﻿810 Mr. A. Stephenson on the Propagation 



The observed magnitude of reflexion when the displacement 

 is at right angles to the plnne of incidence is in agreement 

 with this fact. 



When the displacement is in the plane of incidence * the 

 agreement holds, we shall find, only if the velocity of the 

 irrotational wave is the same in all media. I£ this velocity is 

 relatively large the energy dissipated in irrotational motion 

 is negligible. 



The equations of motion are 



9 dt 2 " dx \dx + dy) n dy \dy dx)' ' ' ' {? '> 



d 2 rj ___ d (d% drj\ _ d rd% __ dn\ ... ) 



p a^~dy\dx + a\j)~ n a\v\dy^dx : r ' ' ' (n * 



with p different in the two media, and at the boundary 



f, 7], -S and ~- are continuous. 

 aoc ax 



The value of p in any material medium may depend upon 

 the nature of the radiation, whether equivoluminal or irro- 

 tational. 



From (i.) and (ii.) 



Jpfdji dri\_(d 2 d 2 \(d% d V \ 

 P dt 2 \dx + dy)' W? + dy 2 )\dx + dyf ' ' * {im) 



^l d A-§l\- (<L d2 \l d % *n\ r \ 



P dt 2 \dy dx)- n \dx 2 ' h ay)\dy dx)' ' ' W 



where p' is the effective density for the irrotational wave. 

 For the incident and reflected waves 



£=— sin# cos « (^cos0-f-ysm 6 — ci) 



-f r sin 6 cos {rc( — xcos6 + y sin — ct) — a) , 



17= cos cos «(.rcos0-}-?/sin#— ct) 



+ r cos 6 cos {«:( — xcos6+y sin — ct) —a}; 



in the transmitted wave 



f 1 = — s sin 6 X cos { k ± (x cos 6 ± + y sin 6 X — c x t) — /3] , 

 7] 1 = scos6 1 cos{K 1 (xcos0i-\-ysin0 l —Cit)—/3} ; 



* The discussion due to Green (1840) is still accepted : bis conclusions 

 are erroneous. 



