INFL. OF Dirrusion ON THE PRoPAG. or BouNpary WAVES. 7 
sinh k(y — ^") 
B" we Be BE ML d 
) ee 097 sinh EAT 
Hi 
DU do g 
+fa(z) = \cosh k(y — e) + zi sinh k(y — 2)] dz 
4 dz ( kV E ) 
A" 
sinh ky [' 
do 11 | I = 
Zn AU a (2) am (cosh k(h" — 2) -- Ey? Sinh Æ(k" — E dz , 
0 
B Sie QUA sinh kh!’ | is 
2 
i do : F 
+fe(2) de (cosh k(y — 2) + 2 sinh k(y — 3) dz 
0 
—R 
DR e (z) = (cosh k(h' + 2) — im sinh E(h' + 2) de . 
0 
It is readily verified that these equations, used for the upper and lower 
fluids respectively, define a function « which is everywhere, except 
possibly for y = 0, a solution of B, and satisfies the boundary condi- 
tions to vanish at y = A" and y = h’. For y = 0, where 9760, 
both equations give the common value c,; which, without loss of 
generality, we may put equal to unity. The only condition, possibly 
: ; é ee da 
not satisfied, is that of a continuous derivative dy for y = 0, and this 
condition is immediately formulated by the equation: 
C) — 0, coth kh” — 
A 
1 G 2 do ft | I . "m 
= eem ji) FE (cosh k(h".— 2) + zys sinh k(h" — 2) dz 
= oy coth kh’ + 
Hö 
| 1 NT. SAT! AL m : STARS 
EE sinn ii) dz (cosh k(h ES 2) — kV? sinh k(h Sim 2) de 
which now serves to determine admissible values V, answering to 
possible free waves. 
