Waves in parallel or swirling stratified shear flows

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Abstract
The dispersion relations for infinitesimal internal gravity waves (A) and axisymmetric waves in swirling streams (B) are considered. In both cases the mainstream may be sheared and density stratified in the transverse (vertical in case A, radial in case B) direction. The following results are proved for either case: If the maximum speed Wmax (or minimum speed Wmin) (in a meridian plane in case B) of the mainstream occurs at an interior point in the fluid, then the phase speed of any mode takes all values from the Wmax (or Wmin) to +∞ ( —∞) as the overall Richardson number λ2 varies from 0 to ∞. If Wmax(Wmin) is attained at a boundary point with finite rate of strain, there is a positive non-zero critical Richardson number below which one or both branches of the dispersion relation terminate. These results employ variational methods and correct erroneous results concerning problem B stated in Chandrasekhar's treatise on hydrodynamic stability. Furthermore, bounds are given on the group velocity for both branches of the dispersion relation. From these bounds it is shown that in the absence of reversals of the mainstream (Wmin > 0) upstream propagation of wave energy is impossible whenever upstream propagation of constant phase surfaces is impossible.
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