Harmonic_Analysis

Harmonic Analysis, in mathematics, the name given by Sir William Thomson (Lord Kelvin) and P. G. Tait in their treatise on Natural Philosophy to a general method of investigating physical questions, the earliest applications of which seem to have been suggested by the study of the vibrations of strings and the analysis of these vibrations into their fundamental tone and its harmonics or overtones.

The motion of a uniform stretched string fixed at both ends is a periodic motion; that is to say, after a certain interval of time, called the fundamental period of the motion, the form of the string and the velocity of every part of it are the same as before, provided that the energy of the motion has not been sensibly dissipated during the period.

There are two distinct methods of investigating the motion of a uniform stretched string. One of these may be called the wave method, and the other the harmonic method. The wave method is founded on the theorem that in a stretched string of infinite length a wave of any form may be propagated in either direction with a certain velocity, V, which we may define as the “velocity of propagation.” If a wave of any form travelling in the positive direction meets another travelling in the opposite direction, the form of which is such that the lines joining corresponding points of the two waves are all bisected in a fixed point in the line of the string, then the point of the string corresponding to this point will remain fixed, while the two waves pass it in opposite directions. If we now suppose that the form of the waves travelling in the positive direction is periodic, that is to say, that after the wave has travelled forward a distance l, the position of every particle of the string is the same as it was at first, then l is called the wave-length, and the time of travelling a wave-length is called the periodic time, which we shall denote by T, so that l = VT.

If we now suppose a set of waves similar to these, but reversed in position, to be travelling in the opposite direction, there will be a series of points, distant ½l from each other, at which there will be no motion of the string; it will therefore make no difference to the motion of the string if we suppose the string fastened to fixed supports at any two of these points, and we may then suppose the parts of the string beyond these points to be removed, as it cannot affect the motion of the part which is between them. We have thus arrived at the case of a uniform string stretched between two fixed supports, and we conclude that the motion of the string may be completely represented as the resultant of two sets of periodic waves travelling in opposite directions, their wave-lengths being either twice the distance between the fixed points or a submultiple of this wave-length, and the form of these waves, subject to this condition, being perfectly arbitrary.

To make the problem a definite one, we may suppose the initial displacement and velocity of every particle of the string given in terms of its distance from one end of the string, and from these data it is easy to calculate the form which is common to all the travelling waves. The form of the string at any subsequent time may then be deduced by calculating the positions of the two sets of waves at that time, and compounding their displacements.

Thus in the wave method the actual motion of the string is considered as the resultant of two wave motions, neither of which is of itself, and without the other, consistent with the condition that the ends of the string are fixed. Each of the wave motions is periodic with a wave-length equal to twice the distance between the fixed points, and the one set of waves is the reverse of the other in respect of displacement and velocity and direction of propagation; but, subject to these conditions, the form of the wave is perfectly arbitrary. The motion of a particle of the string, being determined by the two waves which pass over it in opposite directions, is of an equally arbitrary type.

In the harmonic method, on the other hand, the motion of the string is regarded as compounded of a series of vibratory motions (normal modes of vibration), which may be infinite in number, but each of which is perfectly definite in type, and is in fact a particular solution of the problem of the motion of a string with its ends fixed.

A simple harmonic motion is thus defined by Thomson and Tait (§ 53):—When a point Q moves uniformly in a circle, the perpendicular QP, drawn from its position at any instant to a fixed diameter AA′ of the circle, intersects the diameter in a point P whose position changes by a simple harmonic motion.

The amplitude of a simple harmonic motion is the range on one side or the other of the middle point of the course.

The period of a simple harmonic motion is the time which elapses from any instant until the moving-point again moves in the same direction through the same position.

The phase of a simple harmonic motion at any instant is the fraction of the whole period which has elapsed since the moving-point last passed through its middle position in the positive direction.

In the case of the stretched string, it is only in certain particular cases that the motion of a particle of the string is a simple harmonic motion. In these particular cases the form of the string at any instant is that of a curve of sines having the line joining the fixed 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  
  
(1)

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 + ...

(2)

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  
  
(3)

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  
  
(4)

where

k² = n²ρ/T.

(5)

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

(6)

provided

kl = sπ, [s = 1, 2, 3, ...].

(7)

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  
  
(8)

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

(9)

where u is a solution of the equation

d²u | + | n²ρ | u = 0.  
---|---|---|---  
dx² | T  
  
(10)

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

∫l0 ρur(x) us(x) dx = 0.

(12)

This enables us to determine the coefficients, thus

Cr = ∫l0 ρƒ(x) ur (x)dx ÷ ∫10 ρ {ur(x)}² dx.

(13)

The extension to spaces of two or three dimensions, or to cases where there is more than one dependent variable, must be passed over. The mathematical theories of acoustics, heat-conduction, elasticity, induction of electric currents, and so on, furnish an indefinite supply of examples, and have suggested in some cases methods which have a very wide application. Thus the transverse vibrations of a circular membrane lead to the theory of Bessel’s Functions; the oscillations of a spherical sheet of air suggest the theory of expansions in spherical harmonics, and so forth. The physical, or intuitional, theory of such methods has naturally always been in advance of the mathematical. From the latter point of view only a few isolated questions of the kind had, until quite recently, been treated in a rigorous and satisfactory manner. A more general and comprehensive method, which seems to derive some of its inspiration from physical considerations, has, however, at length been inaugurated, and has been vigorously cultivated in recent years by D. Hilbert, H. Poincaré, I. Fredholm, E. Picard and others.

References.—Schuster’s method for detecting hidden periodicities is explained in Terrestrial Magnetism (Chicago, 1898), 3, p. 13; Camb. Trans. (1900), 18, p. 107; Proc. Roy. Soc. (1906), 77, p. 136. The general question of expanding an arbitrary function in a series of functions of special types is treated most fully from the physical point of view in Lord Rayleigh’s Theory of Sound (2nd ed., London, 1894-1896). An excellent detailed historical account of the matter from the mathematical side is given by H. Burkhardt, Entwicklungen nach oscillierenden Funktionen (Leipzig, 1901). A sketch of the more recent mathematical developments is given by H. Bateman, Proc. Lond. Math. Soc. (2), 4, p. 90, with copious references.

(H. Lb.)

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