number and K iμ ðlrÞ is the Macdonald function with imaginary index satisfying the
modified
parametric
Bessel
equation
LK iμ ðlrÞ = 0,
where
L = r
2 ∂
2
∂r 2 + r
∂
r∂r + ðμ
2
− r
2 l
2
Þ. We note that function K iμ ðlrÞ is real if the values of μ
are real and argument lr is positive. Hence, we write the delta function δðr − r 0 Þ
using a pair of direct and inverse Kantorovich-Lebedev transformations [5, 6]
FðμÞ =
Z + ∞
0
K i μ ðxÞ
f ðxÞ
x
dx, f ðxÞ =
2
π 2
Z + ∞
0
shðπμÞK i μ ðxÞFðμÞμdμ .
This implies the expansion of the delta function (completeness condition) in the
form
δðr − r 0 Þ =
2
r π 2
Z + ∞
0
shðπμÞK i μ ðlrÞ Ki μ ðlr 0 Þμdμ:
ð2:11Þ
We seek the solution of problem (2.7) in the form
Pðr, φ, lÞ =
2q
π 2
Z + ∞
0
shðπμÞK iμ ðlrÞ Kiμðlr0ÞΦμðμÞμdμ,
ð2:12Þ
where the function of the angular variable Φ μ ðφÞ is still unknown. Substituting
(2.11) and (2.12) into (2.7), we obtain the boundary-value problem for determining
this function
d
2 Φ μ ðφÞ
dφ 2
+ μ
2
Φ μ ðφÞ = − δðr − r 0 Þ,
dΦ μ ð0Þ
dφ =
dΦ μ ðφ r Þ
dφ
= 0.
ð2:13Þ
It follows from (2.13) that Φ μ ðφÞ is the angular Green function of the form
Φ μ ðφÞ = −
1
μ 2 φ r
−
2
φ r
∑
∞
n = 1
cosðφμ n Þ cosðφ 0 μ n Þ
μ 2 − μ 2
n
, μ n = 2π n ̸ ln
c + γ
c − γ
, n ≥ 1.
ð2:14Þ
In the expression for Pðr, φ, lÞ in (2.12), we consider a single wave mode ðn ≥ 1Þ
P n ðr, φ, lÞ = −
4q cosðφμ n Þ cosðφ 0 μ n Þ
φ r π 2
Z + ∞
0
shðπμÞ Ki μ ðlrÞK iμ ðlr 0 Þμdμ
μ 2 − μ 2
n
. ð2:15Þ
Internal Gravity Waves in Horizontally Inhomogeneous Ocean
119