Differential equations are the key to the mathematical analysis of continuous systems

I had not intended to discuss the vibration of strings, as too elementary and well-known, but after investigating the rest of Lamb on Sound, I saw that there were interesting observations to be made. This study is surely one that a person wanting to learn about physics and mathematics should take up early on, since it includes so much of value, especially as a concrete example of wave motion, and of how to set up and solve differential equations. Many of the subjects treated here, such as the wave equation, resonance and resonators, radiation and so forth, are explained in more detail in other articles. I'll try to give an introduction here to the whole subject, based on vibrating strings.

Vibrating strings are the basis of an important class of musical instruments, and many of their properties have been known since ancient times. The sounds produced by strings of lengths proportional to 1, 1/2, 1/3, 1/4, ... are harmonious, for which reason this sequence is called the harmonic sequence. Mathematical analysis shows that a vibrating string produces only these harmonic sounds, which provides a reason for the use of strings to make pleasant music. This property is not common in vibrating bodies. Only strings and pipes produce harmonic overtones, which explains their predominance in music.

Strings, by themselves, affect the air very little, so they are feeble sources of sound. In order to sound loudly, they must be connected to a *sounding board* that presents a broad face to the air, and radiates acoustic waves efficiently, or perhaps to a *resonator*, such as a hollow chamber. The string is the vibrating system, with a built-in natural frequency of vibration. Energy is supplied by plucking, striking or bowing, and energy is lost to the sounding board. These elements are found in every stringed musical instrument.

The properties of strings can be learned by experiment. Père Mersenne (1588-1648) asserted that the pitch of a string was inversely proportional to the length and the square root of the density, and directly proportional to the square root of the tension. This is sufficient for practical purposes, but intellectually unsatisfying, and no stepping-stone to understanding more difficult things. It would be better if the properties of strings could be derived by reasoning from some general principles of wide application. Not only would this strengthen the validity of the principles, but would give practice in applying them. As a by-product, we would better know strings.

At the end of the 17th century, Newton showed that changes in motion were only caused by the action of forces, and the acceleration produced by a force was equal to the force divided by the mass of the body. He, and Leibniz, devised a convenient means of working with rates of change--the Calculus--and showed that the phenomena of the Solar System were understandable by mathematics, notably the orbits of the mysterious comets. If x is the coordinate of a body of mass m, then the acceleration is d^{2}x/dt^{2}, and the *equation of motion* is md^{2}x/dt^{2} = F(x), where F(x) is the force acting on the body when it is located at x. This is an ordinary differential equation, whose solution is some function x = x(t) that gives the position of the body as a function of time. By solution we mean, of course, that the equation of motion reduces to an identity if we substitute x(t) and carry out the differentiation.

For example, if F(x) = -kx (such a force would be provided by a spring pulling the body towards x = 0), then d^{2}x/dt^{2} + (k/m)x = 0. If we try a solution Ce^{iωt} (if you are not comfortable with exponentials and the imaginary unit i, use C cos ωt instead), then we find -ω^{2} + k/m = 0, or ω = 2πf = (k/m)^{1/2}. f is the frequency of oscillation, and we have related it to the physical properties of the system. The *kinetic energy* T of the body is m(dx/dt)^{2}/2, and its *potential energy* V is kx^{2}/2. In an oscillator such as this, the energy is continually in transit between the two forms, but its total is constant. Prove this by substituting the solution in T + V. Incidentally, the force F(x) = -dV/dx, and the *momentum* m(dx/dt) = dT/d(dx/dt). These concepts are the stuff of elementary particle mechanics (not elementary-particle mechanics!).

The quantities in these equations must be expressed in a consistent system of units (which is the price we pay for the convenience of precise equations in place of stating everything by proportions). One such system is: m - gram (g), x - cm, t - second (s), F - dyne (dy), T and V - erg. The usual acceleration of gravity g = 981 cm/s^{2}, so a mass of 1 g has a weight of 981 dyne at the earth's surface. 10^{5} dyne = 1 N, and 10^{7} erg = 1 J, in terms of the usual SI units. For conversion to American units, 454 g = 1 lb, and 25.4 mm = 1 in. Here, g = 32.2 ft/s^{2}. Understanding units is better than blindly trying to use one system without understanding.

