'66*4 Mr. R. Hargreaves on the Difference between 



separate parts for incident and reflected waves ; for the 

 .first 



Z,(l}*i(a 1 >+V--ci i )'+i(Xi^+Y 1 J !-Z 1 ^), . (46) 



and an equal value for the second. 



To evaluate this in terms of E x = ^(a r 2 + by + c L 2 ) 

 -+i(X 1 2 + Y 1 8 + Z 1 '0 = X 1 3 + y 1 * + Z 1 2 , use 



c 1 2 + VZ^ = (Z 1 Y 1 - mi X 1 )^ + arX 1 -f W 2 1 Y I ) 2 



= (X l 2 +Y 1 2 )(l-n 1 2 )=(E 1 -Z/)(l^ 1 ^ 

 or c 1 2 + Z 1 2 = E 1 (l-» 1 2 ), and so Z^(l) = w 1 2 E 1 . . . (5) 



The pressure 2E X % 2 of both waves is given by (4a). 

 The flux of energy has a property like Z~ as to its product 

 terms ; for 



X l b 2 -Y 1 a 2 -\X 2 b 1 — Y 2 a 1 = X l b 1 — Y 1 a l — X l b 1 -\-Y 1 a 1 == 0. 



The flux is therefore assessed as the sum for the separate 

 waves : for the incident wave 



V(XA-Y,a 1 ) 



=vx 1 (» 1 x 1 -; 1 z,)-vy 1 (,» 1 z 1 -» 1 Yi)=v»,E„ 



for the reflected wave — Vn 2 E 2 , with a total flax vanishing 

 because there is no loss of energy. 



§ 2. With a moving surface of reflexion there is loss 

 of energy, and the formula? are sensibly modified. The 

 boundary conditions are now 



X—wbJY = 0, Y + wa/y-0; these involve c=0, . (6) 

 -as before, for at the moving surface 



The relative flux of energy, and the modified pressure 

 formula, are shown in the equation 



V(X/>-Ya)-/rE = *^Z r -?<<X6-Yrt)/V}, . . (7) 



valid because the difference between its members is 



Y[(X-wb/Y) (6-«?X/V)- (Y + wa/Y)(a + wY/Y)], 



a quantity which vanishes in virtue of (6). The meaning 

 of (7) is that the relative flux of energy is equal to the work 

 done on the moving face by the pressure. Since 



. Z,-^(X/>-Ya)/V=:L + X(X-R^/Y) 



+ Y(Y + wa/Y)-c\ . (8) 



