##### Document Text Contents

Page 47

r

19.

v/aves, their effective value being zero for a period, only odd

harmonics need he considered. Rewriting the equation and dis-

regerding the even harmonics the following equation is obtained.

2) = ai cos 6" + as cos Z t blq cos 5 <9 . , . , + h, sin e +•

"b^ sin 3 ^ + b5 sin be.,.

It is assumed; that the mean horizontal axis is mid-

way between the highest and lov/est points of the wave; that the

zero point of the v;ave is chosen where the vave crosses this

axis; and that the position and negative portions of the waves

are idenical so that only a half wave - 180<^ - need be considered.

The unknown coefficients a.-i^a.2f^^ . . . .b^^.bg jb^. . . must be found.

These will then be combined to determine R^.Rg.Rg ••• according

to the equation i—~ r

E - /a2+

and the angles ^, * ^ 3 * ^ s >' •' by the equation

= tan -^-^

ai

Where RiyRg etc are the amplitudes of the harraomics, and#i»$2 ^'''O*

I

are the angles of lag with respect to the complex wave. When

i

these are determined the first equation v/ill appear in the form.

Y = R cos - $) R3 cos (3 - ^3) R5 cos (5^ ~ $5)

Professor Runge's scheme was to divide the period into

4 n parts since in one per iod of ar sine v;ave there corresponds

four values, one in each quadrant, for - , of the same sine value

for every angle. These four were grouped together for a single

multiplication. To separate the odd and even orders of harmonics

r

19.

v/aves, their effective value being zero for a period, only odd

harmonics need he considered. Rewriting the equation and dis-

regerding the even harmonics the following equation is obtained.

2) = ai cos 6" + as cos Z t blq cos 5 <9 . , . , + h, sin e +•

"b^ sin 3 ^ + b5 sin be.,.

It is assumed; that the mean horizontal axis is mid-

way between the highest and lov/est points of the wave; that the

zero point of the v;ave is chosen where the vave crosses this

axis; and that the position and negative portions of the waves

are idenical so that only a half wave - 180<^ - need be considered.

The unknown coefficients a.-i^a.2f^^ . . . .b^^.bg jb^. . . must be found.

These will then be combined to determine R^.Rg.Rg ••• according

to the equation i—~ r

E - /a2+

and the angles ^, * ^ 3 * ^ s >' •' by the equation

= tan -^-^

ai

Where RiyRg etc are the amplitudes of the harraomics, and#i»$2 ^'''O*

I

are the angles of lag with respect to the complex wave. When

i

these are determined the first equation v/ill appear in the form.

Y = R cos - $) R3 cos (3 - ^3) R5 cos (5^ ~ $5)

Professor Runge's scheme was to divide the period into

4 n parts since in one per iod of ar sine v;ave there corresponds

four values, one in each quadrant, for - , of the same sine value

for every angle. These four were grouped together for a single

multiplication. To separate the odd and even orders of harmonics