The solutions are derived for the shallow water standing waves of small amplitude that can form in channels or lakes of varying breadth with a concave parabolic bottom. In addition explicit solutions are given for standing waves in a ring-type lake with a parabolic bottom and a central circular island that has vertical walls.

In this paper we obtain a representation of elastic wave fields (i.e. solutions of the equations of motion of classical elastokinetics) in terms of three functions satisfying scalar wave equations. Although the form of the representation implies no restriction upon the choice of coordinate system or upon the shape of the elastic body, it is found that the result can only be applied advantageously to initial-boundary-value problems having spherical polar coordinates as a natural frame of reference. The representation is shown to be complete in the sense that every (sufficiently smooth) elastic wave field in a homogeneous, isotropic body bounded by two concentric spheres can be expressed in the given form.

Under conditions of axial symmetry the representation generates wave fields which can be decomposed into poloidal and toroidal constituents, the former arising from two scalar wave functions and comprising both dilatational and rotational waves, and the latter being associated with a single scalar wave function and a state of pure shear of the elastic solid. Finally, the representation is used to obtain a formal solution describing the elastic pulse generated in an infinite body by the application of time-dependent tractions to the surface of a spherical cavity.

The Linear Theory of an Elastic Cosserat Plate

By A.E. Green and P. M. Naighdi

(Received 23 June 1967)

We are grateful to Dr R. B. Osborn for the following corrections:

Equation (3·19)1 should read

Equation (5·11)3 should read

Owing to an unusual definition of stress couples an error has occurred in one boundary condition at r = a in section 5. The following changes are necessary:

Interchange M12 and M21 in (5·1) and (5·2).

Replace Mrθ by Mθr in (5·12).

In equations (5·26), (5·28), (5·29), (5·30) and (5·31) replace γ by −γ but leave 1 + γ unaltered.

Replace (5·32) by

a result independent of γ.

Replace (5·33) by

A detailed study is made of angular oscillations of small amplitude about a fixed axis of a rigid sphere embedded in an infinite elastic solid. Three modes of vibration of the sphere are considered: steady oscillations arising from the application of a periodic torque; forced oscillations produced by an arbitrary time-dependent torque; and free oscillations excited by an impulsive torque. Due to the transfer of energy to the surrounding material by the radiation of an elastic shear wave, free oscillations of the sphere are damped, the principal parameter affecting the damping being the density contrast between the sphere and its surroundings. Illustrative numerical results, referring to steady and free oscillations of the sphere, are presented in graphical form.

In this paper, which is a continuation of (1), we study steady (i.e. time-harmonic) and transient rectilinear oscillations of small amplitude of a rigid sphere embedded in an infinite elastic solid. Two types of transient motions are considered, forced oscillations in which the sphere is subject to a prescribed time-dependent force, and free oscillations in which the sphere is set in motion by an impulsive force. For each mode of vibration of the sphere the character of the solution is determined by two parameters, the density contrast between the sphere and its surroundings and a parameter related to the Poisson's ratio of the elastic solid. Numerical results referring to transient rectilinear oscillations are presented in graphical form.

This paper deals with wave propagation in inhomogeneous compressible electron plasmas. The waves are described by Maxwell's equations coupled to the linearized single-fluid equations of hydrodynamics. Coupled wave equations are derived which describe the propagation of coupled electromagnetic and electron acoustic waves. Results are obtained for generally inhomogeneous plasmas and are specialized to planar and cylindrically stratified media. Particular attention is given to the planar case and several approximate techniques for treating the equations are discussed. The fields in a region of coupling are investigated.

The problem considered is that of the diffraction of sound pulses by a plane interface separating two stratified media in each of which the refractive index increases with distance from the interface. Two types of diffraction phenomena occur, glancing ray diffraction and total reflection. The solution is obtained as a series of ‘pulse propagation modes’, using a double transform technique. The main results are some formulae for the pulse field near a diffracted wavefront.

The problem of scalar Dirichlet diffraction of a plane wave by an elliptic disc is discussed. A scheme is given whereby the low frequency expansion of the scattered field may be readily obtained. Series expansions are obtained for the far-field amplitude up to and including the second order in the wave number. The first two terms of the scattering cross-section are also derived.

1. In a recent paper Karp and Karal(1) have suggested a generalization of the normal impedance boundary condition which might be applicable to surfaces which can support more than one surface wave and have determined the total field produced by a magnetic line dipole placed above an infinite plane characterized by such a condition. The theory is at this stage purely tentative and arguments concerning its plausibility are given in (1). The validity of the generalized impedance condition has also not yet been experimentally verified. From the point of view of experimental verification it would seem useful to have available a theoretical solution valid for an arbitrary electromagnetic field incident on a plane characterized by a generalized impedance condition and such a solution is given in the present note. By means of a technique used by the author in related problems (2,3) an explicit solution is given for an arbitrary incident field and it is shown that the radiation field and the amplitudes of the surface waves may be expressed in terms of the radiation field of the incident wave.

