A method already developed for the diffraction of scalar waves by a circular disc is generalized so as to be applicable to the diffraction of an electromagnetic wave. The generalization leads, in a straightforward manner, to an asymptotic development of the current density on the disc at high frequencies. The principal complication which arises is in solving three simultaneous linear equations for three constants which occur in the method of solution. Detailed calculations are given for a normally incident plane wave.

1. The problem of determining the distribution of stress in the neighbourhood of a penny-shaped crack defined in terms of cylindrical coordinates (ρ, φ, z) by 0 ≤ ρ ≤ α, z = 0, has been considered by Sneddon ((2)) and Sack ((1)). In the latter paper the solution is derived only in the case in which the stress field is due to the application of constant pressure to the faces of the crack. In the former paper the analysis given applies to an axisymmetric distribution of pressure p(ρ) applied to both the upper and lower face of the penny-shaped cavity, but the calculation of the stress intensity factor

and of the energy W required to open up the crack is a complicated matter even in the case in which p(ρ) is a constant.

This paper considers a family of viscous flows closely related to the exact Jeffery-Hamel solution ((l), (2)) of the two-dimensional Navier-Stokes equations, for diverging or converging flow in a channel. It is known that if the walls of the channel intersect at an angle less than π then there is a unique solution of the Navier-Stokes equations in which the streamlines are straight lines issuing from the point of intersection of the walls and the flow is everywhere diverging or everywhere converging. The flow parameters depend on the total fluid mass M emitted at the point of intersection and the angle 2α between the walls. By taking the Reynolds number R = M/ν, where v is the kinematic viscosity, the stream function can be expanded in a power series in R in which the leading term is a Stokes flow. Alternatively the solution can be developed by perturbing the Stokes flow and is one of very few examples known in which a Stokes flow can be regarded as a uniformly valid first approximation everywhere in an infinite fluid region. The class of flows to be considered is a generalization of the Jeffery–Hamel flow by taking the flow region to be finite and bounded by two circular arcs which intersect at an angle less than π At one point of intersection fluid is forced into the region and an equal amount is absorbed out at the other point. It is found to the first order that the flow at the two points of intersection corresponds to the zero Reynolds number limit for diverging and converging flow, respectively. Now since the flow at these points can be developed by perturbing the Stokes flow solution it is reasonable to assume that the zero Reynolds number flow in the entire finite region bounded by the arcs is a Stokes flow since the most likely region in which this approximation becomes invalid is locally at the points of intersection but here the validity of the approximation is ensured. A comparison of the convection terms with the viscous terms verifies that this conclusion is borne out.

It has been shown in recent years ((5)–(8), (10)) that it is possible to obtain closed form solutions for the time harmonic wave equation when a linear combination of the wave function and its normal derivative is prescribed on the surface of a wedge. Boundary-value problems of this type occur in the problem of diffraction by a highly conducting wedge or by a wedge whose surfaces are thinly coated with dielectric. In certain circumstances such surfaces can support surface waves and one important aspect of the solution of the boundary-value problem is the determination of the amplitude of the surface wave excited.

It is supposed that a viscous incompressible fluid is contained in an oblate spheroidal cavity (major axes a, eccentricity e) in a rigid body, and that, up to time t = 0, both fluid and container rotate together with angular velocity ω about the minor axis LB of the spheroid, which is fixed in space. At t = 0, the axis of rotation is moved impulsively, and is given a motion of precession (angular velocity Ω) about an axis LS fixed in space, which makes an angle α with LB. It is required to find the ultimate state of motion of the fluid relative to the container. In a previous paper (Stewartson and Roberts (4)), this problem was solved for arbitrary α under the assumptions that

where v is the kinematic viscosity. In the present paper, it is shown how the problem may be solved for arbitrary e (including zero) under the assumptions that

The plate is of thickness 2h and occupies the region defined by 0 ≤ x2+y2 ≤ x2, −h ≤ Z ≤ h, x, y, Z being rectangular Cartesian coordinates. A normal load , where p is a constant, is applied to the surface Z = h and the boundary is held so that the displacement on it is zero. Using the results of an earlier paper by Tiffen and Sayer (3), expressions are obtained for the mean stresses and stress couples at any point of the plate.