Let us finally confront the string. We think of it as stretched along the x-axis, with its ends fixed a distance L cm apart. It is uniform, and has a linear density of ρ gm/cm. It has no stiffness--that is, it can bend effortlessly--but perhaps can stretch a little. If it were absolutely rigid, it could not depart from a straight line, and would not vibrate. We allow it enough stretch, or enough give in the supports, to be deflected sideways a small amount. In fact, we assume that it is under a tension P dyne at all times. The configuration of the string can be specified by giving its sideways displacement y as a function of x and t, y = y(x,t). Even if the ends are fixed, a small sideways displacement does not change the tension P appreciably. The idealization of the system we have just carried out is an essential part of our argument, not just hand-waving. An attempt to reproduce exactly a real string would make the problem too complicated to solve. Every problem in mathematical physics is simplified, and simplification without changing the essence of the problem is the mark of the master.

When we attempt to apply Newton's methods, we are initially baffled by the lack of particles of mass m to which we apply forces F. In fact, the first attempts to analyze the string replaced it by a chain of N small masses m = ρL/N. This has some interesting connections with the vibration of bodies considered as composed of molecules, but makes the present problem too fussy. In the 18th century, Leonhard Euler (1707-1783) and Joseph Louis Lagrange (1736-1813) led the advance of mathematical analysis, establishing much of the modern notation, and showing how to treat continuous systems from first principles, not as assemblies of discrete particles.

The key to the new method is to isolate a portion of the continuous system, small enough to pass for a particle, to consider the forces acting on it due to the surrounding parts of the system as well as external forces, and to write its equation of motion. An element of a string is shown in the Figure. This is called a *free-body diagram*, and is a creation of great value. The element shown, of mass ρdx, moves sideways under the components F and F + dF of the tension on its ends. The net upward force is F + dF - F = dF, and this is the change in the vertical component of P in the distance dx, or d(Pdy/dx)/dx, where the sine of the angle with the horizontal is approximated by its tangent, dy/dx. This is, of course, a very good approximation. That P is along the tangent is a consequence of the lack of stiffness of the string. For the situation in the Figure, dF is directed downward, since the element is curved so that d^{2}y/dx^{2} is negative. The magnitude of this second derivative is the reciprocal of the radius of curvature of the string at that point.

Now we can apply Newton's Law to the element, and set its rate of change of momentum, ρdx(d^{2}y/dt^{2}) equal to the net force we have just evaluated. We find d^{2}y/dt^{2} = c^{2}d^{2}y/dx^{2}, with c^{2} = P/ρ. c is the *phase velocity*. This, of course, is the celebrated *wave equation*, first studied by J. le Rond d'Alembert (1717-1783). Its remarkable properties are discussed in another article, Wave Functions. Its general solution, y(x,t) = F(x - ct) + f(x + ct), where F and f are any functions, is called d'Alembert's solution. Waves transport energy without transporting matter. The string is the simplest example of this, and quite concrete, so rewarding to study. This wave equation might best be called 'a' wave equation, since there are others with different properties. However, it describes most of the important kinds of waves found in nature, especially when generalized to make the phase velocity c a function of frequency. It is a *linear* equation, since y appears to the first power in every term, so the sum of two solutions is also a solution. This *principle of superposition* is very important and useful.

We have written the wave equation with the usual symbols for differentiation, but this is not conventional practice. When we have a function of more than one variable, such as y(x,t), it is usual to write the derivative with respect to an individual one of the independent variables with a distinctive round d, to warn that there are other variables involved, and only a partial change is represented. These are then called *partial derivatives*, and their use helps to amaze the public. There is nothing very complicated about them, so they can be used with equanimity. A differential equation in which partial derivatives appear, such as the wave equation, is called a *partial differential equation* or PDE. Such equations have a glorious power, but are in general more difficult to solve than ordinary differential equations with only one independent variable.

The crowning achievement of partial differential equations is the description of the electromagnetic field by James Clerk Maxwell (1831-1879), with eight coupled equations involving the six vector field components and the independent variables x,y,z and t. A search for a mechanical model was unsuccessful, and eventually conceded to be impossible. Nevertheless, some of the models have left traces in the definitions and units of electromagnetism, notably in the Giorgi system of units, which is the one most widely used by engineers.

The kinetic energy in the string is the integral of ρ(dy/dt)^{2}dx/2 and the potential energy the integral of P(dy/dx)^{2}dx/2. The potential energy is the work done in creating the displacement statically, or the work done in stretching the wire as the wire moves sideways. In a travelling wave f(x - ct), so the time and space derivatives are related by df/dx = f' (f' is the derivative of f with respect to its argument), df/dt = -cf', and T = ρc^{2}f'^{2}/2, V = Pf'^{2}/2 per unit length, which are equal. For a wave travelling to the left, df/dt = +cf', and the conclusions are the same. In a travelling wave, the energy is half kinetic, half potential.

