OBJECTIVES

After you have finished this lesson you should understand

	¥	the electromagnetic spectrum.
	¥	scattering and refraction of colored light.
	¥	interference of waves and beats.
	¥	standing waves, harmony, vibrations, and music.
	¥	natural vibrations and resonance.
	¥	electrons as waves.
	¥	an explanation of the Periodic Table.


In the last lesson some of the properties of electromagnetic radiation were described; two were particularly noteworthy: Radiation can travel through a vacuum, and the speed of propagation is the same to all observers no matter what their speed is. This latter property leads to a simple relationship between the wavelength and frequency of light as seen by observers in a fixed medium.


	frequency = F(speed of light,wavelength)	wavelength = F(speed of light,frequency)


	Light is visible to human beings only over a very small range of frequencies, but as these frequencies are directly observable, our understanding of the properties of light are to a large extent based on them. Rainbows have been seen in the sky after rainstorms for as long as there have been human beings to see them. What exactly are they and how are they formed? Rainbows, or spectrums of light, are also formed when white light is shone through glass prisms. In both cases white light passes from air into another medium: in the first case water, in the second glass. The speed of light depends on the medium in which it is travelling; the denser the medium, the slower the speed. This is not surprising if one thinks of light as fluctuating electric and magnetic fields. Denser mediums contain more intense electric and magnetic fields, which interact with the light thus slowing it down. The effect of the change of speed is to cause refraction (bending) of the light. The amount that light is bent is found to depend on the wavelength (hence frequency) of the light. In glass, water, and other materials, higher frequencies are bent more than lower frequencies. This phenomena is called dispersion. Thus blue light, a higher frequency, is bent more than red light.

	White light is not a pure color; it is a mixture of many visible frequencies. When it enters a denser medium, the lower frequencies (red) are bent less than the higher (blue) ones, thus splitting the light into the colors of the rainbow, red through violet. Why is blue bent more than red in denser mediums? It turns out that higher visible frequencies interact more with such materials than lower visible ones. This is also true for air and gives an explanation of the color of the sky at different times of the day. The sun emits all visible frequencies. The most intense frequency lies in the yellow range and is related to the temperature of that celestial body. The air molecules bend and reflect the blue frequencies the most and scatter them around the atmosphere whence they continue their travel to the surface of the earth and our eyes, thus giving the sky a blue color. The removal of the blue frequencies from sunlight enhances the yellow color of our local star. In the early morning and late evening, when the sun is low in the sky, sunlight must travel through more of the earthÕs atmosphere; thus more of its light is scattered, yellow frequencies as well as the blue. The color that is left is redÑthus the red sun at sunrise and sunset and the different shade of blue of the sky. If the pollution of the atmosphere is highÑmore dust, more smokeÑthen scattering is still greater and red light is also scattered, giving spectacular sunsets and sunrises.

	Waves are the propagation of vibrations in the medium, which is constituted of matter in the case of sound and water waves or of electric and magnetic fields in the case of electromagnetic radiation. These vibrations are oscillations of the medium (i.e., a periodic motion about an equilibrium position) so that on the average none of the medium moves. If the motion is perpendicular to the direction of propagation, we have transverse waves, while if it is parallel, we have longitudinal waves. This oscillatory motion is produced by forces on the medium that are being propagated from the source of the disturbance. Two or more waves can meet at the same time and place (like bosons). The forces affecting the medium at this point then combine to form a resultant vibration. This is called interference of waves. If the forces act in the same direction, the wavesÕ amplitude increases and we have constructive interference; if they act in different directions, we have destructive interference. If two waves always displace the medium in the same direction they are said to be in phase, while if they do not, they are said to be out of phase. The direction of displacement about the direction of propagation is conventionally designated by a + or -. This leads to the simple rule that to calculate the amplitudes of the resultant wave, one adds the amplitudes of the component waves, calling the amplitudes + if the displacement is one side and - if it is the other. Thus for two waves to always constructively interfere they must be in phase, while if two waves are exactly out of phase and have the same intensity, they cancel out each other. The following diagram demonstrates these points:


Resultant Wave (schematic)















The positions where the waves exactly cancel, so there is no displacement of the medium, are called nodes.

