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

4. PLANE INTERNAL W AVES
Numerous writers have investigated the properties of Boussinesq (e.g. Lighthill 1978
§ 4.1 and 4.4, Brekhovskikh & Goncharov 1985 § 10.4) and non-Boussinesq (e.g. Tolstoy
1973 § 2.4, Liu & Yeh 1971) plane internal waves and have analysed, via the group velocity
theory, the generation of internal waves by point sources. We summarize here their
conclusions, for the interpretation of forthcoming results.
4.1. Boussinesq case
The dispersion relation for plane monochromatic Boussinesq internal waves (k is the
wavevector, k h its horizontal projection, and k and k h their moduli),
ω = N kh
k
, (4.1)
implies a frequency ω < N, an arbitrary wavelength λ and an inclination θ 0 = arc cos (ω/N) of the
planes of constant phase to the vertical. The phase velocity c φ with which these planes move
and the group velocity c g with which energy propagates are perpendicular, according to
cg = N
k kz
kh
k
k
cφ = ω k k k with c φ = ω k = N k kh
k = N k cos θ0 , (4.2)
× k
k × ez with cg = N2 – ω2 k
= N
k kz
k
= N
k sin θ0 . (4.3)
Thus, energy propagates along the planes of constant phase. Conversely k satisfies
By virtue of
Internal wave generation. 1. Green’s function 12
k = N cg cg
cg
cg
×
cg × ez sgn cgz . (4.4)
v = P
ρ0 cg
cg
cg
, (4.5)
fluid particles move along straight-line paths also parallel to the wavecrests.
A monochromatic point source consequently radiates Boussinesq internal waves along
directions inclined at the angle θ 0 = arc cos (ω/N) to the vertical, on a “St Andrew’s Cross” in
two dimensions and a cone with vertical axis, hereafter called characteristic, in three dimensions
(figure 1). Surfaces of constant phase are parallel to this cone. They move toward the level of