This paper is an investigation of the short-wave asymptotic motion due to a cylinder heaving at the surface separating two infinite liquids. The cylinder is assumed to have a smooth cross-section of an arbitrary shape that intersects the surface of separation at right angles. A non-rigorous argument is used to get the asymptotic expansion of the velocity potentials of the motion in the two liquids in the far field. The asymptotic evaluation of the coefficients describing wave-making, virtual-mass and damping are also obtained in terms of the limit potentials of the motions in the two liquids.

In (5) a weak convergence result for U-statistics was obtained as a special case of a reverse martingale theorem; in (7) Miller and Sen obtained another such result for U-statistics by a direct argument. As they stand these results are not very closely connected, since one is concerned with U-statistics Uk for k ≥ n, while the other deals with Uk for k ≤ n, but if one instead thinks of k as unrestricted and transforms the random functions Xn which enter into one of these results into new functions Yn by setting Yn(t) = tXn(t−1) one finds that the Yn are (aside from variations in interpolated values) just the functions with which the other result is concerned. As the limiting Wiener process W is well-known to have the property that tW(t−1) is another Wiener process it is not too surprising that both results should hold, and part of the purpose of this paper is to provide a general framework within which the relationship between these results will become clear. A second purpose is to illustrate the simplification that the martingale property brings to weak convergence studies; this is shown both in the U-statistic example and in a new proof of the convergence of the empirical process.

In two recent papers (6, 10) a new formulation of the governing equations for finite plane-strain deformations of compressible isotropic elastic solids was presented. This has been used to obtaingeneral solutions to the field equations for a certain class of strain–energy functions, and then closed-form solutions to a large class of boundary-value problems.

In a previous paper this author used elementary mathematics to describe some simple connexions between multi-valued Legendre transformations and those elementary catastrophes having one ‘behaviour’ variable, which belong to a family of polynomials called cuspoids (see Sewell (13)). The Legendre singularities permitted there are isolated and of dimension one, i.e. they exist where a Hessian matrix exceptionally has co-rank one (not more) and elsewhere has full rank (co-rank zero).

Many inhomogeneous systems of practical interest are made up of an inclusion, or several inclusions, of one elastic phase bonded to a surrounding matrix of another phase, and the behaviour of such systems has been studied extensively. A thin layer of some other elastic phase intervenes between an inclusion and the matrix in some important applications, for instance, where coated nuclear fuel particles are bonded in a matrix, or where duplex cylindrical fibres are employed in fibre-reinforcement, or wherever chemical action induces a protective finish either intentionally or inevitably as when an oxide coating covers a metallic surface. Our objective is to show how to take account of the pronounced influence that even a thin coating may exert upon, for instance, the stress concentrations just outside an inclusion or cavity, or upon the overall elastic moduli of a suspension of inclusions (though discussion of this latter topic is omitted here in order that it may be explored more fully elsewhere).

This paper concerns steady plane periodic waves on the surface of an ideal liquid flowing above a horizontal bottom. The flow is irrotational. The volume flow rate is denoted by Q, the velocity potential by ø, the period in ø of the waves by 2L, and the maximum angle of inclination between the tangent to the surface and the horizontal by θm.

Krasovskii (12) established that, at each fixed Q and L, there exist wave solutions for each value of θm strictly between zero and ⅙π. We establish that, at each fixed Q and L, there exist wave solutions for each value of qc strictly between c and zero. Here qc is the flow speed at the crest, and

where g is the acceleration due to gravity. Krasovskii's set of solutions is included in the set that we obtain.

A new method for generating tensorial derivatives of tensor functions is proposed. The method is based on the use of tensors as absolute entities along with the advantages offered by their decomposition on an orthonormal base.

Some aspects are considered of the Einstein vacuum field equations for the diagonal form of the general static metric. A power series expansion is considered in which in the first non-trivial order one gets seven equations for four unknowns. This is reduced to a single Poisson equation in which the source is given in terms of a harmonic function and some arbitrary functions of two variables. A similar reduction is carried out in detail for the next non-trivial order. The approximation scheme can be extended to arbitrarily high orders in principle. The connexion of the diagonal form with the Weyl solutions and the merits of the diagonal and non-diagonal forms are briefly discussed. A general solution of one of the field equations is presented.

It is shown in this paper that the conservation laws of non-local continuum mechanics are equivalent to, and can be derived from, the conservation of energy and the invariance conditions under superposed rigid body motions. Also, the theory of non-local thermoelasticity is reconsidered in the light of recent developments in thermodynamics, taking invariance conditions fully into account.

The initial-value problem of waves generated by a moving oscillatory surface pressure against a vertical cliff is solved and a uniform asymptotic analysis of the unsteady state is given. The same problem with no cliff is solved by Kaplan, and by Debnath and Rosenblat but their solutions are not uniform.

Fluid is contained in a rotating annulus of rectangular cross-section which is subject to a radial temperature gradient along the base and is insulated around its upper surfaces. If E is an Ekman number, which is assumed small, a finite amplitude cellular motion of wavelength ∼ E⅓ develops near the inner sidewall of the annulus by an exchange process if the local Rayleigh number, R, exceeds its critical value Rc and the Prandtl number of the fluid, σ, is greater than a critical value, σc (Daniels and Stewartson(5)). In the present study we consider the remaining range of values 0 < σ < σc for which overstability is preferred, occurring at a value of the Rayleigh number, Rc0, less than Rc. Disturbances to the basic flow near the sidewall now oscillate in time with frequency ∼ E⅓ and are amplified if R exceeds Rc0, the vertical cellular velocity attaining values ∼ E in a region of extent ∼ E near the sidewall. On the time-scale ∼ E disturbances take the form of two wave-like components which travel in opposite directions with the characteristic group velocity of the system. The first is simply convected away from the sidewall into the stably stratified interior and decays, while the second travels towards the sidewall where it is reflected. The reflexion coefficient is determined by the dynamics of an E⅓ boundary layer where an oscillatory motion is generated by the incident wave.