	If two travelling waves of very similar frequency are in the same region of space they will interfere to form a resultant travelling wave. The resultant wave will have the following form:










The amplitude of the combined wave increases and decreases in a regular periodic fashion. This increase and decrease thus has a certain frequency, called the beat frequency, and is equal to the difference of the two frequencies of the constituent waves. The increase and decrease of the amplitude of the beat forms a wave as shown by the above diagram. The speed of propagation of this wave can be calculated by multiplying the frequency and the wavelength of the beat wave. The frequency, wavelength, and speed of the beat wave are not equal (except in very rare situations) to the frequency, wavelength, and speed of the resultant wave. Beats can be heard in sound when two sound waves have similar frequencies and are perceived as regular thumps, thus the phrase Òthe beat of the music.Ó If infinitely many travelling waves are in the same region of space it is possible to produce the following resultant wave packet.


Angle of incidence











It is important to note that the wavelength and frequency of the resultant waves inside the wavepacket are not constant throughout  the extent of the wave packet.

	When a travelling wave, called the incident wave, hits the boundary between two mediums, part of the wave is reflected and part of the wave absorbed. If the second medium does not extend very far, some of the wave may hit the boundary between the second medium and a third where some of the wave is again reflected and a part transmitted into the third medium. The percentage of the wave reflected, absorbed, and transmitted depends on the mediums and the nature of the wave, and we will describe in detail how electromagnetic waves interact with matter in the next lesson. Now we will concentrate on the interference between an incident wave and a reflected wave.

	The angle of reflection of a wave is equal to the angle of incidence, as the following diagram shows:

Angle of reflection











If the incident wave and the reflected wave are along the same path, interference will occur. The form of the resultant wave will depend on the phase relationship between these two waves. If they are in phase, they will constructively interfere forming a wave with increased amplitude but with the same frequency and wavelength; if they are out of phase, destructive interference will take place and a decrease in amplitude will be observed. In order that constructive interference take place, the forces causing the displacement of the incident wave must match the forces causing the displacement of the reflected wave. Thus the reflected wave must meet a later incident wave with the vibrations in both pushing in the same direction, as shown in the following diagram:


Direction of the motion of the medium propogating the wave












When will this occur? The only time it can occur is when the incident wave meets the boundary at precisely the time there is no displacement of the medium. This position of a wave is called a node. Nodes occur every half wavelength. An interesting phenomena occurs when a second reflection occurs where the source of the wave is situated. The incoming incident wave (which has been reflected once) will constructively interfere with the wave coming from the source if they are in phase. Again this is true if the meeting point is a node. The following diagrams show such situations:




























In these cases the distances between the reflecting surfaces B(F(l1,2)), 2B(F(l2,2)), 3B(F(l3,2)), are the same and equal to L, the distance between the two reflecting surfaces, and l1, l2, and l3 are the wavelengths of the waves in each case. We could have continued to draw pictures of patterns formed by incident and reflected waves that are in phase and found that the waves that form such patterns are always integer multiples of half the wavelength.

B(F(l1,2)), 2B(F(l2,2)), 3B(F(l3,2)), 4B(F(l4,2)), 5B(F(l5,2)), 6B(F(l6,2)), 7B(F(l7,2)), 8B(F(l8,2)), É

	Let us concentrate on one point * of the patterns (a), (b), and (c) above, formed by the incident and reflected waves propagating from right to left to right. At this point, but at different times, the vibrating medium is progressively displaced from its equilibrium position to the position +a, then back again through the equilibrium position to the position -a, and then back again. These displacements are due to the combination of the incident and reflected waves. Such patterns can be produced by fixing the ends of a piece of string of length L and causing it to vibrate with a wavelength l1, l2, or l3. These patterns are called stationary or standing wave patterns. The one formed by waves with wavelength l1 is called the fundamental or first harmonic; the one formed by waves with wavelength l2 is called the first overtone or second harmonic; the one formed by waves with wavelength l3 is called the second overtone or third harmonic, and so on. These wavelengths are all related to the distance between the two reflecting surfaces, L. The fundamental wavelength is twice this distance, while the first overtoneÕs wavelength is precisely this distance, and as three halves of the second overtone fit into this distance, the distance is equal to two thirds of the wavelength of the second overtone. We can collect this information in the following table:


