1912] APPLIED TO PHYSICAL PROBLEMS 133
components with somewhat different values of u may limit the periodicity to a comparatively small range. Conversely, a prolonged periodicity is associated with an approach to discontinuity in the values of C or S.
The complete curve representing </> (x) will in general include features of various lengths reckoned along oc, and a feature of any particular length is associated with values of u grouped round a corresponding centre. For some purposes we may wish to smooth the curve by eliminating small features. One way of effecting this is to substitute everywhere for <£ (x) the mean of the values of cf> (x) in the neighbourhood of x, viz.
x+a
tWdx, ..............................(6)
&
the range (2a) of integration being chosen suitably. With use of (5) we find for (6)
1 [x+a l r00 sin ua
^- I d>(x)da}=-\ du-------f0cosux 4- Ssinux}, .......(7)
2ajx_a vv vr.y ua l }
differing from the right-hand member of (5) merely by the introduction of the factor sin ua -f- ua. The effect of this factor under the integral sign is to diminish the importance of values of u which exceed trja and gradually to annul the influence of still larger values. If we are content to speak very roughly, we may say that the process of averaging on the left is equivalent to the omission in Fourier's integral of the values of u which exceed 7r/2a.
We may imagine the process of averaging to be repeated once or more times upon (6). At each step a new factor sin ua ~- ua is introduced under the integral sign. After a number of such operations the integral becomes practically independent of all values of u for which ua is not small.
In (6) the average is taken in the simplest way with respect to x, so that every part of the range 2a contributes equally (fig. 1). Other and perhaps
Fig. 1. Fig. 2. Fig. 3.
better methods of smoothing may be proposed in which a preponderance is given to the central parts. For example we may take (fig. 2)
+*)+*(*-*)}# ................... (8)
o
From (5) we find that (8) is equivalent to
2 f * , 1 - cos ua ,n a . , ,0v
— du - — : — \Ccosux+ S sm ux , ................. (9)
TrJo v?a? l j /ue of u, infer of course from (5) a correspondingly important periodic term in If the large value of C on: S is limited to a very small range of u, t periodicity of <f> extends to a large range of as; otherwise the interference