424 Diffraction by Screen bounded by Straight Edge. 



not be fundamental, and it should be possible to modify the 

 mathematics to allow for an infinite train of waves. The 

 most important c;ise is that of a simple harmonic disturbance, 

 and it is found that in this case our solution reduces to 

 Sommerfeld's and is therefore justified. 



If the approaching waves are represented by 



V = cos *(<#-#) (14) 



we have to investigate 



We have 



If 2 

 /= - I cos /c(ct — y — f sec 2 ct)da. . . (15) 



77 Jo 



/=Real Part of - P >«*-*-* «"**> da 

 71 Jo 



& \ -let; sec -2 a 



Now 



e 



l o 



7? JO 6 



i 



I „ — ikv ^-ikv tan 2 a 



Jo 



ia. 



2a =l-4*l e- tKVSec2a sec 2 adv 



= 1-**1 e- tKV e- tKVt&na sec 2 adv. 

 'o 

 Hence 



£ V^7T 



— 7/Ct 1 " 



2 V /cu 



r/ ^ 



J ^ 7t n 



# b rta = — — ik l 

 ^ *o 



and, finally, 

 f=\^ K (ct-y)-\^/^o,\j^ K {ct-y-v)] 



dv 



• • (16) 

 As this formula can be identified with that of Sommerfeld 

 as quoted by Mr. Har greaves *, the demonstration is 

 complete. 



The result is of interest, not only as containing the first 

 complete solution of a diffraction problem but also as 

 showing that FresnePs integrals, devised for an approxi- 

 mate solution of the problem, suffice for the complete one. 

 This aspect of the subject is discussed at length in Drude's 

 ' Optics.' 



* The change in the direction of the axis of y should he noted. 