l1 = 2L, l2  = L, l3 = F(2L,3), l4 = F(L,2), l5  = F(2L,5), É



If we express all of these wavelengths in terms of the fundamental wavelength, we get 


l2 = F(l1,2), l3 = F(l1,3), l4 = F(l1,4), l5  = F(l1,5), É


	What are the frequencies of these standing waves? We can calculate these by remembering the relationship between the wavelength, frequency, and speed (i.e., frequency equals speed divided by the wavelength). The speed of a wave through a constant medium is the same all the time; thus if the frequency times the wavelength is equal to the same number each time, the frequencies must equal


u2 = 2u1,  u3 = 3u1,  u4 = 4u1,  u5 = 5u1, É


This sequence of frequencies is called a Harmonic Series. What do these frequencies depend on? We see that both the wavelength and the speed of the wave played a role in the above discussion. The wavelength depended on the distance between the reflecting surfaces (i.e., the geometry of the medium), while the speed depended on the nature of the medium; thus the harmonic series of frequencies depends on both. The various overtones in a harmonic series can be characterized by the number of nodes of the standing wave pattern; the fundamental has zero nodes, while the first overtone has one node, and so on.

	It is easy to construct physical realizations of standing waves. In fact you can easily perform the following experiment. Fix a string tautly between two points so that it can move freely in the air. Pluck the string. If you pluck at the correct rate, then a standing wave pattern will appear in the string and a humming sound will be heard. Clearly, you have produced two things: (a) a primitive musical instrument and (b) an example of a standing wave pattern. The standing wave pattern that you produced is probably not a simple one, but a mixture of harmonic frequencies for the string that you have used. These frequencies depend on the length of the string (the geometry of the medium) and the tension and thickness of the string. (These control the speed of the wave propagation in the string.) Each of the harmonic frequencies, called partial tones, are produced with different amplitudes, and they interfere with each other to form a composite standing wave. The amplitude of the fundamental is invariably the greatest, thus its frequency dominates the composite wave, as shown in the following example where the fundamental interferes with the fourth harmonic, which can also be called the third overtone.










	It should be noted that the medium in which these waves propagate to form the standing wave patterns is the string not the air. It is true that the string moves in the air, but the vibrations that produce the waves are between the atoms of the string. The propagation of these vibrations forms a transverse wave since the oscillatory motion of these vibrations are perpendicular to the direction of their propagation (i.e., the direction of propagation of the wave). The vibrations of the standing wave pattern of the string affect the surrounding air molecules and cause them to start vibrating, thus producing a longitudinal sound wave that produces the humming sound heard by your ear.

	Stringed musical instruments are in general more sophisticated than a single taut string, but the basic principle is the same. Consider a six string guitar. Each of the strings has a different thickness. Let us designate the harmonic family of frequencies of each string as a,b,c,d,e,f with decreasing thickness, so that a is the harmonic family of the thickest and f the harmonic family of the thinnest. If plucked, each string will vibrate with its own harmonic family of standing wave patterns, each string producing a tone that will add with the tones from the other strings to form the total sound. The thicker string will vibrate more slowly than the thinner one, thus the pitch of the sound produced by the thicker string will be lower than the one produced by the thinner. The six families of harmonic frequencies can be symbolically listed as
	1a, 2a, 3a, 4a, 5a, 6a, 7a, 8a,	....................
	2b, 3b, 4b, 5b, 6b, 7b, 8b,	....................
	3c, 4c, 5c, 6c, 7c, 8c,	....................
	4d, 5d, 6d, 7d, 8d,	....................
	6f, 7f, 8f,	....................

