## Discussion

### introduction

Maybe you've noticed or maybe you haven't. Sometimes when you vibrate a string, or cord, or chain, or cable it's possible to get it to vibrate in a manner such that you're generating a wave, but the wave doesn't propagate. It just sits there vibrating up and down in place. Such a wave is called a standing wave and must be seen to be appreciated.

A traveling wave in action | A standing wave in action |

I first discovered standing waves (or I first remember seeing them) while playing around with a phone cord. If you shake the phone cord in just the right manner it's possible to make a wave that appears to stand still. If you shake the phone cord in any other way you'll get a wave that behaves like all the other waves described in this chapter; waves that propagate — traveling waves. Traveling waves have high points called crests and low points called troughs (in the transverse case) or compressed points called compressions and stretched points called rarefactions (in the longitudinal case) that travel through the medium. Standing waves don't go anywhere, but they do have regions where the disturbance of the wave is quite small, almost zero. These locations are called nodes. There are also regions where the disturbance is quite intense, greater than anywhere else in the medium, called antinodes.

Standing waves can form under a variety of conditions, but they are easily demonstrated in a medium which is finite or bounded. A phone cord begins at the base and ends at the handset. (Or is it the other way around?) Other simple examples of finite media are a guitar string (it runs from fret to bridge), a drum head (it's bounded by the rim), the air in a room (it's bounded by the walls), the water in Lake Michigan (it's bounded by the shores), or the surface of the Earth (although not bounded, the surface of the Earth is finite). In general, standing waves can be produced by any two identical waves traveling in opposite directions that have the right wavelength. In a bounded medium, standing waves occur when a wave with the correct wavelength meets its reflection. The interference of these two waves produces a resultant wave that does not appear to move.

Standing waves don't form under just any circumstances. They require that energy be fed into a system at an appropriate frequency. That is, when the driving frequency applied to a system equals its natural frequency. This condition is known as resonance. Standing waves are always associated with resonance. Resonance can be identified by a dramatic increase in amplitude of the resultant vibrations. Compared to traveling waves with the same amplitude, producing standing waves is relatively effortless. In the case of the telephone cord, small motions in the hand result will result in much larger motions of the telephone cord.

Any system in which standing waves can form has numerous natural frequencies. The set of all possible standing waves are known as the harmonics of a system. The simplest of the harmonics is called the fundamental or first harmonic. Subsequent standing waves are called the second harmonic, third harmonic, etc. The harmonics above the fundamental, especially in music theory, are sometimes also called overtones. What wavelengths will form standing waves in a simple, one-dimensional system? There are three simple cases.

### one dimension: two fixed ends

If a medium is bounded such that its opposite ends can be considered fixed, nodes will then be found at the ends. The simplest standing wave that can form under these circumstances has one antinode in the middle. This is half a wavelength. To make the next possible standing wave, place a node in the center. We now have one whole wavelength. To make the third possible standing wave, divide the length into thirds by adding another node. This gives us one and a half wavelengths. It should become obvious that to continue all that is needed is to keep adding nodes, dividing the medium into fourths, then fifths, sixths, etc. There are an infinite number of harmonics for this system, but no matter how many times we divide the medium up, we always get a whole number of half wavelengths (^{1}⁄_{2}λ, ^{2}⁄_{2}λ, ^{3}⁄_{2}λ,…, ^{n}⁄_{2}λ).

There are important relations among the harmonics themselves in this sequence. The wavelengths of the harmonics are simple fractions of the fundamental wavelength. If the fundamental wavelength were 1 m the wavelength of the second harmonic would be ^{1}⁄_{2} m, the third harmonic would be ^{1}⁄_{3} m, the fourth ^{1}⁄_{4} m, and so on. Since frequency is inversely proportional to wavelength, the frequencies are also related. The frequencies of the harmonics are whole-number multiples of the fundamental frequency. If the fundamental frequency were 1 Hz the frequency of the second harmonic would be 2 Hz, the third harmonic would be 3 Hz, the fourth 4 Hz, and so on.

