points for its axis, and passing through these two points, and therefore
having for its wave-length either twice the length of the string
or some submultiple of this wave-length. The amplitude of the
curve of sines is a simple harmonic function of the time, the period
being either the fundamental period or some submultiple of the
fundamental period. Every one of these modes of vibration is
dynamically possible by itself, and any number of them may coexist
independently of each other.
By a proper adjustment of the initial amplitude and phase of each of these modes of vibration, so that their resultant shall represent the initial state of the string, we obtain a new representation of the whole motion of the string, in which it is seen to be the resultant of a series of simple harmonic vibrations whose periods are the fundamental period and its submultiples. The determination of the amplitudes and phases of the several simple harmonic vibrations so as to satisfy the initial conditions is an example of harmonic analysis.
We have thus two methods of solving the partial differential equation of the motion of a string. The first, which we have called the wave method, exhibits the solution in the form containing an arbitrary function, the nature of which must be determined from the initial conditions. The second, or harmonic method, leads to a series of terms involving sines and cosines, the coefficients of which have to be determined. The harmonic method may be defined in a more general manner as a method by which the solution of any actual problem may be obtained as the sum or resultant of a number of terms, each of which is a solution of a particular case of the problem. The nature of these particular cases is defined by the condition that any one of them must be conjugate to any other.
The mathematical test of conjugacy is that the energy of the system arising from two of the harmonics existing together is equal to the sum of the energy arising from the two harmonics taken separately. In other words, no part of the energy depends on the product of the amplitudes of two different harmonics. When two modes of motion of the same system are conjugate to each other, the existence of one of them does not affect the other.
The simplest case of harmonic analysis, that of which the treatment of the vibrating string is an example, is completely investigated in what is known as Fourier’s theorem.
Fourier’s theorem asserts that any periodic function of a single variable period p, which does not become infinite at any phase, can be expanded in the form of a series consisting of a constant term, together with a double series of terms, one set involving cosines and the other sines of multiples of the phase.
Thus if φ(ξ) is a periodic function of the variable ξ having a period p, then it may be expanded as follows:
| φ(ξ) = A0＋Σ∞1 i Ai cos | 2iπξ | ＋Σ∞1 i Bi sin | 2iπξ | . |
| p | p |
The part of the theorem which is most frequently required, and
which also is the easiest to investigate, is the determination of the
values of the coefficients A0, Ai, Bi. These are
| A0＝ | 1 | ∫p0 φ(ξ)dξ; Ai = | 2 | ∫p0 φ(ξ) cos | 2iπξ | dξ; Bi = | 2 | ∫p0 φ(ξ) sin | 2iπξ | dξ. |
| p | p | p | p | p |
This part of the theorem may be verified at once by multiplying both sides of (1) by dξ, by cos (2iπξ/p)/dξ or by sin (2iπξ/p)/dξ, and in each case integrating from 0 to p.
The series is evidently single-valued for any given value of ξ. It cannot therefore represent a function of ξ which has more than one value, or which becomes imaginary for any value of ξ. It is convergent, approaching to the true value of φ(ξ) for all values of ξ such that if ξ varies infinitesimally the function also varies infinitesimally.
Lord Kelvin, availing himself of the disk, globe and cylinder integrating machine invented by his brother, Professor James Thomson, constructed a machine by which eight of the integrals required for the expression of Fourier’s series can be obtained simultaneously from the recorded trace of any periodically variable quantity, such as the height of the tide, the temperature or pressure of the atmosphere, or the intensity of the different components of terrestrial magnetism. If it were not on account of the waste of time, instead of having a curve drawn by the action of the tide, and the curve afterwards acted on by the machine, the time axis of the machine itself might be driven by a clock, and the tide itself might work the second variable of the machine, but this would involve the constant presence of an expensive machine at every tidal station. (J. C. M.)
For a discussion of the restrictions under which the expansion of a periodic function of ξ in the form (1) is valid, see Fourier’s Series. An account of the contrivances for mechanical calculation of the coefficients Ai, Bi . . . is given under Calculating Machines.
