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

5. EXACT GREEN’S FUNCTION
5.1. Monochromatic Green’s function
For ω > N, rescaling the coordinates according to
rh' = ω
N rh and z' = ω2 – N2 1/2
N
z (5.1)
reduces equation (3.15) for the monochromatic Green’s function G(r, ω) to a Helmholtz-like
equation, whose Green’s function is given for instance by Bleistein (1984 p. 177). Returning
to the original coordinate system described in figure 5 yields
G(r, ω) = 1
4πr
1/2
–
e
βr
2 ω2 – N2 cos2θ ω2 – N2 ω2 – N2 1/2
ω2 – N2 cos2θ 1/2 . (5.2)
As implied by the Pierce-Hurley radiation condition, the analytic continuation of this result over
the lower half of the complex ω plane provides the value of the Green’s function on the whole
real ω axis. The branch cuts emanating from the branch points ± N, ± N⏐cos θ⏐ are taken as
extending vertically upwards, as shown in figure 6. Thus, the phase of complex square roots
of the form (ω 2 – ω 0 2 ) 1/2 has the following behaviour on the real axis:
ph ω 2 – ω 0 2 1/2 = 0 ω > ω0 , (5.3a)
= – π/2 ω < ω0 , (5.3b)
= – π ω < – ω0 . (5.3c)
Denoting from now on complex square roots, defined by (5.3), as powers 1/2 and real
square roots by a square root symbol, we obtain for the Green’s function at every real
frequency
G(r, ω) = 1
4πr e
– βr ω
2
2 – N2 cos2θ ω2 – N2 ω2 – N2 ω2 – N2 cos2θ = i
Internal wave generation. 1. Green’s function 15
sgn ω
4πr e
– i βr
2
ω 2 – N 2 cos 2 θ
N 2 – ω 2
N 2 – ω 2 ω 2 – N 2 cos 2 θ
sgn ω
ω > N , (5.4a)
N cos θ < ω < N , (5.4b)