## Abstract

The form of the exact solution for the diffraction of a two-dimensional plane harmonic wave by a semi-infinite plate of thickness d is found. The solution involves constants which satisfy an infinite set of equations, and these equations are solved when d is small compared with the wave-length. It is shown that, in the neighbourhood of the shadow, the field is that of a single semi-infinite plane occupying the nearer face of the plate, whatever d, if terms of O(R$^{-\frac{1}{2}}$) are neglected, R being the distance of the point of observation from the edge. It is further shown that, when d is less than wave-length/10, the plate behaves as a semi-infinite waveguide whose sides project beyond the end of the plate by an amount 0$\cdot $11d together with, when the plane of polarization of the incident wave is perpendicular to the plate, a two-dimensional magnetic dipole at the end of the guide. When terms of O(kd) can be neglected, it appears from this result and Hanson's (1930) work on a plate with a cycloidal end that the exact shape of the end of the plate is of no importance; the plate behaves as a semi-infinite wave-guide. The extension of the theory to the diffraction by a thick plate of finite length is briefly discussed. The theory is also extended to incident scalar waves whose direction of propagation does not lie in the plane perpendicular to the plate and, from this, the field due to an incident electromagnetic wave is deduced. It is found that, for all values of d, the diffracted electromagnetic wave at any point is effectively travelling along a cone of semi-angle $\theta _{0}$ and axis the nearer edge, where $\theta _{0}$ is the angle between the edge and the direction of propagation of the incident wave. When d is small compared with the wave-length the plate acts as two parallel planes together with a line of magnetic dipoles at the end of the planes.