HARMONIC ANALYSIS. Many physical phenomena, such as sound waves, alternating electric currents, tides, machine motions, etc., may be periodic in character. Such motions can be measured at a number of successive values of the independent variable, usually the time; and these data, or a curve plotted from these data, will represent a function of that independent variable. Thus the ordinate of the curve at any point is Generally the mathematical expression for f (x) will be 'unknown.
However, with the periodic functions to be found in nature, f(x) can be expressed as the sum of a number of sine and cosine terms. Such a sum is known as a Fourier series (q.v.) and the determination of the coefficients of these terms is called harmonic analysis. The term harmonic analysis can also be used in the broader sense of the analysis into any kind of periodic com ponents, such as spherical har monics, cylindrical harmonics, tesseral harmonics, etc. How ever, in this treatment we shall confine ourselves to the develop ment into Fourier series, which development has found more ex tensive practical application than any of the others. One of the terms of such a series has a period equal to that of f(x) and is called the fundamental. Other terms have shorter periods, which are aliquot parts of the funda mental. These are called harmonics.
Harmonic analysis is of great value in mathe matics, in physics, and in engineering. A sound wave, for instance, can cause the vibration of a thin diaphragm, whose motion is photographed in the form of a continuous curve by the use of the Phonodeik of D. C. Miller, or by other methods. The shape of the resulting curve will depend on the quality of the sound and will generally be quite complicated. However, it can be analysed into its fundamental and harmonics or overtones by use of Fourier series. It is the number and magnitude of these har monics which determine the quality of the sound, and therefore such investigations are of great value in the scientific study of music and speech. Similar methods are used extensively by the electrical engineer in the study of alternating currents. In me chanical engineering they can be used in the investigation of valve motion and other mechanical movements. Harmonic analysis is also used in the study of tidal records ; and, by the use of such information as it affords, the prediction of tides is possible. (See TmES. ) All the above applications have been for cases where the move ments are truly periodic. In a large class of other phenomena, such as the weather, sun spots, magnetic deviations, river flow, atmospheric strays in radio reception, etc., the fundamental period is usually not evident and the periods of the harmonics are not aliquot parts of the fundamental. Even in such cases, however, modified methods are being used to some extent, the analysis still being made into a series of sine or cosine terms.
Any ordinary non-periodic curve of finite length can also be analysed by the harmonic method, the scale being changed in the i-direction so that the length is 2 it units. An example taken from D. C. Miller's excellent book, The Science of Musical Sounds, is shown in fig. I a, where a profile is analyzed and found to have the equation : These harmonics were then combined, giving the result shown in fig. Ib.
In 1822 Fourier showed that a func tion y = f(x) could be expressed between the limits of x = o and x = 27 by the series: where k = I, 2, 3, • • • Equation (I) can be written in the alternate form: y =
sin (x+01) +c2 sin 2 (x+4 2) +ca sin 3
+ . . ., (3) where ck =1i
, Ok = tanl
(4) ak Assume that a record has been obtained of some periodic phenomenon expressed as a curve or as a set of data which can be plotted and called f (x). Even although f (x) cannot be ex pressed as a simple function, equation (I) can be used to repre sent it, and the coefficients ak and bk can be determined. It will be necesary first to find the period of the function; that is, the distance between corresponding points on successive waves. This distance will be called 2irradians or 36o°, and can then be divided into any convenient number of parts, say n. The first n ordinates are measured and their values substituted in equation (I), giving n equations in the n undetermined coefficients. These equations can be solved to obtain ak and bk. The n equations have the form: yk = bo+bl cosxk+b2cos 2xk+ • • .
