Waves generated in rotating fluids by travelling forcing effects

Abstract

The two-dimensional wave pattern produced in a homogeneous rotating fluid by a forcing effect oscillating with a frequency σ′₀ and travelling with a uniform speed U along a line inclined to the axis of rotation at an arbitrary angle α is studied following Lighthill's technique. It is shown how the far field changes with α and σ′₀.

For all σ′₀ < 2Ω, except for σ′₀ = 2Ω sin α (Ω being the angular velocity of the fluid), the forcing effect excites two systems of waves. When σ′₀ → 2Ω sin α one of these systems spreads out, influencing the upstream side while the other shrinks in the downstream direction. This upstream influence is to the left or to the right of the line of motion of the forcing effect (the forcing line) according as σ′₀ − 2Ω sin α[lg ] 0 and increases as σ′₀ − 2Ω sin α decreases. For σ′₀ > 2Ω there is only a single system propagating downstream. As α varies these systems undergo a kind of rotation retaining the main features. α ≠ 0 or ½π makes the pattern asymmetric about the forcing line while a non-zero σ′₀ splits the steady-case identical wave systems into two, which are otherwise coincident.

When σ′₀ = 2Ω sin α the forcing effect excites straight unattenuated waves of fixed frequency travelling both ahead and behind in a ‘column’ parallel to the forcing line and enclosing it. Also there are two other systems, which propagate without penetrating into an upstream wedge. It is shown that this ‘column’ is the counterpart of the ‘Taylor column’.