On Motions Near the Lagrange Equilibrium Point L 4 in the Case of Routh's Critical Mass Ratio

We deal with the planar restricted circular problem of three bodies. We study trajectories in a small neighborhood of the Lagrange equilibrium point L ₄ when mass ratio is close to Routh's value. In particular, we show that the case of proper degeneracy takes place and for most initial conditions trajectories are conditionally-periodic. We obtain an approximate representation of families of periodic solution emanating from the equilibrium point L ₄. We also show that in the case of instability of L ₄ the trajectories starting in a vicinity of L ₄ remain in a finite domain forever. We give an upper bound of this domain. To carry out our investigation, we analyze the dynamics of a general Hamiltonian system with two degrees of freedom in the case of 1 : 1 resonance in detail.This revised version was published online in October 2005 with corrections to the Cover Date.     

Model of wind-generated ambient noise in stratified shallow water

In order to build a rapid ocean ambient noise model adapted for a stratified shallow water, a hybrid model of normal mode method (for far field) and ray method (for near field) is suggested which combines the advantages of both methods. Since the near field of wind-generated noise is not sensitive to the sound speed profile, the sound speed profile is regarded as a constant; which makes the model rapid and accurate. The simulation results are in agreement with those of the wave model.     

Generalized Born scattering of elastic waves in 3-D media

It is well known that when a seismic wave propagates through an elastic medium with gradients in the parameters which describe it (e.g. slowness and density), energy is scattered from the incident wave generating low-frequency partial reflections. Many approximate solutions to the wave equation, e.g. geometrical ray theory (GRT), Maslov theory and Gaussian beams, do not model these signals. The problem of describing partial reflections in 1-D media has been extensively studied in the seismic literature and considerable progress has been made using iterative techniques based on WKBJ, Airy or Langer type ansätze. In this paper we derive a first-order scattering formalism to describe partial reflections in 3-D media. The correction term describing the scattered energy is developed as a volume integral over terms dependent upon the first spatial derivatives (gradients) of the parameters describing the medium and the solution. The relationship we derive could, in principle, be used as the basis for an iterative scheme but the computational expense, particularly for elastic media, will usually prohibit this approach. The result we obtain is closely related to the usual Born approximation, but differs in that the scattering term is not derived from a perturbation to a background model, but rather from the error in an approximate Green's function. We examine analytically the relationship between the results produced by the new formalism and the usual Born approximation for a medium which has no long-wavelength heterogeneities. We show that in such a case the two methods agree approximately as expected, but that in a media with heterogeneities of all wavelengths the new gradient scattering formalism is superior. We establish analytically the connection between the formalism developed here and the iterative approach based on the WKBJ solution which has been used previously in 1-D media. Numerical examples are shown to illustrate the examples discussed.     

Effective Hamiltonian for the D'Alembert Planetary Model Near a Spin/Orbit Resonance

The D'Alembert model for the spin/orbit problem in celestial mechanics is considered. Using a Hamiltonian formalism, it is shown that in a small neighborhood of a p:q spin/orbit resonance with (p,q) different from (1,1) and (2,1) the 'effective' D'Alembert Hamiltonian is a completely integrable system with phase space foliated by maximal invariant curves; instead, in a small neighborhood of a p:q spin/orbit resonance with (p,q) equal to (1,1) or (2,1) the 'effective' D'Alembert Hamiltonian has a phase portrait similar to that of the standard pendulum (elliptic and hyperbolic equilibria, separatrices, invariant curves of different homotopy). A fast averaging with respect to the 'mean anomaly' is also performed (by means of Nekhoroshev techniques) showing that, up to exponentially small terms, the resonant D'Alembert Hamiltonian is described by a two-degrees-of-freedom, properly degenerate Hamiltonian having the lowest order terms corresponding to the 'effective' Hamiltonian mentioned above.     

First-order asymptotics for the eigenfrequencies of the Earth and application to the retrieval of large-scale lateral variations of structure

Summary . We present a variety of examples, showing systematic fluctuations as a function of angular order of measured eigenfrequencies for given normal modes of the Earth. The data are single station measurements from the GEOSCOPE network. Such fluctuations are attributed to departures from the lowest order asymptotic expression of the geometrical optics approximation. We derive first-order asymptotic expressions for the location parameter for all three components of the Earth's motion, by a method based on the stationary phase approximation and geometric relations on the unit sphere.
We illustrate the sensitivity of the fluctuations to the different parameters involved (source parameters, epicentral distance, laterally heterogeneous earth model) with synthetic examples corresponding to GEOSCOPE observations. Finally, we show the results of first attempts at inversion, which indicate that, when the fluctuations are taken into account, more accurate estimates of the great circle average eigenfrequencies can be obtained, and additional constraints put on the structure in the neighbourhood of the great circle.