INFL. OF DIFFUSION ON THE PRopac. or BouNDARY WAVES. 9 
Proceeding to the case of a gradual transition, our equations 
S) and 9) are at once simplified because of our assumption of a trans- 
€) 
ition layer of thickness small compared with the wave-length T. 
Since d is appreciable only within levels y for which ky is very small, 
it follows that 
ky . d 
is always very small, and that approximately 
feo de =/e"“0 de = KK): (E à 
Thus we put the integration variable 2 equal to zero in any exponen- 
tial multiplied by 9 (or :)J. Hence 8) reduces to: 
, 
, Sinh k(y — A") SO HN S RE fö do 
Dan nr (on mm) 
+ (sinh. ky + 25 cosh k fo de 
| S Y m ky? cosn t athe 
0 
x 
| . m da 
+ sinh ky(coth OM re d) J^: de dz 
x" 
0 
— k sinh kylt IE iw coth kh") Jo cee 
0 
To resolve this equation, we put: 
sinh k(y—h") _ E 
PSA San + cosh ky.g + sinh ky. v , 
where q and v are numerically small functions, which are to repre- 
sent the slight deviation from the state considered by Stokes. Observ- 
ing that within the integrals we write: 
lo dq 
= eap = coth kh” + kw + = 5 
