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

= – 1
4πr e
– βr N
2
2 cos2θ – ω2 N2 – ω2 N2 – ω2 N2 cos2θ – ω2 ω < N cos θ . (5.4c)
In accordance with the group velocity theory, internal waves propagate for frequencies N⏐cos
θ⏐ < ⏐ω⏐ < N and possess there the phase (4.13). Out of this frequency band they are
evanescent. Under the Boussinesq approximation their wavy character disappears, apart
from a phase jump of π/2 on the characteristic cone ⏐cos θ⏐ = ω/N where the Green’s function
diverges. Although this phenomenon is consistent with the confinement of Boussinesq internal
waves on the characteristic cone, it also shows the inadequacy of the Green’s function to
represent them there.
The preceding method has been introduced by Pierce (1963), later rederived by
Hurley (1972) and recently developed by Appleby & Crighton (1986, 1987). “Classical”
investigations of the monochromatic Green’s function of internal waves involve, however, a
somewhat different Fourier analysis in the space domain. G(k, ω), defined by
readily follows from (3.15) as
G(r, t) = 1
2π 4 dω d3 k G(k, ω) e i ωt – k.r , (5.5)
G(k, ω) =
1
ω 2 k 2 + β 2 /4 – N 2 k h 2
, (5.6)
but remains indeterminate for frequencies and wavenumbers related by the dispersion
relation (4.9). The radiation condition makes it determinate, by imposing that
G(k, ω) = 1
ω – iε 2 k2 + β 2 /4 – N2 lim
ε → 0+
2
kh = pv
, (5.7a)
1
ω2 k2 + β 2 /4 – N2 2
kh + iπ δ ω2 k2 + β2
4 – N2 kh 2 sgn ω . (5.7b)
The principal value of the singular generalized function (5.6) is consequently added a
combination of free waves represented by the Dirac delta, so as to eliminate any wave which
does not originate at the source.
Internal wave generation. 1. Green’s function 16
G(k, ω) is first inverted along one space direction by the joint use of the residue