Jökull - 01.12.1977, Page 83

Horizontal fluid surface be X- We consider
noqtion of the fluid with the velocity vector
v (P,t) = (u,v,w) (P,t) and the pressure p (P,t) at
the point P = (x,y,z) in F and time t. Under
motion the free íluid surface is t). The Eulerian
equations of motion for the fluid F take the
forrn
pDv/Dt = -Vp + pg (16)
'vhere D/Dt is the material derivative and g is
the acceleration of gravity. Assuming no fluid
sources these equations have to be adjoined by
the condition of incompressibility
V-v = 0. (17)
Disregarding surface tension, the boundary
conditions are
Dp/Dt | j_0 = 0, Ponfl (18)
and
n • v = 0, P on r, (19)
where n is the outward normal to T-
In the case of small amplitude motion, we
linearize (16) by neglecting the product terms
on the left and assume that p = pb + p where
ph is the hydrostatic pressure and p (P,t) is a
small perturbation pressure such that |p|«|p|,|.
We obtain on the basis of (16)
p3tv = -VP (20)
and applying (17) to (20) the standard result
-V2P = 0 (21)
follows revealing tliat p is a harmonic function
in 15. The boundary condition for p is
dp/dn = 0, P on r. (22)
Moreover, since 3zph = gp, the linearized sur-
face condition (18) reduces to
9tP + wpg = °’ p on fi- (23)
As a final step of linearization, we move the
condition (23) from Í1 to X by assuming that Í1
deviates from X by a small vertical amplitude
h (S,t) which is positive up. Tlie perturbation
pressure on X can then be approximated by
P = pgh-
The value of the vertical fluid velocity w is
assumed to be the same on X as on fl and hence
(23) reduces to
3th + w = 0. (25)
Inserting w from (25) into the vertical com-
ponent of (20) we finally obtain
p9tth = 9ZP> P °n X’ (26)
which in conjunction witli (21) and (22) is an
equation for gravity waves of infinitesimal am-
plitude. This form is, however, unsatisfactory
since it includes two dependent variables h and
p. The situation can be rectified by observing
tliat because p is liarmonic in B its cross surface
derivative at X can He expressed in terms of
the values of P on X with the help of the sur-
face operator H defined by (6). Hence, using
(24) and (6) we can eliminate p from (26). In
carrying out this final step we are also able to
generalize (26) by including an impressed pres-
sure source term in (24) represented by an
impressed surface amplitude ho (S,t). The re-
sulting equation is
(9tt + gH) h = f, S in X’ (27)
tvhere the source term is
f = —gHh0. (28)
Using the inverse of H defined by (15), we can
also restate (27) in the following form
(K9tt + g) h = —gh0. (29)
The integro-differential equation (27) is our
main result, the gravity wave equation to be
derived. As stated, it applies to waves of in-
finitesimal amplitude on deep water.
NORMAL MODES, IMPULSE RESPONSE
AND DISPERSION RELATION
The gravity wave normal modes are obtained
lty assuming for the jth mode
hj = iq exp (icojt) (30)
and inserting in (27) with f = 0 we obtain the
simple mode relation
-cof + gXjV2 = 0 (31)
JÖKULL27. ÁR 81
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