But among all these diverse phenomena, what is the common element that characterizes them all as waves? It is that they all display some cyclical or repetitive pattern or behavior over space and time. Thus, the ripple in the lake displays repeating patterns of crests and troughs; people doing the wave in the stadium stand up, wave their arms and sit down in turns in a pattern that goes around the stadium; sound is caused by alternate compression and decompression of air.
Figure 9.1 Some examples of different kinds of waves are shown. The common feature of all waves is that a certain pattern repeats itself.
. The simplest and, therefore, the most important wave of all is called a sine wave, and that comes from the name of the trigonometric function that describes it mathematically. In fact, if you were to ask kids to draw waves, they would often draw something that resembles a sine wave; it is the natural shape we all associate with waves. And because it is so easy to visualize, we will use a sine wave to illustrate the general features of waves.
Since waves are cyclical, the fundamental feature of a wave is the distance over which it repeats itself—that distance is called the wavelength—and for a sine wave, that happens to be just the separation between the successive crests, shown in . Waves can be strong or weak—is it a tsunami or just a ripple? The strength of a wave is measured by its amplitude, which is the difference between the height of the crest (the maximum) and the bottom of the trough (the minimum); the larger the crest-to-trough difference, the stronger would be the wave—ask any big-wave surfer, or check out a big-wave surfing video online, what’s big is the amplitude!
Figure 9.2 The two defining characteristics of a wave are its wavelength, which is the shortest distance over which the wave repeats itself, and its amplitude, which is the maximum vertical extent of the wave from its highest point to its lowest point.
where sine waves with three different frequencies (and wavelengths) are shown. If we know one, then we can deduce the other, so the wavelength and frequency convey the same information and may be used interchangeably.
With just these basic features, we are ready to map people’s personalities to waves, as a first step toward understanding how people match up in a relationship. In principle, this is quite simple—with every individual’s personality, we can associate a unique waveform. Think of our state of mind, our feelings, our preferences, and our personalities as a superposition of repetitive wave patterns unique to each one of us; those patterns determine who we are. If you are biologically inclined, you could think of all those patterns akin to some sort of brain waves, but the actual biology of it is not relevant or even necessary here.
. They are called harmonics: the first one—the fundamental frequency—sine(1) is the first harmonic, sine(2) the second harmonic, sine(3) the third harmonic, and so on. If all this sounds very musical, that’s no coincidence, because these waves describe musical notes as well, and we could conjure a more poetic image by dubbing our entire analysis here as the music of human relationships.
With the building blocks at hand, we can now create any kind of wave we like, to match any personality type. To do that, we associate one specific frequency with every quality of a person, and we are free to pick and choose them as we like. For example, sine(1) could represent taste in music, sine(2) work habits, sine(3) culinary tastes, and so on. Once we have made the associations, it is just a matter of determining how much of each elemental wave we need to add in to the personality blend, and that just happens to be the amplitude of each component. If a quality is strong in a person, that harmonic would have a large amplitude; if it is weak, it would have a small amplitude;, and if absent altogether, the amplitude would just be zero. And for every choice and combination of amplitudes of the relevant harmonics, we will have a different waveform corresponding to a distinct personality. The choices are infinite, as they should be in order to describe infinitely different personalities possible.
, the result already looks a bit complicated, and that is with just three components! The figure explains how we add up waves: We line up the waves and, at every position left to right, we add the heights and subtract the depths as measured from the horizontal median lines of all the contributing harmonics. Thus, at positions where all the waves rise up, their sum is higher still, but elsewhere if some waves rise while others fall, there will be mutual cancellations and the sum will be reduced. It would certainly be very tedious to add up the waves drawing them by hand this way, but that’s no problem because computers can do it instantaneously!
As we add more and more components for the various facets of human personalities, we will end up with some really complicated waveforms. That is no problem, however, because no matter how complicated it gets, we can always break them down into the simple, individual sine wave components with the Fourier rules. But even without going into all those details, we can already deduce what the wave patterns would look like for a range of different personality types:
An interesting person has a lot of different Fourier components, corresponding to many-faceted interests and character traits.
, and we ask, are they “in sync” or “out of sync” with each other? Do they rise and fall together? If they do, then we say they are in phase. But if they are out of sync, meaning that the crests of the two waves are displaced relative to each other, then they are said to be out of phase. And exactly how much out-of-sync they are is given by their phase difference, which is just the distance between corresponding crests of the two waves, as we can see in . The minimum phase difference of course would be zero, when the waves are perfectly in sync. But there is also a maximum possible phase difference (“out-of-sync-ness”) that occurs when the crests of one wave are exactly halfway (or one half-wavelength) in between the crests of the other wave, so that one wave has a crest exactly where the other has a trough like in (c).
Figure 9.5 Phase of a wave is important for comparing waves. When two waves rise and fall together, their phase difference is zero as in (a); otherwise not, as in (b) and (c). Particularly if one wave reaches its peak exactly where the other one is at its lowest point like in (c), then the two waves have maximum phase difference.
, and once we have calibrated the dial by assigning phase values for all tastes in music, then your own particular musical taste will correspond to some specific phase value or range of phase values on that dial.
Figure 9.6 Different phases of a wave (measured by how much the first crest is shifted from the reference line) can be calibrated to represent variations of a particular facet of life, for example, musical tastes. In this example, we assume that a taste for classical music is completely out of phase with taste for heavy metal; therefore, the waves representing them have the maximum phase difference.
, so that when you add them together, the combined amplitude gets larger still, the troughs get deeper and crests get higher, and that is a happier, stronger wave. In the language of quantum physics, your musical tastes are in constructive interference. As far as music is concerned, you are a perfect match, because you are in phase, and your joint amplitude is stronger and more enhanced, and you would say you share common musical tastes. In your relationship future, I see you going to a lot of concerts together, spending evenings listening and even dancing to music together, sharing your musical discoveries with each other, leading to an overall musical enriching of your relationship.
, your combined amplitude is very much diminished. In the jargon of quantum physics, this facet of your relationship is in destructive interference. The implication for your relationship is that in being together, you would have to give up on the prospects of ever enjoying a good meal together as a couple, and over time you could go from bemused toleration to utter disgust of the other person’s choices of food, and it could be the cause of a serious rift in the long run.
Electromagnetism refers to the related phenomena of electricity and magnetism. With both phenomena there is a region of influence; for example, a magnet can attract iron objects in its neighborhood. That region of influence around a magnet or an electrically charged object contains an invisible electromagnetic field by which such influence is exerted. Any disturbance in that field propagates like ripples in water and is called an electromagnetic wave. Light originates in the disturbance of the electromagnetic fields associated with moving electrons inside atoms, as we saw in .