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

so that
Internal wave generation. 1. Green’s function 38
ξ 2 ~ ω N2 – ω 2
N 2
Then the internal wave field is given by
ψ(r) ~ i
r a e– i arc cos σ+/a . (8.23)
aU0
ω 1/2 N 2 – ω 2 3/4 a r e i/2 arc cos σ+/a , (8.24)
P(r) ~ ρ0aU0 ω 1/2 N 2 – ω 2 1/4 a r e i/2 arc cos σ+/a , (8.25)
v (r) ~ U0
2
ω 1/2
N 2 – ω 2 1/4 a r a
a 2 – σ+ 2 r r e i/2 arc cos σ+/a . (8.26)
This expression makes Appleby’s & Crighton’s (1987) result for a pulsating sphere more
explicit, and is similar to Lighthill’s (1978 § 4.10) result for a distributed mass source.
The phase Φ = ωt + (1/2) arc cos (σ + /a) varies transversely, in agreement with the
experiments of McLaren et al. (1973). From its differentiation with respect to σ + (cf. (7.10)) we
deduce the transverse wavelength λ = 4π (a 2 – σ + 2 ) 1/2 ,and the phase and group velocities
cg = N2 – ω 2
k
c φ = ω k = 2 ω a2 – σ+ 2 , (8.27)
= 2 N 2 – ω 2 a 2 – σ+ 2 , (8.28)
which are zero at the edges σ + = ± a of region III and maxima halfway between them. The
definition of λ is, however, of purely academic interest, since not even a single oscillation of the
phase takes place between these edges. Only a continuous monotonic variation of π/2 in Φ is
observed, which replaces the phase jump obtained in § 5.1 between regions II and IV.
Interpreting it as a quarter of an oscillation we rather introduce, as did Appleby & Crighton
(1987), an effective wavelength “ λ” = 8a.
The pressure and velocity oscillate in phase and verify, consistently with (4.5),
v ~
P
2 ρ0 N 2 – ω 2 a 2 – σ+ 2 r r
. (8.29)
Their decrease as 1/√r, conformable to the conservation of energy for conical waves, coincides
with the experiments of McLaren et al. (1973) again. The velocity singularity at the edges of