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Aviation History
1917
1917 - 1336.PDF
DECEMBER 20, 1917. reception.^and the average frequency of the notes emitted _ note as heard to the average frequency of the note emitted is and heard [would be the same. As it is, however, the path'if- f~ F \ V(Sj - s) ,., hpcnmpc, " hpn'- thf. nath E0.(- r) isBhorter than the path DO (=n). so that the final ' y(Sl _s) +•„(,, _ r) which becomes, ^hen^tht path isis 2'0 SCCONDS •5 10 S TIME. : - impulse overtakes the initial impulse by a period of time r* ~ r-. The time of arrival of the whole note is shortened by this amount so that the N waves emitted reach the observer ii a time Sl~ s - t~^-. The average frequency of tne^note NvV 1-0 IS Z'O infinitesimally short, so as to approximate in magnitude tu a point, , . -;-,'.-. heard is therefore---- The ratio of the r, average frequency as heard to the average frequency as emitted is therefore Producing OF to K and putting <f> for the angle KFG, we have cos CFO = cos *. Substituting, expression (5) becomesy -- which is identical with (4). This ratio is greater than1 unity, i.e., the note heard is higher than the note emitted. When D is so close to E that DE is infinitesimal, sx — s) becomes ds and (rl —;) becomes dr. Expression (1) then becomes V + v cos(p With a little reflection the reader will see that we cannot, with accuracy, talk of the frequency of the note emitted by the source in passing a point in its path, because the time the source is at any point is so short that only an infinitesimal part of a single wave-length can be emitted there. Expression (5) does not therefore, strictly speaking, represent the ratio of V-v ds Now 1 +h*, whence r dr = s ds or*,ds •4a). 1; CEO. r t (3) Substituting this value, (2) becomes V-v cos CEO Now if we produce OE to H and put <p for the1 angle HEC between the line OE produced and the line EC drawn from E in the direction in which the source is moving, we have cts * — — cos CEO. We may therefore write (3) in the form V V + v cos <(> ___Now consider the case in which*the source is moving away from the pedal point, the note being emitted between two points F. G. Let CG = S,, CF = S, OG ~»-, and OF «= r. In this case the termination of the note has to traverse a path rt loager than the path r traversed by the initial impulse. The termination of the note therefore lags behind and the time of arrival is increased by the period—^. The time of emission the frequency as heard to the frequency as emitted, of a note emitted at a point in the path of the source. It represents the limiting value of the ratio time of emission when the,time of ci" t f ltime of arrival i* and-therefore the time of arrival is v V Proceeding as before, the ratio of the average frequency of the emission becomes vanishingly small. For this reason we shall call expression (5) the Doppler Ratio and denote it by the letter R. When we are dealing with a finite portion of the path of the source, such as DE, expression (r) is the average value of the Doppler Ratio for the portion of the path under consideration. When the finite portion of the path con-1336
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