INFL. or DIFFUSION on THE PRoPAG. or BouNDARY WAVES. 5 
ait ys ae dU 
= p TE Dna V dy 
ga do d 
k(U — Y) dy dy 
(BOUM — he = Viens 
as a system of differential equations determining the way in which 
the wave-motion depends on the vertical co-ordinate. For simplification 
we write: 
a = aU — V), B- Bu m: 
MT EM : | ao Che = 
N Tr 
and by eliminating f we have: 
d , dû £ FAN one 
A) 4; CU D) — Bove = 9 qa. 
where now U is written instead of U — V. 
With regard to the boundary-conditions to be combined with the 
differential equation A), we remark that in the preceding investiga- 
tion vertical displacement is proportional to « = a(U — V), whereas 
U —V da 
p= De en is proportional to the variable part of the horizontal 
velocity. Thus, for instance, the condition of a limiting, rigid, hori- 
zontal wall is expressed in the form « = 0. 
Among problems included in the equation A) may be cited the 
stability problem offered by two currents of different densities and dif- 
ferent velocities, separated by a thin layer, through which there is a 
eontinuous transition of density and current velocity. Another problem, 
frequent in practical hydrography, demands the calculation of the ve- 
locity of propagation of boundary-waves in the interior of a fluid moving 
with a variable horizontal velocity, whose vertical variation as a rule 
may be assumed to be very small compared with the maximum den- 
sity gradient. In this latter case, the variable current velocity is ge- 
nerally determined by cireumstances escaping theoretical treatment, 
but may be considered to be known practically from actual observation. 
Nova Acta Reg. Soc. Sc. Ups., Ser. 4, Vol. 4, N. 2. Impr. *°/s 1916. 2 
