INFL. or DIFFUSION ON THE PRoPAG. oF BovNpARY Waves. 15 
hence h’ and A" are expressed in terms of h: 
h h 
IL. h' = eee m T N — :j dus — Eoo Ic E (uS 
! 11 Se 
O0, 0 9 o OU NO Fae 
which expressions, as is necessary, are identically satisfied for 
0 57 (ores wt EM 
ON ce) eal NE ENTER 
With the definitions contained in I) and II) the equation determining 
the modified wave-velocity is finally: 
9 
9? 
ILI. y kV 
g 
— gps (e' — e") +! coth kh’ + g" coth kh” = 0 , 
an equation whose validity is expressly confined to cases of such slight 
j 
diffusion that the value v: ealeulated from it shows only a slight rela- 
tive difference from the value 10). 
6. The only root of III which is of interest to us is easily com- 
puted approximately. We may, for instance, introduce the constant: 
k(g' coth kh' + e" coth kh") 
D-—-. 7 NOSA 
: (Gi = 
IU) 
Ars g 
= 0 — p" [25 É 
where V, is the uncorrected wave-velocity. The corrected velocity V 
is then determined by the formula: 
g  k(g' coth kh' + o" coth kh") | g — 1 
[AC Bone) Fort 
There is consequently a diminution of velocity in the ratio VI — I’, 
a result which may be summed up as follows: 
Waves on a slightly diffused boundary are propagated as if moving 
according to the law of Stokes in a fluid of the same total depth h as the 
