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

8. INTERNAL W AVE RADIA TION BY A PULSA TING SPHERE
As the simplest example of a source of internal waves with finite dimensions we
consider in this section a sphere of radius a, on which surface the normal velocity U(t) is
imposed. Particular attention will be paid to the far-field modelling of the sphere by the
equivalent point mass source, a monopole of strength 4πa 2 U(t) (cf. Pierce 1981 § 4.1 and
4.3). From the preceding section, non-Boussinesq effects upon internal wave generation
appear to be satisfactorily accounted for by the consideration of point sources. Accordingly,
our investigations will be restricted to the Boussinesq case.
Although Hendershott (1969) has already treated both monochromatic and transient
Boussinesq pulsations of a sphere, his results are invalidated by the error mentioned in
appendix C. Appleby & Crighton (1987) also dealt with monochromatic internal waves, and
Grimshaw (1969) with transient internal waves, generated by a sphere undergoing
monopolar or dipolar motion. The present work is complementary to theirs, in that the link
between point and extended sources is investigated.
8.1. Exact solution
Under the Boussinesq approximation the internal wave equation (3.10), and the
condition of fixed radial velocity at the surface of the sphere, become (subscripts B will from
now on be omitted)
∂ 2
∂
r
∂t2 ∂r
∂ 2
∂t 2 Δ + N2 Δh ψ(r, t) = 0 , (8.1)
+ N 2 rh ∂
∂rh
ψ(r, t) = aU(t) at r = a . (8.2)
We solve them by the method we used to calculate the Green’s function; then we deduce the
pressure and velocity fields from the internal potential, by differentiating it according to the
Boussinesq versions of (3.11)-(3.12). Thus we consider first monochromatic waves, described
by
Internal wave generation. 1. Green’s function 34
ω 2 – N 2 rh ∂
ω 2 – N 2 Δh + ω
∂rh
+ ω 2 z ∂
∂z
2 ∂2
ψ(r, ω) = 0 , (8.3)
∂z2 ψ(r, ω) = – aU(ω) at r = a , (8.4)