energy from one place to another. And, since this energy can travel faster than a person can, it can be used as a rapid signaling means. What an excitement it must have been when people began to understand that a wave could be used to carry voice information to them from a distance they never could hope to traverse, or cover. That is why the old masters were interested in waves! Look at TV today. You can see people being interviewed in Russia and China. In those days It took three months to get to China.
When a person shouts, the sound he/she creates starts with an impulse of air directed over his/her vocal cords. The impulse generates a sound which travels about 1100 ft./sec. A shout used to warn a person walking close to the end of a cliff illustrates an important use of waves as a rapid signaling means. Without the shout, the person would have to climb the cliff and actually show the climber the imminent, or impending, danger.
Light waves from fires have been used as a signaling means for countless generations. These signals traversed distance far more rapidly than a man could run the same distance.
When someone shouts to warn another person that she is too close to the edge of a cliff, the person shouting is using fast traveling sound waves as a rapid signaling means to send her warning.
In 1887, Heinrich Hertz (1857_1894) proved electromagnetic waves (radio waves) could be generated by a spark jumping across a gap at the ends of a loop of wire. Upon learning of this discovery a lot of people became excited over the its possibilities. There was good reason for this excitement. At that time, the Morse code was being used to send words in the form of a code sent through wires by interrupting an electric current, and voice was being sent by telephone, by means of a microphone which varied an electric current in accordance with the variations of a person's voice. People in the know realized that it was just a matter of time before Hertz's electric waves would be used to send information, or modulation, using electric waves as a courier (now called a carrier wave) that was capable of traveling great distances. The device that would send the information signal into space was called a transmitter. The word transmit originates from the Latin mito, meaning to send, and trans, meaning across. The device that would acquire the signal was called a receiver.
We see, then, that the study of waves is fundamental to our understanding of radio and television antennas, transmitters and receivers, cellular and laser communications devices, etc. The waves are the means by which energy is sent from one place to another. The transmitter and receiver are only devices which process the signals to suit the user's needs.
1.2 The Formation of Waves
Elastic materials are substances that are deformed when a sufficiently strong external force is applied to them. Every elastic material has a limit to the length it can be stretched. For example, a rubber band can be pulled just so far before it either breaks or weakens in it's elastic pulling power. In the latter case, this is called exceeding the elastic limit of the material. Many materials are "springy" only if they are pushed or pulled within certain limits. Forcing a spring beyond a certain distance can stretch the material out of form and alter its springiness. When we speak of the elastic properties of a material we always refer to the material when it is below it's elastic limit, so that we can know that the material is "springy" when we push or pull it.
A force applied to an elastic material deforms the material causing it to exert a restoring force called a counter force. If an elastic material is given a sudden push or pull and suddenly let go a counter force develops a velocity and consequent momentum that moves the molecules of the elastic material to a distance beyond it's original equilibrium, or rest, position. For example, a percussionist's drum membrane is in it's equilibrium position when his drum is not being played. If you were to put a nail in the center of the membrane, reach under the drum and pull the nail toward the floor and suddenly let the nail go you would observe that the surface of the drum membrane would be forced by it's restoring force to go beyond the equilibrium, or rest position of the membrane. A drum membrane that is forced above it's equilibrium position will have sufficient restoring force within it's molecules to pull itself below it's equilibrium position. The interchange of potential energy between the velocity of the membrane molecules and the elasticity of the membrane accounts for the back and forth motion, or vibration of a drum head. The process continues until the vibrations weaken, or dampen, due to the dissipation of energy from the frictional forces within the drum membrane. Another example of an elastic vibrating mass is a diving board on which a diver has just jumped.
The elasticity of the material comprising a diving board supplies the restoring force that propels a diver upward when she jumps on the board .
A LOOK INTO THE FORMATION OF WATER WAVES
Common examples of waves are the waves observed on water surfaces. Water waves have been the subject of intensive study through the years and are difficult to interpret in a complete manner, but we will describe some basic facts about them. A stone dropped into a pool of water creates a series of concentric circles spreading out from the point where the stone entered the water. We may imagine that when a heavy stone hits the water, it pushes some water below it's original, equilibrium level of the rest of the water. Water cannot be compressed. This means that the forced water must go somewhere. If the water is in a swimming pool, the pushed water cannot go past the pool boundaries, i.e., the pool floor or sides. (See Fig. 1.2). The only direction that any water can go is up, for when the water that was originally forced down hits the pool floor, or else, hits against water resting on the pool floor, other water in the pool is forced upward. _ Since the stone occupies space originally occupied by water, water under the stone is pushed downward, and eventually, upon reflection, or displacement, emerges as water flowing above the original equilibrium level of the water. This is because the surface of the water is not a restrictive boundary for the flowing water, as are the pool floor, or sides.
Water forced above the original water level will go up until its potential energy equals its kinetic energy. At this point in the analysis it is easy to see why a larger stone dropped into water creates a larger wave: a larger stone hits the water with more impact than a smaller stone, it displaces more water and, eventually, causes a greater amount of water to be forced above the original water level because of the water's restoring force.
In this analysis of a water wave, all we have learned so far is how a stone dropped into water can result in an upward push that causes a mass of water to move above the original equilibrium level of the water surface. The height that a wave crest attains above it's equilibrium, or zero, level is called the amplitude. Now, we must investigate how this generates a moving water wave.
A water wave is formed when water is pushed below the normal surface (equilibrium) level of the water. This causes other water to be pushed above the normal water level. Water that is above the equilibrium level of the surface must fall down with a force. So, the water falling down causes other water to go up and the water going up eventually falls and the process continues on and on generating a water wave.
