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From the 18th century onward, huge strides were made in the application of mathematical ideas to problems arising in the physical sciences: heat, sound, light, fluiddynamics, elasticity, electricity, and magnetism. The complicated interplay between themathematics and its applications led to many new discoveries in both. The main unifying theme in much of this work is the notion of a partial differential equation.
Musical origins
Harmony
The problem that sparked the entire development was deceptively simple, and it was surprisingly far removed from any serious practical application, coming not so much from the physical sciences but from music: What is the appropriate mathematical description of the motion of aviolin string? The Pythagorean cult of ancient Greece also found inspiration in music, especially musicalharmony. They experimented with the notes sounded by strings of various lengths, and one of their great discoveries was that two notes sound pleasing together, or harmonious, if the lengths of the corresponding strings are in simple numerical ratios such as 2:1 or 3:2. It took more than twomillennia before mathematics could explain why these ratios arise naturally from the motion of elastic strings.
Normal modes
Probably the earliest major result was obtained in 1714 by the English mathematicianBrook Taylor, who calculated the fundamental vibrational frequency of a violin string in terms of its length, tension, and density. The ancient Greeks knew that a vibrating string can produce many different musical notes, depending on the position of the nodes, or rest-points. Today it is known thatmusical pitch is governed by the frequency of the vibration—the number of complete cycles of vibrations every second. The faster the string moves, the higher the frequency and the higher the note that it produces. For the fundamental frequency, only the end points are at rest. If the string has a node at its centre, then it produces a note at exactly double the frequency (heard by thehuman ear as one octave higher); and the more nodes there are, the higher the frequency of the note. These highervibrations are called overtones.
The vibrations produced are standing waves. That is, the shape of the string at any instant is the same, except that it is stretched or compressed in a direction at right angles to its length. The maximum amount of stretching is the amplitude of the wave, which physically determines how loud the note sounds. The waveforms shown are sinusoidal in shape—given by the sinefunction from trigonometry—and their amplitudes vary sinusoidally with time. Standing waves of this simple kind are called normal modes. Their frequencies are integer multiples of a single fundamental frequency—the mathematical source of the Pythagoreans’ simple numerical ratios.
Partial derivatives
In 1746 the French mathematicianJean Le Rond d’Alembert showed that the full story is not quite that simple. There are many vibrations of a violin string that are not normal modes. In fact, d’Alembert proved that the shape of the wave at timet = 0 can bearbitrary.
Imagine a string of lengthl, stretched along thex-axis from (0, 0) to (l, 0), and suppose that at timet the point (x, 0) is displaced by an amounty(x,t) in they-direction (seefigure). The functiony(x,t)—or, more briefly, justy—is a function of two variables; that is, it depends not on a single variablet but uponx as well. If some value forx is selected and kept fixed, it is still possible fort to vary; so a functionf(t) can be defined byf(t) =y(x,t) for this fixedx. Thederivativef′(t) of this function is called the partial derivative ofy with respect tot; and the procedure that produces it is called partialdifferentiation with respect tot. The partial derivative off with respect tot is written ∂y/∂t, where the symbol ∂ is a special form of the letterd reserved for this particular operation. Analternative, simpler notation isyt. Analogously, fixingt instead ofx gives the partial derivative ofy with respect tox, written ∂y/∂x oryx. In both cases, the way to calculate a partial derivative is to treat all other variables as constants and then find the usual derivative of the resulting function with respect to the chosen variable. For example, ify(x,t) =x2 +t3, thenyt = 3t2 andyx = 2x.
Bothyx andyt are again functions of the two variablesx andt, so they in turn can be partiallydifferentiated with respect to eitherx ort. The partial derivative ofyt with respect tot is writtenytt or ∂2y/∂t2; the partial derivative ofyt with respect tox is writtenytx or ∂2y/∂t∂x; and so on. Henceforth the simpler subscript notation will be used.
D’Alembert’s wave equation
D’Alembert’s wave equation takes the formytt =c2yxx. (9) Herec is aconstant related to the stiffness of the string. The physical interpretation of (9) is that the acceleration (ytt) of a small piece of the string is proportional to the tension (yxx) within it. Because theequation involves partialderivatives, it is known as a partialdifferential equation—in contrast to the previously described differential equations, which, involving derivatives with respect to only one variable, are called ordinary differential equations. Since partial differentiation is applied twice (for instance, to getytt fromy), the equation is said to be of second order.
In order to specify physically realistic solutions, d’Alembert’s wave equation must be supplemented by boundary conditions, which express the fact that the ends of a violin string are fixed. Here the boundary conditions take the formy(0,t) = 0 and
y(l,t) = 0 for allt. (10) D’Alembert showed that the general solution to (10) isy(x,t) =f(x +ct) +g(x −ct) (11) wheref andg are arbitrary functions (of one variable). The physical interpretation of this solution is thatf represents the shape of a wave that travels with speedc along thex-axis in the negative direction, whileg represents the shape of a wave that travels along thex-axis in the positive direction. The general solution is a superposition of two traveling waves, producing the complex waveform shown in thefigure.
In order to satisfy the boundary conditions given in (10), the functionsf andg must be related by the equationsf(−ct) +g(ct) = 0 and
f(l −ct) +g(l +ct) = 0 for allt. These equations imply thatg = −f, thatf is an odd function—one satisfyingf(−u) = −f(u)—and thatf is periodic with period 2l, meaning thatf(u + 2l) =f(u) for allu. Notice that the part off lying betweenx = 0 andx =l is arbitrary, which corresponds to the physical fact that a violin string can be started vibrating from any shape whatsoever (subject to its ends being fixed). In particular, its shape need not be sinusoidal, proving that solutions other than normal modes can occur.