Internal Wave Generation in Uniformly Stratified Fluids. 1 ... - LEGI

G(r, t) = 1
8π 2 r
+∞
– ∞
e
iωt – βr
2 ω2 – N2 cos2θ ω2 – N2 ω 2 – N 2 1/2 ω 2 – N 2 cos 2 θ
1/2
1/2 dω
. (5.13)
Then, separating by (5.4) the contributions of propagating and evanescent waves, we find
G(r, t) = 1
4π 2 r
+ 1
4π 2 r
N
N cos θ
N
∞
–
0
sin βr
2
N cos θ
ω 2 – N 2 cos 2 θ
N 2 – ω 2
N 2 – ω 2 ω 2 – N 2 cos 2 θ
e
– βr
2
– ωt
dω
ω 2 – N 2 cos 2 θ
ω 2 – N 2 cos ωt
ω 2 – N 2 ω 2 – N 2 cos 2 θ
dω
. (5.14)
Closing for t < 0 the integration contour of (5.13) by an infinite semicircle in the lower
half-plane where G(r, ω) is analytic, and applying Jordan’s lemma, we recover the causality of
the Green’s function: G(r, t) = 0 for t < 0. Separating its odd and even parts with respect to
time, we have
and (5.14) accordingly becomes
Internal wave generation. 1. Green’s function 19
G(r, t) = – H(t)
2π 2 r
G(r, t) = 2 H(t) Godd(r, t) = 2 H(t) Geven(r, t) , (5.15)
N
N cos θ
cos βr
2
ω 2 – N 2 cos 2 θ
N 2 – ω 2
N 2 – ω 2 ω 2 – N 2 cos 2 θ
sin ωt
dω
. (5.16)
Evanescent waves consequently merge with propagating waves and transform them into the
standing waves appearing in (5.16). Under the Boussinesq approximation, the relative
contributions of propagating and evanescent waves are even equal, but it must be reminded
that their propagating or evanescent character is lost in that case.