where we have labelled the fundamental frequency of that a-family Ò1aÓ, of the b-family Ò2bÓ, the c-family Ò3cÓ, the d-family Ò4dÓ, and so on. We have done this to emphasize that the fundamental frequency of the a-family is lower than the b-family, which is lower than the c-family. Each of these fundamental frequencies, and hence the overtone frequencies, is determined by the length of the string, the material the string is made of, and the tension in the string. Once the guitar is made, the material of the strings cannot be changed, but the length and tension in each string can. In fact the act of playing music is changing the lengths of each of the strings in an orderly fashion to produce notes and chords. The tension of the strings is altered to vary the pitch of the notes, which are dominated by the pitch of the fundamental harmonic frequencies. The overtones add to the quality or timbre of the tones. The overtones, even though they are generally at frequencies inaudible to the human ear, are very important to the quality of the sound. If they are absent, the sound is flat, two dimensional, while if they are present, the sound has depth and is more three dimensional. The other parts of the guitar, in particular the sound box, affect the quality and amplification of the sound waves produced by the vibrating strings before they leave on their journey to the listenerÕs ear.

	There are three traditional families of musical instruments: stringed, wind, and percussion. The stringed family includes such instruments as guitars and violins, the wind family organs and trumpets, and the percussion drums and symbols. All of these families function by forming standing wave patterns. For stringed instruments it is one dimensional transverse standing waves in the string; for wind instruments it is one dimensional longitudinal standing waves in air formed in a partially enclosed space in the instrument; for percussion instruments it is two dimensional standing waves on a surface of the instrument. Nowdays there is a fourth class of instruments: electronic synthesizers. These emulate the frequencies of the traditional instruments; however, the less expensive ones are not very successful in producing overtones and the quality can sometimes sound a bit artificial.

	How are the frequencies of the harmonic families related to the terminology usually used in the description of music? Pythagoras (about 580-500 B.C.) was the first documented person to explore the relationship between the lengths of a string and the pitch of its tone. He found that if the ratio of string lengths was a whole number, then the sound of this string combination was pleasing to the ear. A musical scale is a sequence of tones of increasing pitch. If two tones sound equivalent to our ear and there is no other equivalent tone with a pitch between these tones, then these tones are said to be an octave apart. Galileo in the 16th century related frequencies to musical scales and found that the frequency of the higher of two tones that are one octave apart is double that of the lower frequency. The second tone is thus a higher harmonic than the first tone. If the fundamental tone of a string is f, then the tone with frequency 2f (first overtone) is one octave above the fundamental and the tone with frequency 4f (third overtone) is one octave above that. The most common scale in western culture is based on the notes

	do	re	mi	fa	sol	la	ti	do
	A	B	C	D	E	F	G	A¢

The second do is one octave above the first; the names do, re, mi, fa, sol, la, ti stand for first, second, third, fourth, fifth, sixth, and seventh. The note A is defined to be at 440 Hz. Traditionally western culture has used a scale called the ÒJust major scale,Ó where middle C is at 264 Hz and thus the next higher equivalent note C¢ is at 528 Hz. Another scale, called the ÒEqual tempered scale,Ó is used more commonly now days. The frequency of A is the same, but the intervals between the other notes in the same octave is different and more regular. This leads to a different frequency for middle C. The interested reader should consult texts devoted to this subject for more details.

	All objects can be made to vibrate in standing wave patterns, which are called natural vibrations, and the frequencies of these are called natural frequencies. These frequencies are, of course, the harmonic frequencies for the object. In general the standing wave patterns are three dimensional, and unlike musical instruments they tend to attenuate very quickly. Thus to make such objects vibrate one must continue to apply an external source of vibration, called the forcing vibration. In general objects around us are not vibrating, so to produce natural vibrations, the forcing vibration must be at a natural frequency for that object. If it is not, the vibrations forced into the object will quickly die out due to destructive interference of waves. If the forcing vibration is at a natural frequency and it is continuously applied, the object will vibrate with a bigger and bigger amplitude. (The applied vibration reinforces the vibration in the object.) This phenomena is called RESONANCE (which IÕve capitalized due to its importance). What happens if the object is already vibrating at a natural frequency? The forcing vibration will force the object to vibrate at a higher natural frequency if the frequency of the forcing vibration equals the difference between the higher natural frequency and the frequency at which the object is currently vibrating; if not, destructive interference will occur and the vibrations will be damped out.

