Jökull - 01.12.1977, Qupperneq 84

and hence the angular frequency
a>, = (32)
The pressure impulse response hp (S,t) of an
unperturbed static fluid is obtained by assum-
ing causal conditions and that the fluid surface
2 is hit at t = 0+ by a 8-like pressure pulse
which is centered at R, viz.,
Po (S.t) = 8 (S—R) 8+ (t) (33)
or
MS.t) = (l/gp)8(S-R)8+(t) (34)
and hence
K (S.t) = (l/gp) 8+ (t) 2Uj (S) Uj* (R) (35)
i
where 8+ (t) is the timelike delta function
centered at t = 0 + .
Let the impulse hp be expanded
hp = S aj (t) uj (S), t > 0 (36)
j
and insert this and h0 into equation (27). The
operation with H given by (14) is straightfor-
ward and yields for each mode
D2aj + gXj'^aj = - (1/p) XjV2Uj* (R) 8+ (t), t > 0
(37)
aj = 0, t ^ 0
where D = d/dt and the factor exp (—Xj1/5z)
I z X 0
has been omitted.
The solution for aj is elementary and yields
aj = — (1 /p) (Xj%/g)V2Uj* (R) sin (gV*XjMt), t > 0 (38)
and hence the impulse response
hp (S,t) = - (1 /pg%) lim 2 exp (—Xj,/2z) Xjw sin (g*Xj" t) Uj (S) Uj* (R), t > 0 (39)
z f 0 J
The basin of infinite extent is of special
interest. In this case we substitute the continu-
ous variable k for the discrete j’s and the eigen-
functions and eigenvalues are
</>k (x>y) = (1/2-tt) exp [-i(kix+k2y)],
(40)
Xk = k2 = ki“ + k22.
The dispersion equation for the infinite basin
follows directly from equation (31) above by
inserting Xj’/2 = k and hence
o>2 — gk = 0 (41)
This expression gives the well known phase
velocity of gravity waves
c = (g/k),/2- (42)
The impulse response for the infinite basin is
obtained by inserting (40) into (39) and ob-
serving that the summation over j becomes an
integration over k. Placing the impulse at the
origin and taking advantage of the radial sym-
metry in k-space we obtain the well known
result (Stoker, 1957, page 160)
t > 0 (43)
hp (r.t)
■ (1 / 27rpg'4) lim J exp (—kz) J0 (kr) sin [t(gk)V2j k3/2dk,
z 10
where r is the radial coordinate in 2-
82 JÖKULL 27. ÁR