INFL. OF DIFFUSION on THE PRoPAG. or BouNpARY Waves. 13 
2) The diffusion is slightly advanced, so that 
x" M 
f? dz n 
0 0 
h (o' LE e") ? h(o' DE o") 
where h is the smallest of h’ and A", are very small fractions of unity. 
5. Returning now to the velocity-equation 9), and approximating 
on the same principles, we find: 
"s är 
TH EISE | g i^. de M z | J 7 e 
LS == (coth Ke, ae NE dz da -- k(1 En ky? coth kh ) ox dz 
where the integrals are easily caleulated from our expressions above. 
Hence: 
A x" 
JU k(coth HUE xix) e"] : dz + (x = una) [ode 5 
16) ee Uu 
I'z-— = coth kh' + am) ef = dz — (x —- le dz. 
The sum /'-- /" gives the correction to be applied to Stokes' 
equation 10). The expression may be simplified: to begin with we 
note that the terms k/d dz, kfedz are to be neglected because of 
the long wave-length. Further simplification is attained by disposing 
conveniently of the zero-level y = 0. This may be done in different 
ways; we proceed as follows: 
Suppose that, when finally calculated, the percentage difference 
between yz and Stokes value is slight (a supposition which conse- 
quently is more rigorous than that introduced above); we may write: 
o" (coth EI E = s oe’ coth kh' + o' En (coth kh = = : 
