In the geometrical optics limit on an aspherical earth, the 2l+1 singlet eigenfunctions of an isolated multiplet nSl or nTl with n≪l are associated with the stable closed orbits of propagating surface waves. The long-term evolution of surface-wave trajectories is in turn governed by the great circular average ϵ¿ of the local relative phase velocity perturbation Δωlocal/ω0, where ω0 is the degenerate multiplet eigenfrequency and Δωlocal is the local eigenfrequency perturbation. Correct to first order in ϵ, the normal t to a surface wave orbital plane, averaged over an orbit, satisfies the fast gyroscope equation dt/dΔ=∇1ϵ¿¿t¿, where Δ is the angular arc length along the orbit. This equation stipulates that t¿ precesses around the contours of ϵ¿ at a rate proportional to the contour spacing.

The only paths that remain closed in the presence of the perturbation ϵ are those corresponding to the critical or stationary points where ∇1ϵ¿=0. Stable closed paths correspond to local minima and maxima of ϵ¿, while unstable paths correspond to saddle points. A singlet eigenfunction is composed of waves propagating around all the great circle paths belonging to a single precessing family, and the associated asymptotic eigenfrequency perturbation Δω/ω0 is just the value of the perticular contour of ϵ¿ around which the normal t¿ is precessing. Since the contours are closed about the local minima and maxima of ϵ¿, the singlet eigenfunctions are confined to the vicinity of the locally slow and fast great circle paths on the surface of the earth. Caustics that are the envelopes of the precessing trajectories separate regions of oscillatory behavior from regions of exponential decay. The quantization condition that selects the eigenfrequencies or eigencontours from the continuum of contours of ϵ¿ is derived from the requirement that the WKBJ singlet eigenfunctions be single valued and properly connected along the caustics. correct to first order in ϵ the condition is that the length of either caustic be equal to 2&pgr;m/(l+1/2). Alternatively, if L=(l+1/2)t¿ is regarded as the orbital angular momentum of a precessing surface wave, the condition is that ⟨Lz⟩=m where the zˆ axis coincides with a local minimum or maximum of ϵ¿ and ⟨Lz⟩ is averaged over a full precession cycle. This latter form of the quantization condition emphasizes the gyroscopic analogy and obviously resembles the definition of angular momentum eigenstates in quantum mechanics.

All but one of the 2l+1 asymptotic eigenfrequencies on a nonrotating earth are doubly degenerate because of the even parity of ϵ¿, but this degeneracy is removed by the earth's rotation, which splits each doublet by an additional amount Δω≈2mβ&OHgr;z, where β is the rotational splitting parameter and &OHgr;2 is the zˆ component of the angular velocity of rotation.