In this paper we consider sound pulses generated in the fluid outside an impermeable sphere by axisymmetrical time-dependent deformations of the sphere. Following an account of some analytic techniques applicable to this class of problems, a detailed study is made of the case in which the movement of the spherical boundary is confined to a cap of angle α (0 < α < π). In this case a rather complicated pattern of wavefronts develops consisting of a direct (spherical) wavefront, diffracted wave-fronts and a toroidal wavefront having as its centre circle the edge of the cap. Approximations to the velocity potential near the various wavefronts are obtained and some numerical results are given.

The unsteady two-dimensional boundary-layer flows are investigated, using the equations linearized as by Lighthill. It is assumed, unlike separable forms and power functions taken by Sarma, that the perturbations which cause the unsteadiness in the flow are arbitrary functions of the distance along the flow and the time. In addition to the Reynolds number, it is explicitly assumed that the flow for large times is defined by a set of parameters like

ξ = φ/U0, U0(x) and φ (x are functions associated with the main stream in steady flow, λ(x, t) represents a perturbation, t time and x is the distance along the surface. This particular assumption is essentially an idea that is suggested by the work of Sarma. In solving the equations a number of free constants and arbitrary functions are used, which will be specified according to the given physical situation. For large times series solutions are assumed in terms of the above parameters and ultimately sets of differential equations are obtained in a single variable. Thus the theory makes the problem ready for computational work. For small times using the steady state solutions given in this paper, we proceed along the same lines as given in the work of Sarma. The velocity as well as thermal boundary layers are analysed in this paper.

Existence and uniqueness proofs for a boundary-value problem associated with a magnetohydrodynamic Falkner–Skan equation are presented. Relevant special cases of the problem herein considered include the magnetohydrodynamic rear stagnation point flow, and the non-magnetic ‘backward boundary layers’ of Goldstein(2).

The motion of a solid in a viscous conducting fluid permeated by a magnetic field has been the subject of a number of papers. Using the Stokes approximation Chester (3) first derived an expansion in powers of the Hartmann number for the drag on a sphere translating uniformly through a fluid, parallel to the applied magnetic field. In a subsequent paper (4) the same author treated the high Hartmann number limit. Later, Chang (2) considered, in the low Hartmann number regime, the motion of an axisymmetric body translating slowly along the axis of a fluid filled tube, and was able to calculate a drag formula which includes the lowest order wall effect term. An alternative treatment of this class of problems has recently been given by Williams (13), using an intergral equation formulation.

This paper is principally concerned with the two-dimensional time harmonic vibrations of an elastic half-space y ≥ 0, containing a submerged cavity in the form of an infinite circular cylinder. Two sequences of line source potentials are obtained which are singular along the axis of the cylinder, satisfy the free surface conditions on y = 0, and represent outgoing waves at infinity. (The radiation condition.) It is proved that any solution of the governing equations which satisfies the free surface conditions and consists of outgoing waves at infinity is expansible as a sum of these fundamental source potentials, with coefficients to be determined from the boundary conditions on the cylinder only.

The requirement of outgoing waves is carefully discussed and it is shown that the conditions taken give rise to, and are satisfied by, potential fields which would be regarded intuitively as representing outgoing waves.

Tiffen and Lowe in two recent papers (4,5) have developed exact theories for the bending and stretching of generally loaded elastic plates using the moments of the fundamental equations of infinitesimal elasticity. The two theories depend on the choice of independent variables so that a pair of functions, denoted by and are independent. The choice is correct in (4) but not in (5). This paper pro-poses a modification to the forms of the functions and in (5) and then shows that the modified forms are independent. The subsequent alteration to (5), as will be seen, involves only the biharmonic function χ Apart from a minor algebraic adjustment the rest of the theory, containing the real and complex eigenfunctions and particular integral, is preserved. It should be noted that the modified theory like the original is obtained entirely from the moment equations.

In this paper the mixed problem for an isotropic, elastic half-space is considered. Boundary conditions are prescribed interior and exterior to a circular region of unit radius, and the state of stress is assumed to be axially symmetric. Several authors have treated this problem. Mossakovskii(1) considered a punch adhering to and indenting an elastic half-space. Has solution was obtained by introducing certain operators that transformed the half-space problem into a problem in plane potential theory. The method of linear relationship was used to solve this auxiliary problem and inverse operators returned the plane to the half-space. The general case of a circular, rigid punch adhering to a half-space was treated by Ufliand (2,3) and a solution was obtained through the use of toroidal coordinates and the Mehler-Fok integral transforms.

Correction to article ‘Asymmetric inequalities for non-homogeneous ternary quadratic forms’, by Vishwa Chander Dumir.

Proc. Camb. Phil. Soc. 63 (1967), 291–303.

The first sentence on page 292 should read:

The object of this paper is to prove the corresponding result for indefinite quadratic forms with positive determinants.