THE WAVE EQUATION - MATHEMATICS AND THE VIEW OF THE WORLD IN EARLY MODERN TIMES - The Remarkable Role of Evolution in the Making of Mathematics - Mathematics and the Real World

Mathematics and the Real World: The Remarkable Role of Evolution in the Making of Mathematics (2014)

CHAPTER III. MATHEMATICS AND THE VIEW OF THE WORLD IN EARLY MODERN TIMES

22. THE WAVE EQUATION

The laws of nature set out in the form of the relation between functions and their rates of change, that is, using differential equations, have, since Newton, become the major instrument used to understand nature through mathematics. Newton laid the foundation, and from his days until today scientists use his method and propose new equations to describe additional natural phenomena. If and when the experimental results corroborate the correctness of the equation, it is customary to name the equation after the scientist who proposed it. The following is a partial list of equations mentioned in the titles of lectures given recently in mathematical conferences in which I participated: Euler equation, Riccati equation, Navier-Stokes equation, Korteweg-de Vries equation, Burgers equation, Smoluchowski equation, Euler-Lagrange equation, Lyapunov equation, Bellman equation, Hamilton-Jacobi equation, Lotka-Volterra equation, Schrödinger equation, Kuramoto-Sivashinsky equation, Cucker-Smale equation, and so on. Each equation has its story and its use. Generally, the equations are formulated in terms of the relation between a function or a set of functions and their derivatives or integrals. The equations implement Newton's laws, sometimes in conjunction with other laws of nature, such as the law of conservation of energy or the law of conservation of matter. The names of the equations show that scientists have been using differential equations since Newton and continue to do so, and much remains to be done.

In this section we will present one equation that connects the distant past with the current era. This equation played a crucial role in the additional revolution in the relation between mathematics and nature, a revolution that we will discuss in the next chapter. The equation is not named after its formulator but after what it describes, namely, the wave equation. We are particularly interested in one specific aspect of it, and that is the string equation. No mathematical background is needed to understand what follows, but it does call for patience and tolerance for the mathematical symbols (and these can also be skipped without impairing the understanding of the text).

Undulations, or wavelike motion, can be seen in many of the materials around us, from the actual waves in the sea to the vibrations of string or membranes stretched to different extents. As we stated in the previous section, the equation that describes the motion of a spring or a pendulum was put forward as early as in Newton's days. In discussing the oscillation of a stretched piece of string or the movement of the waves in the sea, the situation is slightly more complex as the height of the wave changes from one place to another, and it also changes with time. That is, to describe a wave requires a function of time and place. The height of the wave may be notated by u, and for every location x and time t, the quantity u(x, t) will represent the height of the wave at that location and at that time. We can relate to the rate of change of the height of the wave according to the time at a fixed location, and according to the location at a fixed point in time. The rate of change of the function according to location at a fixed time (imagine the profile of a wave at a given moment in time) is written in mathematics as ∂xu(x, t), and it is called the partial derivative by location (the notation ∂ was introduced to mathematics by the Marquis de Condorcet in 1770; some attribute it to the mathematician Adrien-Marie Legendre in 1786). Similarly, the expression ∂tu(x, t) indicates the rate of change of the function according to time (imagine the increase and decrease in the height of the wave at a given location). The second derivatives, that is to say the rate of change of the function of the rate of change, are written in mathematics as ∂xxu(x, t), and ∂ttu(x, t), respectively (see the diagram below of an advancing wave). The height of the wave varies with time and location. The law of its change will be given by a differential equation.

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Early versions of differential equations describing waves were first suggested by Leonhard Euler (1707–1783) in 1734 and by Jean D’Alembert (1717–1783) in 1743. The breakthrough in the understanding of waves in general, and the vibrations of a taut string in particular, came in 1746 and 1748, when D’Alembert and then Euler published different solutions to the equation. It eventually transpired that the two solutions were in fact the same solution in different forms. For the sake of completeness we will give the equation itself:

ttu(x, t) = c2xxu(x, t)

In ordinary language this says that the acceleration of the height of the wave according to time changes in proportion to the speed at which the rate of change of the height of the wave changes at the location where it is measured.

When the solutions to this equation were being studied, a surprising link was found with discoveries made by the Greeks, of course with no mathematical explanation, regarding the length of a piece of string and the note it produces when it is plucked. Without going into the technical details, we will just state that the relevant solutions of the equation describing the vibrations of a piece of string of length L, whose ends are fixed and unmoving, are a mixture of the sine and cosine functions that we have already encountered and that were known to the Greeks. Again for completeness, we set out a general form of solutions:

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There is an infinite number of solutions, as for every natural number n (and for all coefficients αn and βn) the formula provides a particular solution. It can be proved mathematically that every solution of the equation will be the sum of such particular solutions. The formula itself will not interest readers for whom mathematics and its use is not their profession, but two conclusions drawn from it are related directly both to the distant past and to future developments as follows.

Note that the length L of the string appears in the above expressions only in the denominator. The practical significance of this mathematical fact is that when the length of the string is halved, the frequency of the vibrations (the speed at which the sine and cosine functions change) is doubled. Thus, with a delay of some two thousand years, mathematics solved the mystery of the source of the finding dating back to Pythagoras, that when the length of the string is shortened by a half, the note it produces rises by exactly a full octave. The string equation connects a full octave, which the ear can identify naturally, with doubling the speed of vibration of the string. In a similar fashion, it is possible to obtain other findings from Pythagoras's time regarding the pitch of a note and the speed of vibration of the string.

Moreover, the equation shows that the vibration of the string consists of the sum of the vibrations shown in the formula for every value of n equal to 1, 2, 3, and so on. These vibrations have a “clean” frequency given by images for natural n. These frequencies are called natural, or characteristic, frequencies of the string. They will play a central role in the modern description of nature, which we will present in the next chapter.