In a stratified wave guiding system, it is often possible to adjust parameters, such as the wave frequency or some dimension of the guide, so that the propagation constants of two modes become equal. These modes are then said to be degenerate. In a tapered guide some of the parameters are slowly varying functions of distance x measured along the guide, and the conditions of degeneracy can be satisfied for some value x0 of x, possibly complex. Here the two modes are said to be critically coupled, and x0 is called a ‘coupling point’ in the complex x-plane. Cases are known where two modes in the earth-ionosphere wave guide could approach critical coupling in the twilight region of the earth, where the guide is tapered. The x dependence of the composite field for the two modes can then be expressed in terms of an Airy Integral function. The theory is closely analogous to that used for studying critical coupling of Ordinary and Extraordinary radio waves within a horizontally stratified ionosphere.

A geometrical representation of the transition matrices of a non-homogeneous chain with N states, in terms of certain convex subsets of , is used to describe aspects of the chain. For example, an important theorem of Cohn on the structure of the tail σ-field is a simple corollary. The embedding problem is shown to be entirely geometrical in character. The representation extends to Markov processes on quite general state spaces, and the tail is then represented by the projective limit of these convex sets.

A uniformly valid asymptotic solution is found for waves refracted by a shearing current or by bottom topography in the case where two straight caustics are present. The ratio of the amplitude of the reflected wave to that of the incident wave increases monotonically from 1/√2, when the caustics are close together, to unity when they are widely separated. A slight modification of the theory gives an expression for the frequencies of waves trapped between two caustics.

1. In this paper we shall use the methods of (6) to give a short proof of Wunderlich's result (7, 8) on the distribution of twin ‘primes’ for the Hawkins random sieve. We shall also obtain a sharp estimate on the gaps between successive ‘primes’.

We introduce the idea of dissociatedness for multiply-subscripted random variables: this is a weaker property than independence in general, coinciding with it in the singly-subscripted case; but it is still strong enough to allow a suitable version of the Strong Law of Large Numbers to hold. This result can be applied to prove a weak convergence result of interest in data analysis.

1. In this paper, the author derives a formula of inversion for the integral transform

where a > 0, k > 0. The actual formula obtained appears as equation (8) of the present paper and is deduced from an integral formula constructed in (1).

In a recent paper(14), Talenti gave a complete characterization of the spectrum of the maximal realization Hop of − Δ in Lp(Rn), for p in the range

A systematic method is given for finding all of the singularities in the stress fields associated with self-similar elastodynamic problems for half-spaces, directly from representations developed in (1). No particular symmetry is assumed, and both two- and three-dimensional problems are discussed for arbitrarily anisotropic half-spaces. Expressions for body wave arrivals are obtained, reproducing results given in (1), and head wave arrivals, not discussed in (1), are found. Arrivals produced by interfacial dislocations and cracks are found by the same method. As an example, an account of the first arrival produced by a penny-shaped crack expanding under shear on an interface is completed, by finding the head wave that precedes the body wave, already found in (1), in a region exterior to a cone in one of the half-spaces.

In a recent paper Koiter (5) discussed a principle of stationary complementary energy for the finite deformation of elastic materials. The complementary energy functional he uses depends only on the components of the nominal stress, and not on the displacement field. He, incorrectly, attributes the principle to Zubov (14).

This paper continues (and concludes) the mathematical analysis begun in (8) of a formal theory of viscous flow in channels with slowly curving walls. In that paper, the theory was shown to yield strict asymptotic expansions, in powers of the small curvature parameter, of exact solutions of the Navier-Stokes equations, but the proofs were restricted to a set of Reynolds numbers and wall divergence angles that is distinctly smaller than the set on which the formal approximation is defined. In the present paper, we study in more detail a certain linear, partial differential operator TN, the invertibility of which is essential to the proofs. This operator is shown to be invertible (and the formal theory is thereby justified) on a parameter domain that is much larger than and may well be the whole of . A key step is to associate with TN a family of operators that approximate TN locally and have much simpler coefficients.

Suppose that an elastic half-space, which contains certain surface defects, inclusions and cavities, is in free, two-dimensional, time-harmonic vibration, with the wave field at infinity ‘outgoing’ in character. It is shown that the elastic potentials representing such a ‘standing mode’ can be expressed in the form of contour integrals, for instance

U(t) being an analytic function of t. By considering the far field of these potentials, it is shown that U(t) is zero on a certain arc in the t-plane and is therefore identically zero. It follows that ø(r) is zero everywhere and this proves the non-existence of such standing modes in these configurations.

This uniqueness theorem justifies the solution given by the author (Gregory (2)) for the problem in which time harmonic stresses act on the walls of a cylindrical cavity lying beneath the surface of an elastic half-space. It is also shown that if a Rayleigh surface wave is incident on any system of surface defects, inclusions and cavities, then energy must be transferred from the surface wave to scattered outgoing body waves of both P and S types.

It is shown that, at an arbitrary point in a general elastic material and regardless of the prevailing state of deformation, there exists in every plane at least one direction in which a transverse acceleration wave may propagate. The possibility of a transverse wave actually being able to progress in such a specific direction depends upon the associated local wave speed being real and conditions are obtained under which this requirement is satisfied.

The concept of ideal strength of perfect crystals as an instability phenomenon is critically reviewed in the context of generalized moduli associated with arbitrary measures of stress and strain. Further aspects of elastic response to finite strain are discussed in relation to the classical model of an atomic lattice.

The basic state considered in this paper is a parallel flow of a jet-like character with the centre of the jet being at or near a free surface which is horizontal. Stationary surface gravity waves may exist on such a flow, and a number of examples are looked at for small amplitude waves. Explicit solutions are given for ‘top-hat’ profile jets and for two-dimensional flows. Asymptotic solutions are developed for stationary waves of large wave-number.

A β-plane ocean is considered, and dissipative processes are modelled by using a coefficient of lateral friction. Attention is focused on the singularity which occurs if the angle that the western boundary makes with the north approaches ½π. The dimensions of the singular region are found using boundary layer techniques, and the solution for the transport stream function valid within it is obtained in closed form.