To search for standing wave solutions, try functions like y = y(x)e^{iωt} (again, if you don't like the exponential, use cos ωt instead). Now, y(x) satisfies y" + ω^{2}/c^{2}y = 0, and this ordinary differential equation has the general solution y = A cos (ωx/c) + B sin (ωx/c). If we take x = 0 at the left-hand end, then B = 0, and sin (ωL/c) = 0. We have applied the *boundary conditions* y(0) = y(L) = 0. From the roots of sin x = 0, we find ω = nπc/L = (nπ/L)(P/ρ)^{1/2}, or f = (nc/2L), where n = 1,2,3,.... These results include Mersenne's rules, as well as the basis for the harmonic series. The displacement of the string is sinusoidal. Each value n corresponds to a different *mode* of vibration. In the nth mode there are (n - 1) points that do not move at all, and these are called *nodes*. The points of maximum amplitude between the nodes are called *loops*. Each point on the string moves with simple harmonic motion in time, corresponding to its own amplitude. Any motion can be expressed as the superposition of normal modes with the proper amplitudes. Very pretty!

Once we understand the solutions in the ideal case, we can go back and investigate our assumptions, asking what modifications would result if the string had a little stiffness, if the ends were not held absolutely fixed, if the string were nonuniform, if gravity were taken into account, and so on. This investigation may well suggest results and extensions that would escape our notice in a purely empirical study. We are also better-equipped to make quantitative predictions of what would happen in different cases. All this is a result of the insight provided by a mathematical analysis. Paraphrasing a famous computer, the purpose of mathematics is insight, not numbers.

When a string is *plucked*--that is, pulled aside and released--a number of modes are excited, but mainly those of lower frequency, especially the *fundamental*, or lowest frequency. In this case, the string is given an initial displacement. When a string is *struck*, a sidewise momentum is given to a small interval, producing an initial velocity. If the disturbance is very sharp and localized, many modes are excited, with a more even distribution than in plucking. A mode with a node at the point of plucking or striking is not excited to any degree. The higher modes give the sound produced its *quality*, and so long as only a few higher modes are excited, the result is pleasing. These higher modes are called *harmonics* for this reason. Another name is *overtone*. The tone made by a string is always changed by changing the length of the string, since changing the tension is difficult, and changing the linear density impossible.

A piano must cover a wide range of tones, say from 40 to 4000 Hz, and this would require the strings for the lowest tones to be 100 times longer than the strings for the highest tones, if they were all the same, and had the same tension. The strings for the graver tones are wrapped with a coil of wire to make them heavier without an increase in stiffness. Multiple strings are used to produce greater volume. The hammers strike near the ends of the strings so that harmonics are excited, but the harmonics above the sixth are discouraged (they are not pleasing). The best place for striking seems to be at about 1/7 of the length. The hammers themselves are padded to spread the impulse and ensure the predominance of the fundamental tone. The control of string tension is very important; any yielding causes the instrument to go 'flat.'

A whistling is sometimes heard when standing near a telegraph line in the wind. The ignorant once thought this was a message going down the wire. The pitch increases with the speed of the wind, and decreases for thicker wires. These sounds are *aeolian tones*, produced by a dynamic instability in the flow around a wire. Friction at the boundary layer at the wire causes vortices to be shed, first to one side, and then to the other. An empirical formula for the frequency is f = 0.195V/D, where V is the speed of the wind, and D is the diameter of the wire. The shedding of the vortices causes a force on the wire, first to one side, then to the other. The force and oscillation are not in the direction of the wind, as is often believed. The motion is communicated to the wire support, and may be radiated from there. The vortices themselves cause more disturbance than the mere oscillation of the wire would without resonance. In cases of resonance, the amplitude of oscillation can become large. For a wire of 1/6" diameter, the frequency is 20.6V(mph) Hz. A 30 mph wind would make a sound of about 600 Hz. Strouhal's experiments gave a constant of 0.185. Rayleigh shows that the constant is more generally a function of VD/ν (the Reynolds number), where ν is the kinematic viscosity.

H. Lamb, *The Dynamical Theory of Sound*, 2nd ed. (London: Edward Arnold, 1925), Chapter II.

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Composed by J. B. Calvert

Created 27 June 2000

Last revised 2 July 2000