The magnetic field observable outside a body of conducting fluid in which field is imbedded may be considerably altered by convection currents in the fluid. One possible explanation of the geomagnetic secular variation foci is that localized convection cells in the earth's core disturb the main field present. An analytic solution for such a process is readily obtained by assuming the form and dimensions for such a cell, and shows that the magnitude of the secular variation cannot easily be explained on these lines without the presence of a subsurface toroidal magnetic field of some hundreds of gauss which is ‘convected through’ the surface of the core.

Introduction. The effect of a vibrating point force on the surface of a semi-infinite elastic solid was investigated in a classic paper by Lamb ((8)). When the point force is replaced by a vibrating rigid disc the problem is more complicated, because the displacement is specified under the disc and the remainder of the surface is stress-free. This gives rise to a mixed boundary-value problem.

In this paper a fundamental problem of thermal transport in a channel-gas flow has been studied. The most general solution in terms of integral formulae has been obtained when the duct is bounded by an arbitrary closed contour, the prescribed normal temperature gradient at the wall varies arbitrarily along the periphery, and the specified additional heat generation varies in an arbitrary manner from point to point in a cross-section. The power series solution is also discussed, and is given in a separate section. One of the objects of the present study is to look forward to the qualitative as well as quantitative effects of viscous dissipation and work of compression. Several results and all the graphs of Tao ((13)) have also been corrected.

A time harmonic Rayleigh wave, propagating in an elastic half-space y ≥ 0, is incident on a certain impedance boundary condition on y = 0, x > 0. The resulting field consists of a reflected surface wave, scattered body waves, and a transmitted surface wave appropriate to the new boundary conditions. The elastic potentials are found exactly by Fourier transform and the Wiener-Hopf technique in the case of a slightly dissipative medium. The ψ potential is found to have a logarithmic singularity at (0,0), but the φ potential though singular is bounded there. Analytic forms are given for the amplitudes of the reflected and transmitted surface waves, and for the scattered field. The reflexion coefficient is found to have a simple form for small impedances. A uniqueness theorem, based on energy considerations, is proved.

The effect of a vertical barrier, fixed in an infinitely deep sea, on normally incident surface waves of small amplitude was first considered by Ursell (1) and generalizations which retain the two-dimensional aspects of the problem have subsequently been considered by John (2) and Lewin (3). The fluid motion due to the flexural vibrations of a barrier of finite depth has been considered by Alblas (4), the motion in this case being three-dimensional.