2xk+ • • • , where k=o, I, 2, 3, . . . (n—I) successively; yk is the kth ordinate of the curve, and xk is the corresponding abscissa ex pressed in degrees. From these it can be shown that A curve plotted from (I) using the coefficients (5) will pass through all the values yk exactly, but probably will not coincide with the experimental curve at other points. Obviously, the use of a larger number of terms will increase the accuracy. In some applications the function can be very closely approximated by a few terms. In other cases, particularly if the wave has sharp corners, a large number of terms is necessary to get a sufficiently accurate expression. It is often possible to tell something about the coefficients by inspection, thus simplifying the mathematical work of analysis. If positive and negative loops of the curve are the same, then even harmonics are absent. This is practically always the case with alternating currents. Also, if the curve for x < o is thz, reflection in the y-axis of the curve for x > o, there will be no sine terms; while if the curve is symmetric about the origin, there are no cosine terms.
The use of equation (5) is the basis of numerous so-called schedule methods of analysis. These are merely short cuts for solving the equations (5) , many of the routine multiplications being combined and tabulated in a schedule. Of these, the best-known is probably that of Runge. It can best be explained by an example. For this a six-point schedule has been selected. Only odd harmonics are considered, and the zero is taken where the curve crosses the x-axis. Then the six equations are : 3
y1 cos 3o° 4-
cos 6o° +
cos 9o° +
cos I
+
cos I
3b3 = yl cos
cos
cos
cos
cos
355 = y1 cos I
cos
cos
cos
cos 7 5o°,
sin 30°+
sin 6o°+
sin 90°+
sin I 20°+
sin I 5o°,
sin
sin
sin
sin
sin 45o°,
sin
sin
sin 75o°.
can be expressed as functions of 3o, 6o, and 90 degrees: 3b1=
sin
sin 60°, 3b3 = — (y2 — y4) sin 9o°, 3b5=
sin
sin 6o°, 3a1=
sin
sin
sin 9o°, 3a3=
sin
3a5 = (y1+y5) sin
— (y2+y4) sin
sin 9o°.
It will be noticed that, except for
all the coefficients occur as sums or differences. In the schedule given above,
has been added so that the curve does not have to cross the x-axis at x = o. If f(x) = o when x = o, as in the previous equations, then of course yo disappears. The work can be tabulated as shown above. A numerical example given by Grover is as follows: This has been plotted in fig. 2. The heavy curve is the one to be analyzed. The six points used in the above schedule are enclosed by circles. Since the wave is nearly sinusoidal, the analysis results in a large fundamental sine curve and very small higher har monics. Synthesis gives the light curve of fig. 2. It will be noted that this passes through all six points, but obviously it cannot be expected to coincide with the original curve throughout. By This motion is communicated to the cylinder. Pure rolling motion is used throughout, a principle which Kelvin considered very important.
In the Kelvin harmonic analyser the sphere S is rolled back and forth along the disc by an amount equal to the ordinates of the curve being traced. Therefore, if the sphere is a distance r taking a larger number of points and, therefore, a different schedule, a much closer approximation would be obtained.
A method of selected ordinates has been devised by Fischer Hinnen. This appears to require slightly less computation than the Runge method, but has the disadvantage that a new set of equally-spaced ordinates must be measured for each pair of coefficients. Other methods of computation of the coefficients have been worked out by Steinmetz, S. P. Thompson and others. Various graphical methods have also been devised. C. S. Slichter, for instance, uses a special graph paper which introduces the sine or cosine factor without computation. The area under the curve is measured with an ordinary planimeter. It requires the replotting of the curve for each coefficient, and it is doubtful if any time is saved over the schedule method. Other graphical methods have been used by Clif ford, Perry, Harrison, Ashworth, Beattie and Rottenburg.
Mechanical Methods.—The above methods all require a large amount of labour, espe cially if many coefficients are to be determined. If much of the work is to be done, therefore, some machine method is advis able. Equation (2) is used in most of the mechanical an alyzers.
y = f (x) of fig. 3a is to be analyzed, it is merely necessary to multiply the ordi nates by sin x (fig. 3b) and ob tain the area under the resulting curve (fig. 3c). This gives f(x) sin x dx, which is proportional to
The other coefficients are obtained similarly. Most of the machines, therefore, consist of some means of multiplying the ordinates of the curve by sin kx or cos kx and integrating the product. (See MATHEMATICAL
