  Linear waves in bounded inviscid fluids do not generally form normal modes
with regular eigenfunctions. Examples are provided by inertial waves in a
rotating fluid contained in a spherical annulus, and internal gravity waves in
a stratified fluid contained in a tank with a non-rectangular cross-section.
For wave frequencies in the ranges of interest, the inviscid linearized
equations are spatially hyperbolic and their characteristic rays are typically
focused on to wave attractors. When these systems experience periodic forcing,
for example of tidal origin, the response of the fluid can become localized in
the neighbourhood of a wave attractor. In this paper I define a prototypical
problem of this form and construct analytically the long-term response to a
periodic body force in the asymptotic limit of small viscosity. The vorticity
of the fluid is localized in a detached shear layer close to the wave attractor
in such a way that the total rate of dissipation of energy is asymptotically
independent of the viscosity. I further demonstrate that the same asymptotic
dissipation rate is obtained if a non-viscous damping force is substituted for
the Navier-Stokes viscosity. I discuss the application of these results to the
problem of tidal forcing in giant planets and stars, where the excitation and
dissipation of inertial waves may make a dominant, or at least important,
contribution to the orbital and spin evolution.
