[1] See G. P. Harsdorfer, _Math. und philos. Erquickstunden_
(Nuremberg, 1677), ii. 147.
[2] _Musicus_ [Greek: autodidaktos] (Erfurt, 1738), p. 70.
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 1/2l 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 (S
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 [phi]([xi]) is a periodic function of the variable [xi] having
a period p, then it may be expanded as follows:
__[oo] 2i[pi][xi] __[oo] 2i[pi][xi]
[phi]([xi]) = A0 + \ ^i A_i cos ---------- + \ ^i B_i sin ----------. (1)
/__1 p /__1 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, A_i, B_i. These are
_ _
1 /p 2 /p 2i[pi][xi]
A0 = -- | [phi]([xi])d[xi]; A_i = -- | [phi]([xi]) cos ---------- d[xi];
p _/0 p _/0 p
_
2 /p 2i[pi][xi]
B_i = -- | [phi]([xi]) sin ---------- d[xi].
p _/0 p
This part of the theorem may be verified at once by multiplying both
sides of (1) by d[xi], by cos (2i[pi][xi]/p)/d[xi] or by sin
(2i[pi][xi]/p)/d[xi], and in each case integrating from 0 to p.
The series is evidently single-valued for any given value of [xi]. It
cannot therefore represent a function of [xi] which has more than one
value, or which becomes imaginary for any value of [xi]. It is
convergent, approaching to the true value of [phi]([xi]) for all
values of [xi] such that if [xi] 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 [xi] in the form (1) is valid, see FOURIER'S
SERIES. An account of the contrivances for mechanical calculation of
the coefficients A_i, B_i ... 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 [f](t) is known theoretically to be
of the form
[f](t) = A0 + A1 cos n1t + B1 sin n1t + A2 cos n2t + B2 sin n2t + ... (2)
where the periods 2[pi]/n1, 2[pi]/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 [f](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
2[pi]/n1, 2[pi]/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
[dP]^2y [dP]^2y
[rho] ------- = T -------, (3)
[dP]t^2 [dP]x^2
where T is the tension, and [rho] the line-density. In a "normal mode"
of vibration y will vary as e^(int), so that
[dP]^2y
------- + k^2y = 0, (4)
[dP]x^2
where
k^2 = n^2[rho]/T. (5)
If [rho], 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 = s[pi], [s = 1, 2, 3, ...]. (7)
This determines the various _normal modes_ of free vibration, the
corresponding periods (2[pi]/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 = [f](x), can be reproduced by a series of
the type
[pi]x 2[pi]x 3[pi]x
[f](x) = B1 sin ----- + B2 sin ------ + B3 sin ------ + ... (8)
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 [rho] 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) e^(int) (9)
where u is a solution of the equation
d^2u n^2[rho]
---- + -------- u = 0. (10)
dx^2 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
[f](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
_
/l
| [rho] u_r(x) u_s(x) dx = 0. (12)
_/0
This enables us to determine the coefficients, thus
_ _
/l / /l
C_r = | [rho][f](x)u_r(x) dx / | [rho] {u_r(x)}^2 dx. (13)
_/0 / _/0
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. Poincare, 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.)
HARMONICHORD, an ingenious kind of upright piano, in which the strings
were set in vibration not by the blow of the hammer but by indirectly
transmitted friction. The harmonichord, one of the many attempts to fuse
piano and violin, was invented by Johann Gottfried and Johann Friedrich
Kaufmann (father and son) in Saxony at the beginning of the 19th
century, when the craze for new and ingenious musical instruments was at
its height. The case was of the variety known as _giraffe_. The space
under the keyboard was enclosed, a knee-hold being left in which were
two pedals used to set in rotation a large wooden cylinder fixed just
behind the keyboard over the levers, and covered with a roll-top similar
to those of modern office desks. The cylinder (in some specimens covered
with chamois leather) tapered towards the treble-end. When a key was
depressed, a little tongue of wood, one end of which stopped the string,
was pressed against the revolving cylinder, and the vibrations produced
by friction were transmitted to the string and reinforced as in piano
and violin by the soundboard. The adjustment of the parts and the
velocity of the cylinder required delicacy and great nicety, for if the
little wooden tongues rested too lightly upon the cylinder or the
strings, harmonics were produced, and the note jumped to the octave or
twelfth. Sometimes when chords were played the touch became so heavy
that two performers were required, as in the early medieval organistrum,
the prototype of the harmonichord. Carl Maria von Weber must have had
some opinion of the possibilities of the harmonichord, which in tone
resembled the glass harmonica, since he composed for it a concerto with
orchestral accompaniment. (K. S.)
HARMONIUM (Fr. _harmonium_, _orgue expressif_; Ger. _Physharmonika_,
_Harmonium_), a wind keyboard instrument, a small organ without pipes,
furnished with free reeds. Both the harmonium and its later development,
the American organ, are known as free-reed instruments, the musical
tones being produced by tongues of brass, technically termed "vibrators"
(Fr. _anche libre_; Ger. _durchschlagende Zunge_; Ital. _ancia_ or
_lingua libera_). The vibrator is fixed over an oblong, rectangular
frame, through which it swings freely backwards and forwards like a
pendulum while vibrating, whereas the beating reeds (similar to those of
the clarinet family), used in church organs, cover the entire orifice,
beating against the sides at each vibration. A reed or vibrator, set in
periodic motion by impact of a current of air, produces a corresponding
succession of air puffs, the rapidity of which determines the pitch of
the musical note. There is an essential difference between the harmonium
and the American organ in the direction of this current; in the former
the wind apparatus forces the current upwards, and in the latter sucks
it downwards, whence it becomes desirable to separate in description
these varieties of free-reed instruments.
[Illustration: By courtesy of Metzler & Co.