### one dimension: two free ends

If a medium is bounded such that its opposite ends can be considered free, antinodes will then be found at the ends. The simplest standing wave that can form under these circumstances has one node in the middle. This is half a wavelength. To make the next possible standing wave, place another antinode in the center. We now have one whole wavelength. To make the third possible standing wave, divide the length into thirds by adding another antinode. This gives us one and a half wavelengths. It should become obvious that we will get the same relationships for the standing waves formed between two free ends that we have for two fixed ends. The only difference is that the nodes have been replaced with antinodes and vice versa. Thus when standing waves form in a linear medium that has two free ends a whole number of half wavelengths fit inside the medium and the overtones are whole number multiples of the fundamental frequency

### one dimension: one fixed end — one free end

When the medium has one fixed end and one free end the situation changes in an interesting way. A node will always form at the fixed end while an antinode will always form at the free end. The simplest standing wave that can form under these circumstances is one-quarter wavelength long. To make the next possible standing wave add both a node and an antinode, dividing the drawing up into thirds. We now have three-quarters of a wavelength. Repeating this procedure we get five-quarters of a wavelength, then seven-quarters, etc. In this arrangement, there are always an odd number of quarter wavelengths present. Thus the wavelengths of the harmonics are always fractional multiples of the fundamental wavelength with an odd number in the denominator. Likewise, the frequencies of the harmonics are always odd multiples of the fundamental frequency.

The three cases above show that, although not all frequencies will result in standing waves, a simple, one-dimensional system possesses an infinite number of natural frequencies that will. It also shows that these frequencies are simple multiples of some fundamental frequency. For any real-world system, however, the higher frequency standing waves are difficult if not impossible to produce. Tuning forks, for example, vibrate strongly at the fundamental frequency, very little at the second harmonic, and effectively not at all at the higher harmonics.

### filtering

The best part of a standing wave is not that it appears to stand still, but that the amplitude of a standing wave is much larger that the amplitude of the disturbance driving it. It seems like getting something for nothing. Put a little bit of energy in at the right rate and watch it accumulate into something with a lot of energy. This ability to amplify a wave of one particular frequency over those of any other frequency has numerous applications.

- Basically, all non-digital musical instruments work directly on this principle. What gets put into a musical instrument is vibrations or waves covering a spread of frequencies (for brass, it's the buzzing of the lips; for reeds, it's the raucous squawk of the reed; for percussion, it's the relatively indiscriminate pounding; for strings, it's plucking or scraping; for flutes and organ pipes, it's blowing induced turbulence). What gets amplified is the fundamental frequency plus its multiples. These frequencies are louder than the rest and are heard. All the other frequencies keep their original amplitudes while some are even de-amplified. These other frequencies are quieter in comparison and are not heard.
- You don't need a musical instrument to illustrate this principle. Cup your hands together loosely and hold them next to your ear forming a little chamber. You will notice that one frequency gets amplified out of the background noise in the space around you. Vary the size and shape of this chamber. The amplified pitch changes in response. This is what people hear when the hold a seashell up to their ears. It's not "the ocean" but a few select frequencies amplified out of the noise that always surrounds us.
- During speech, human vocal cords tend to vibrate within a much smaller range that they would while singing. How is it then possible to distinguish the sound of one vowel from another? English is not a tonal language (unlike Chinese and many African languages). There is very little difference in the fundamental frequency of the vocal cords for English speakers during a declarative sentence. (Interrogative sentences rise in pitch near the end. Don't they?) Vocal cords don't vibrate with just one frequency, but with all the harmonic frequencies. Different arrangements of the parts of the mouth (teeth, lips, front and back of tongue, etc.) favor different harmonics in a complicated manner. This amplifies some of the frequencies and de-amplifies others. This makes "EE" sound like "EE" and "OO" sound like "OO".
- The filtering effect of resonance is not always useful or beneficial. People that work around machinery are exposed to a variety of frequencies. (This is what noise is.) Due to resonance in the ear canal, sounds near 4000 Hz are amplified and are thus louder than the other sounds entering the ear. Everyone should know that loud sounds can damage one's hearing. What everyone may not know is that exposure to loud sounds of just one frequency will damage one's hearing at that frequency. People exposed to noise are often experience 4000 Hz hearing loss. Those afflicted with this condition do not hear sounds near this frequency with the same acuity that unafflicted people do. It is often a precursor to more serious forms of hearing loss.

### two dimensions

The type of reasoning used in the discussion so far can also be applied to two-dimensional and three-dimensional systems. As you would expect, the descriptions are a bit more complex. Standing waves in two dimensions have numerous applications in music. A circular drum head is a reasonably simple system on which standing waves can be studied. Instead of having nodes at opposite ends, as was the case for guitar and piano strings, the entire rim of the drum is a node. Other nodes are straight lines and circles. The harmonic frequencies are not simple multiples of the fundamental frequency.

The diagram above shows six simple modes of vibration in a circular drum head. The plus and minus signs show the phase of the antinodes at a particular instant. The numbers follow the (D, C) naming scheme, where D is the number of nodal diameters and C is the number of nodal circumferences.