	Some amusing and not so amusing examples of resonance can be found. If a forcing vibration is at a resonance frequency for an object, the amplitude of the induced natural frequency will increase more and more; that is, the atoms and molecules in the object will oscillate with bigger and bigger amplitudes about an equilibrium point. Thus if the object is at all rigid, there will come a time when it breaks apart! This time might be sooner or later depending on how rigid the object is. Glass is a fairly rigid material; thus if a glass object is continuously forced to vibrate at a resonance frequency, the amplitude of this vibration will increase and very soon the glass object will shatter. Think about the opera singer shattering windows and wine glasses! A less amusing result of resonance is the collapse of bridges or buildings due to forcing vibrations of wind or passing cars. An even more bizarre example would involve human beings if they happened to wander into an area where there was a forcing vibration that happened to match a resonance frequency of theirsÉ

	All things have properties of both waves and particles; we will study the evidence for this in the following units. This is especially pronounced and significant for microscopic objects such as electrons. In order to encapsulate these two properties (wave and particle) in a pictorial model of such objects (a model that might or might not be exactly valid, but satisfies the criterion of being useful) one can view such entities as wave packets. The particle-like property is then due to the wavepacket as a whole and the wavelike properties due to the resultant wave or waves forming the packet. A stream of electrons can thus be viewed as a wave train (i.e., a whole sequence of wave packets). It is clear from this picture of electrons that they do not have a single definite position, or a single frequency (therefore energy), or a definite single speed. What happens when such an electrically negative electron finds itself in a region of space around an electrically positive nucleus? It becomes trapped in this region; that is, it is forced to exist in a fixed region of space. This determines a geometry or shape for the electron, just as length does for a vibrating string with fixed ends. The electron, however, is a wave, and waves cannot exist in a fixed region of space unless they form standing wave patterns. Thus in order to exist around a nucleus, the electron must be in a standing wave pattern. These patterns are three dimensional and various harmonic families of such patterns can be formed. These families are called s, p, d, f, g, h,É The fundamental of each family is denoted by 1s, 2p, 3d, 4f, 5g, 6h, and so on. The energy, or equivalently the frequency, of the first overtone of the s family is usually of the order of the 2p fundamental frequency, and the frequencies of the second overtone of the s family and first overtone of the p family are usually of the order of the fundamental of the d familyÑthus the naming convention for the fundamentals of the various families. The form of the fundamental three dimensional standing or stationary wave patterns are shown in the following picture:

	The higher harmonics of each family have different numbers of nodes, for three dimensional standing wave patternsÕ nodes form a two dimensional surface. There are several differences between a 1s and 2s wave: (a) the size/volume differÑthe radius of a 1s wave is less than a 2s wave; (b) the 2s wave has a nodal surface that separates the wave into two regions of space; and (c) the frequency of the 2s is higher. These differences are repeated for 2p and 3p, 3d and 4d, and so on. That different harmonic families can be formed is analogous to the different harmonic families found in a guitar. The properties of the region of space in which the electron standing wave patterns are formed are determined by the amount of positive charge on the nucleus and the number of electrons surrounding the nucleus. Different amounts of positive charge and total number of electrons change the properties of the region and hence the frequencies and thus energies of the harmonic series of standing wave patterns. This is analogous to the properties of the material forming the strings in a guitar determining the frequencies of the different strings.

	Electrons of different spins can be in the same region of space; thus both can form 1s, 2s, 2p, 3s, 3p, 3d standing waves patterns. The different standing wave patterns correspond to the different subshells of electrons discussed in the previous lesson. This now explains (i.e., gives very substantial reasons) why electrons form subshells around nuclei. If you refer to the end of the last lesson, you will see that this was the crucial unexplained assumption that was needed in order to explain the periodic table and chemical reactions. In the following units we will discuss again the evidence that all things have both particle and wave properties.