1.3 The Theory of Oscillation
Any particle attached to an elastic material that is a distance, d, beyond it's equilibrium level has a potential energy, with respect to the normal, or equilibrium level, of the particle . An example of potential energy is the energy in a book raised to a distance, d, above a table. When the book is dropped, say, from a three-foot distance above a table it will hit the table with considerably more force than when it is dropped from a one-foot distance above the table. In like manner, water above its equilibrium level, having mass, is pulled downward by gravity and the tensile forces within the envelope of the water waveform. But, when the falling water reaches it's original equilibrium level it strikes more water and acts the same as any other mass hitting the water, eventually causing other water to be forced above the original undisturbed water level. The process continues, like a pendulum _ water above the surface comes down, with inertia, and hits other water in the pool, thereby forcing water to it's boundary, causing reflected water to move upward, beyond the original water equilibrium level. The process, known as harmonic motion, continues with a back and forth motion, just like a weight on a coil spring. Any vibrating object is called a harmonic oscillator if the elastic material follows Hooke's law. In the process of vibration, the wave is only a manifestation of energy propagation through the material. Individual particles of water do not move horizontally with the wave, away from the original disturbance, rather, they only move up-and-down. The motion of both objects have a likeness, or correspondence, to a sine-wave. This means that the motion of a water wave follows the form of a sine-wave, which, in turn, can be interpreted to follow the form of an object traveling around a circle. Because a water wave has a form similar to that of a sine-curve, the mathematical relations that describe a sine-curve can be used to describe water waves. A mathematical relationship that matches, or compares, one kind of variation with another is called an isomorphism (Greek, ISO means same and morphos means form).
A cork floating on top of water through which a wave is traveling only moves up-and-down. It does not move in the direction of the wave propagation. In like manner, A particle of water, P, suspended under the water surface, like the cork, only goes up-and-down when a wave passes through.
Max Planck (1858-1947)
Max Planck was born in Kiel, Germany and later lived in Munich. He subsequently moved to Berlin where at the University of Berlin he studied under two great physicists Kirchoff and von Helmholtz. While at the University of Berlin he wrote "Treatise on Thermodynamics" in 1897 which demonstrated his intense understanding of the practical aspects of physics. During the 1900's one of the problems that mystified physicists was that of blackbody radiation. Finding a solution to this problem turned out to be essential to understanding why a heated object (like a hot iron stove poker) changed colors from white hot to dark red as it's temperature decreased. This change of color was referred to as the energy distribution, or the spectral distribution, of the cooling body. By using statistics, another investigator of the blackbody problem, Wien, developed a "displacement law" which was quite a close solution to the radiation problem, but, still was recognized by Planck as incorrect. Another attempt at the problem made by Lord Raleigh in 1900 and by James Jeans in 1905. Using the law of equipartition of energy they computed the number of modes, or ways of vibration, the atoms in the hot object were assumed to have. They presumed each mode of vibration to be an actual oscillator existing within the atoms of the radiating object. The Rayleigh-Jeans solution was based on classical interpretations of wave phenomena existing within the radiating object and was shown not to be a solution to the problem. Planck, who through his extensive work in thermodynamics, had heavy experience in statistics, had devoted an intensive effort toward resolving the blackbody radiation problem. Max Planck used the laboratory measured values of the spectral distribution of a hot body made by Lummer and Pringsheim to arrive at an empirical formula (i.e., a formula developed by observing what was happening and, then, making up a formula that matched the observed results) that fit the observed spectral distribution of a cooling body. The only thing was that his results did not make sense in terms of the, then known, theories (the classical theories) that had been accepted as valid in the past. So, in order to substantiate his empirically derived formula for blackbody radiation Planck originated his own unique interpretation of the phenomena of blackbody radiation. This caused a whole new field of modern physics to be born. Planck's main contribution to physics was that the energy of a radiating object depends on the wavelength of the radiation multiplied by a number called Planck's constant. His work resulted in the fundamental idea that the energy of the particles like the electrons, protons, etc. within an atom can only take on specific values, called quanta.
A. H. Compton (1892-1962)
Several experiments were crucial in developing support for the photon theory of radiation. In 1923 Compton, working with X rays, showed that short wavelength rays could be scattered from materials. Compton discovered that incident radiation beamed on a material resulted in a scattered radiation beam being given off at a slightly lower frequency. This indicated, according to Planck's theory, that the incident energy had lost energy because of a change in scattering angle, 2. Compton was able to show, through the use of conservation of energy and momentum equations, that the lower frequency of the scattered radiation was due to a loss of photon energy resulting from collisions between the incident photons and the electrons within the material. Thus, the Compton Effect clearly demonstrated that photons possess energy and that a change of photon energy resulted in a change in the photon's wavelength. In other word, a beam of radiation was observed to change frequency because some of it's energy was transferred to electrons within a material. This is comparable to observing that a beam of light changed color upon passing through a material. The effect was accounted for by Planck's theory and, therefore, confirmed his concept.
So, we may conclude by stating that, because actual waves are observed as present in a great variety of physical phenomena and circumstances, the mathematics of waves (formulating how they are formed and how they actually move away from their source) forms a most important part of the study of mathematics and science.
People study waves because the ideas learned from a vibrating string are the same as those found within atoms and molecules, electonic apparatus and antennas, so learning the vibrating string helps people know the inner operation of these other fields of study.
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