Standing waves in two dimensions have been applied extensively to the study of violin bodies. Violins manufactured by the Italian violin maker Antonio Stradivari (1644–1737) are renowned for their clarity of tone over a wide dynamic range. Acoustic physicists have been working on reproducing violins equal in quality to those produced by Stradivarius for quite some time. One technique developed by the German physicist Ernst Chladni (1756–1794) involves spreading grains of fine sand on a plate from a dismantled violin that is then clamped and set vibrating with a bow. The sand grains bounce away from the lively antinodes and accumulate at the relatively quiet nodes. The resulting Chladni patterns from different violins could then be compared. Presumably, the patterns from better sounding violins would be similar in some way. Through trial and error, a violin designer should be able to produce components whose behavior mimicked those of the legendary master. This is, of course, just one factor in the design of a violin.

91 Hz | 145 Hz | 170 Hz | 384 Hz |

### three dimensions

In the one-dimensional case the nodes were points (zero-dimensional). In the two-dimensional case the nodes were curves (one-dimensional). The dimension of the nodes is always one less than the dimension of the system. Thus, in a three-dimensional system the nodes would be two-dimensional surfaces. The most important example of standing waves in three dimensions are the orbitals of an electron in an atom. On the atomic scale, it is usually more appropriate to describe the electron as a wave than as a particle. The square of an electron's wave equation gives the probability function for locating the electron in any particular region. The orbitals used by chemists describe the shape of the region where there is a high probability of finding a particular electron. Electrons are confined to the space surrounding a nucleus in much the same manner that the waves in a guitar string are constrained within the string. The constraint of a string in a guitar forces the string to vibrate with specific frequencies. Likewise, an electron can only vibrate with specific frequencies. In the case of an electron, these frequencies are called eigenfrequencies and the states associated with these frequencies are called eigenstates or eigenfunctions. The set of all eigenfunctions for an electron form a mathematical set called the spherical harmonics. There are an infinite number of these spherical harmonics, but they are specific and discrete. That is, there are no in-between states. Thus an atomic electron can only absorb and emit energy in specific in small packets called quanta. It does this by making a quantum leap from one eigenstate to another. This term has been perverted in popular culture to mean any sudden, large change. In physics, quite the opposite is true. A quantum leap is the smallest possible change of system, not the largest.

Some Probability Densities for Excited Electrons in a Hydrogen Atom | ||

|2,0,0⟩ | |2,1,0⟩ | |2,1,1⟩ |

|3,0,0⟩ | |3,1,0⟩ | |3,1,1⟩ |

|3,2,0⟩ | |3,2,1⟩ | |3,2,2⟩ |

### mathematics

In mathematics, the infinite sequence of fractions ^{1}⁄_{1}, ^{1}⁄_{2}, ^{1}⁄_{3}, ^{1}⁄_{4},… is called the harmonic sequence. Surprisingly, there are exactly the same number of harmonics described by the harmonic sequence as there are harmonics described by the "odds only" sequence: ^{1}⁄_{1}, ^{1}⁄_{3}, ^{1}⁄_{5}, ^{1}⁄_{7},…. "What? Obviously there are more numbers in the harmonic sequence than there are in the 'odds only' sequence." Nope. There are exactly the same number. Here's the proof. I can set up a one-to-one correspondence between the whole numbers and the odd numbers. Observe. (I will have to play with the format of the numbers to get them to line up correctly on a computer screen, however.)

01, 02, 03, 04, 05, 06, 07, 08, 09,…

01, 03, 05, 07, 09, 11, 13, 15, 17,…

This can go on forever. Which means there are exactly the same number of odd numbers as there are whole numbers. Both the whole numbers and the odd numbers are examples of countable infinite sets.

There are an infinite number of possible wavelengths that can form standing waves under all of the circumstances described above, but there are an even greater number of wavelengths that can't form standing waves. "What? How can you have more than an infinite amount of something?" Well I don't want to prove that right now so you'll have to trust me, but there are more real numbers between 0 and 1 than there are whole numbers between zero and infinity. Not only do we have all the rational numbers less than one (^{1}⁄_{2}, ^{3}⁄_{5}, ^{733}⁄_{2741}, etc.) we also have all the possible algebraic numbers (square root 2, 7 minus square root 13, etc.) and the whole host of bizarre transcendental numbers (π, e, e^{π}, Feigenbaum's number, etc.). All of these numbers together form an uncountable infinite set called the real numbers. The number of whole numbers is an infinity called aleph null (א_{0}) the number of real numbers is an infinity called c (for continuum). The study of infinitely large numbers is known as transfinite mathematics. In this field, it is possible to prove that aleph null is less than c. There is no one-to-one correspondence between the real numbers and the whole numbers. Thus, there are more frequencies that won't form standing waves than there are frequencies that will form standing waves.