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

the source, but disappear as soon as they have left the cone within which all the energy is
confined. The motion of fluid particles is radial and also located on the cone. A confirmation of
these features comes from the experiments of Mowbray & Rarity (1967), McLaren et al.
(1973) and Kamachi & Honji (1988). Of particular importance are internal waves of near-
buoyancy frequency. They are confined at the vertical from the source and do not propagate,
for their group velocity vanishes. They instead are vertical oscillations of fluid particles. This too
is confirmed by the experiments of Gordon & Stevenson (1972).
The Boussinesq internal waves generated by a point impulsive source propagate
away from it at the group velocity c g = r/t , where r represents the position with respect to the
source and t the time elapsed since the impulse. The frequency and wavevector in a direction
inclined at an angle θ to the upward vertical follow from (4.4) and (4.1), as
ω = N cos θ and k = Nt
r r r × r r × ez sgn z . (4.6)
The surfaces of constant phase Φ = ωt – k.r = Nt⏐cos θ⏐ are conical (figure 2) and move
toward the level of the source at the decreasing phase speed
c φ = r
t
cotan θ , (4.7)
in agreement with experiments by Stevenson (1973). The wavelength
λ = 2π
k
= 2π
Nt r
sin θ
is constant on toroidal surfaces with vertical axes and radii Ntλ/2π. It decays with time, as the
wavecrests multiply. Again, the motion of fluid particles is radial.
4.2. Non-Boussinesq case
Non-Boussinesq effects transform the dispersion relation into
ω = N
kh
k 2 + β 2 /4
(4.8)
. (4.9)
The wavenumber surface is no more a cone but an hyperboloidal surface of revolution, with
asymptotes making the angle π/2 – θ 0 with the vertical (figure 3). The group velocity points
along its normal and is given by
Internal wave generation. 1. Green’s function 13