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We can see waves in water, and we know they exist in other media, but they also exist as mathematical models. The purest form of wave is the sine wave. It is pure in the spectral sense, meaning that it contains energy at one frequency only. This means that it has no bandwidth and is not capable of conveying any information. It follows that it can easily be compressed. That much is obvious because all of the cycles look exactly the same and we cannot tell if the wave has been delayed by one cycle or by several million. Such a pure sine wave does not exist except as a concept, because purity depends upon the wave having always existed as well as continuing to exist. Fourier analysis tells us if the wave ever began or ended it would have further frequencies due to that modulation.

Despite its purity, the sine wave turns out to be a bit of a nuisance because it is constantly changing. As an electrical signal, the voltage changes with respect to time. There is nothing constant about it, as the slope and the acceleration also change constantly. In order to draw a sine wave, we need to consult a set of trigonometric tables. The phase angle is ωt, where ω is the frequency and t is the time. Drawing a sine wave that way, looking up the voltage every fraction of a radian, is what mathematicians would call a numerical method, another way of saying it’s the use of brute force because nothing more elegant exists.

Fig.1 shows that we can be smarter than that, if we realize that a sine wave is one dimension of a rotation. Rotations that describe a circle are a lot simpler to arrange, because the radius is constant and the angular velocity is constant. That is at least a start, because constant quantities are easier to deal with. It should be pointed out that the phase angle used by mathematicians is the radian, which is the angle subtended by unit circumference of a circle at unit radius. It’s about 57 degrees and there are 2Π of them in a complete circle.

That fundamental connection between waves and rotations tells us that some combination of the two lies behind most of the cyclic phenomena we come across. Days, years seasons, tides, Milankovich cycles and ice ages, ocean waves, radio waves, bow waves, light waves, Kondatriev waves and so on.

Not much we do in broadcasting or elsewhere is free of the influence of waves and rotations, but learning about this important topic is greatly eased because the scientists have done much of the work. Waves in the real world operate according to nature and we have no say in what happens. The best we can do is to observe nature in action and try to make sense of it. It is close observation and acceptance of nature that makes science what it is.

After a number of experiments have been observed, it may be possible to find a common principle that underlies all of them. That common principle may lead to the adoption of a natural law or theory that can be used to predict what will happen under similar circumstances. The more times the prediction turns out to be right, the more trustworthy the theory is held to be.

Ohm’s Law has been tested so many times in so many places, that the chances of it being found wrong are pretty small. On the other hand theories regarding, for example, the future of global warming have never been tested because we are living in the experiment. Many of these theories have the same status as astrology, which exists because a demand has been created.

The technologist has a different problem to the scientist, because he is faced with providing functional solutions to practical problems. The ability of the technologist is enhanced if he is able to predict the performance of his designs using scientific knowledge. Sometimes lives depend on what the technologist does. If that bridge or that building collapses, people die. Alongside that, a scientist finding that a theory is invalid doesn’t feel so bad.

Where does the mathematician fit into this? Firstly, mathematicians do not observe the real world using experiments, so mathematics is not a science. That is not a criticism, by the way, because mathematics is just as rigorous as science and its results can be relied on just as much as those of science. Possibly even more, because mathematics evolved using no more than rational thinking and pencil and paper. The voltage on a wire is one dimensional, whereas rotation is two dimensional. As, we shall see, mathematics will resolve that problem with some help from Leonhard Euler.

The strength of mathematics is that it is possible to build models that replicate the observations of the scientist. The behavior of matter can be copied by suitable equations. Once that becomes possible, then predictions can be made. For the technologist, the equations allow, for example, the strength of a proposed structure to be estimated, so that suitable safety margins can be adopted. For the physicist, the equations allow certain behaviors to be predicted. Those can be checked by new experiments. If the experiments confirm the predictions, then the theories the math is modelling become more trustworthy.

The biggest problem with mathematical models is not the math. Instead it is when equations are put forward that do not model the phenomenon of interest. The equations are solved and give the wrong answers. Philosophy suggests one should always be suspicious of arguments that begin with an equation. It is far better to begin with an understanding of the underlying phenomenon and then show why the equations apply to that.

Philosophy can help to illustrate what rational thinking, sometimes called critical thinking, is and means to the scientist, the mathematician and to the technologist. The scientist is trying to see how the world is, and the good scientist is permanently in doubt about the accuracy of his view of the world. Richard Feynman explained that it was necessary to have a certain amount of integrity about one’s knowledge, including the possibility of being wrong. He shared that philosophy with Bertrand Russell, who said he would not lay down his life for his beliefs because he might be wrong.

The Greeks had some fine philosophers, but they made the mistake of thinking they could explain the world just by thinking about it. They couldn’t; which is why science took over, however, the concepts of rigorous thought that the philosophers left are important.

Critical thinking is no more than looking at existing information with the goal of establishing how reliable it is. Let us take two very different examples. The story of Noah’s Ark is so well known it need not be repeated. When the ark was built, Australia had not been discovered and marsupials were unknown. But if the whole world flooded, how did kangaroos survive? And if all of today’s life is descended from just two survivors of the flood, how is it that all life is not desperately inbred? Yet the story continues to be told and the deeper meaning is that critical thinking is a minority activity.

Moving forward in time, we can look at the excess mortality in various European countries as collated by EuroMOMO. We see that in early 2020, the excess mortality curves for the UK and Sweden occurred at the same time, had the same shape and duration. Yet UK had lockdown and Sweden didn’t. Swedes were allowed to travel, schools and universities remained open.

Understanding, then, stems from looking at the world using science with its integrity, to find what laws are known, looking at technology to see what has been made that works, looking at mathematics, to see how well the world has been modelled and calling on philosophy in the shape of critical thinking to see how rigorous it all was.

That is the approach I intend to take in this look at waves in the hope that all readers will be able to take something away that can be relied upon. The astute reader will already have realised that this approach has absolutely nothing in common with the approach of the politician. One of the most serious problems of our technological age is that the politician is ill-equipped to deal with it and literally cannot communicate with scientists, technologists or critical thinkers.

Nevil Shute wrote about this in 1954, in connection with the deadly crash of the R101 airship in 1930, but what he said remains true.