272 Passage of Waves through Apertures in Plane Screens, 



A comparison of these conditions with those by which (46) 

 was determined shows that in the present case 



When log i in the denominator of (59) may be omitted, the 

 realized form is that expressed by (47), and this corresponds 

 to 



Xm = X P = cos (nt-kx) (60; 



Various Applications. 



Of the eight problems, whose solutions have now been 

 given, four have an immediate application to aerial vibrations, 

 viz. those in which the condition on the walls is d<f>/dn = 0. 

 The symbol </> then denotes the velocity-potential, and the 

 condition expresses simply that the fluid does not penetrate 

 the boundary. The four problems relating to two dimensions 

 have also a direct application to electrical vibrations, if we 

 suppose that the thin material constituting the screen (or the 

 blade) is a perfect conductor. For if R denote the electro- 

 motive intensity parallel to z, the condition at the face of the 

 conductor is R=0 ; so that if R be written for yjr in (53) , (59), 

 we have the solutions for a narrow aperture in an infinite 

 screen, and for a narrow reflecting blade respectively, cor- 

 responding to the incident wave H = e~ ikx . A narrow aper- 

 ture parallel to the electric vibrations transmits very much 

 less than is reflected by a conductor elongated in the same 

 direction. 



The two other solutions relative to two dimensions find 

 electrical application if we identify <£ with c, the component 

 of magnetic intensity parallel to z. For when the other com- 

 ponents a and b are zero, the condition to be satisfied at the 

 face of a conductor is dc/dn = 0. Thus (46), (57) apply to 

 incident vibrations represented by c = e~ ikx . In this case the 

 slit transmits much more than the blade reflects. 



It may be remarked that in general problems of electrical 

 vibration in two dimensions have simple acoustical analogues*. 

 As an example we may refer to the reflexion of plane electric 

 waves incident perpendicularly upon a corrugated surface, the 

 acoustical analogue of which is treated in ' Theory of Sound,' 

 2nd ed. § 272 a, and to the reflexion of electric waves from a 

 conducting cylinder (§ 343). 



* The comparison is not limited to the case of perfect conductors, but 

 applies also when the obstacles, being non-conductors, differ from the 

 surrounding medium in specific inductive capacity, or in magnetic 

 permeability, or in botb properties. 