A more general form of the problem of harmonic analysis presents itself in astronomy, in the theory of the tides, and in various magnetic and meteorological investigations. It may happen, for instance, that a variable quantity ƒ(t) is known theoretically to be of the form
ƒ(t)＝A0＋A1 cos n1t＋B1 sin n1t＋A2 cos n2t＋B2 sin n2t＋. . .
where the periods 2π/n1, 2π/n2, . . . of the various simple-harmonic constituents are already known with sufficient accuracy, although they may have no very simple relations to one another. The problem of determining the most probable values of the constants A0, A1, B1, A2, B2, . . . by means of a series of recorded values of the function ƒ(t) is then in principle a fairly simple one, although the actual numerical work may be laborious (see Tide). A much more difficult and delicate question arises when, as in various questions of meteorology and terrestrial magnetism, the periods 2π/n1, 2π/n2, . . . are themselves unknown to begin with, or are at most conjectural. Thus, it may be desired to ascertain whether the magnetic declination contains a periodic element synchronous with the sun’s rotation on its axis, whether any periodicities can be detected in the records of the prevalence of sun-spots, and so on. From a strictly mathematical standpoint the problem is, indeed, indeterminate, for when all the symbols are at our disposal, the representation of the observed values of a function, over a finite range of time, by means of a series of the type (2), can be effected in an infinite variety of ways. Plausible inferences can, however, be drawn, provided the proper precautions are observed. This question has been treated most systematically by Professor A. Schuster, who has devised a remarkable mathematical method, in which the action of a diffraction-grating in sorting out the various periodic constituents of a heterogeneous beam of light is closely imitated. He has further applied the method to the study of the variations of the magnetic declination, and of sun-spot records.
The question so far chiefly considered has been that of the representation of an arbitrary function of the time in terms of functions of a special type, viz. the circular functions cos nt, sin nt. This is important on dynamical grounds; but when we proceed to consider the problem of expressing an arbitrary function of space-co-ordinates in terms of functions of specified types, it appears that the preceding is only one out of an infinite variety of modes of representation which are equally entitled to consideration. Every problem of mathematical physics which leads to a linear differential equation supplies an instance. For purposes of illustration we will here take the simplest of all, viz. that of the transversal vibrations of a tense string. The equation of motion is of the form
| ρ | ∂2y | = T | ∂2y | , |
| ∂t2 | ∂x2 |
where T is the tension, and ρ the line-density. In a “normal mode” of vibration y will vary as eint, so that
| ∂2y | ＋k2y＝0, |
| ∂x2 |
where
k2＝n2ρ/T.
If ρ, and therefore k, is constant, the solution of (4) subject to the condition that y＝0 for x＝0 and x＝l is
y＝B sin kx
provided
kl＝sπ, [s＝1, 2, 3, . . .].
This determines the various normal modes of free vibration, the corresponding periods (2π/n) being given by (5) and (7). By analogy with the theory of the free vibrations of a system of finite freedom it is inferred that the most general free motions of the string can be obtained by superposition of the various normal modes, with suitable amplitudes and phases; and in particular that any arbitrary initial form of the string, say y＝ƒ(x), can be reproduced by a series of the type
| ƒ(x)＝B1 sin | πx | ＋B2 sin | 2πx | ＋B3 sin | 3πx | ＋. . . |
| l | l | l |
So far, this is merely a restatement, in mathematical language, of an argument given in the first part of this article. The series (8) may, moreover, be arrived at otherwise, as a particular case of Fourier’s theorem. But if we no longer assume the density ρ of the string to be uniform, we obtain an endless variety of new expansions, corresponding to the various laws of density which may be prescribed. The normal modes are in any case of the type
y＝Cu(x)eint
where u is a solution of the equation
| d 2u | ＋ | n2ρ | u＝0. |
| dx2 | T |
The condition that u(x) is to vanish for x＝0 and x＝l leads to a transcendental equation in n (corresponding to sin kl＝0 in the previous case). If the forms of u(x) which correspond to the various roots of this be distinguished by suffixes, we infer, on physical grounds alone, the possibility of the expansion of an arbitrary initial form of the string in a series
| ƒ(x) ＝ C1u1(x)＋C2u2(x)＋C3u3(x)＋. . . | (11) |
It may be shown further that if r and s are different we have the conjugate or orthogonal relation
| ρur(x) us(x) dx＝0. | (12) |
