SPECTRUM ANALYSIS OF WAVES 385 



+ Jo (k) COS {c -\- 2v) t + ■■■ 

 and sin (ct) sin {k sin cl) — Ji (k) cos (c — v) t 



- Ji (k) cos {c -\- v) t 

 + /s (k) COS {c - 3v) I 



- /s (^) COS (c + 3v) t + ••• 

 .'. COS (f/ — k sin ?'/) = Jq (k) COS (c/) 



+ 7] (^) COS (c — z;) / 



- /i (y^) cos (c + v) t 

 + /z (/^) cos (c - 2tO / 

 + J2 (k) cos (c + 2tO / 

 + /s (k) cos (c - 3z') t 



- J3 (k) cos (c + 3zO / H 



APPENDIX C 



In this Appendix the spectrum of a train of rectangular pulses of length 

 2(L + AL) recurring every T seconds, will be found from the spectrum of a 

 single pulse of this train. 



For the single pulse at any frequency/, 



gin ^ -.sin 2^f{L + AL). (1) 



x/ 



For a series of such pulses recurring with a spacing T — 1/c, then the sum of 

 spectra of the individual pulses form a Fourier series of harmonics of c. Thus 



e(t) = ^0 + Z) ^n cos liritd, (2) 



n = l 



where An is the sum of an iniinite number (one from each pulse) of infinitesi- 

 mal terms g(;/c) and g{ — nc), shown in (1). Thus 



^„ ^ 22 — sin 2Trnc{L + AL) (3) 



Tvnc 



Now to put an absolute value to the amplitudes g(/) shown in equation (1), 

 it is necessary to average them over the recurrence period of the single pulse, 

 making them infinitesimals. However, in the train of pulses recurring 

 every T — \/c seconds, the amplitude of An can be determined by averaging 

 the terms in (1) over an interval T. Then 



An = ^^sin 2Tvnc{L + AZ). (4) 



irncT 



When T = 4L = l/c, (4) reduce to 



2E . 

 — sm _, 

 wn 2 



, 2E . n-K (. . aA ... 



y4„ = — sm — ( 1 + —- j (5) 



