Diffraction by a Semi-Infinite Screen With a Rounded End

Suppose a cylindrical wave is incident upon a semi-infinite thin screen with a rounded end, such as that shown in Figure 1. The rounded end is obtained by placing a cylindrical tip of radius a on the end of the screen. Figure 1 A semi-infinite thin screen, represented by the heavy line with its end at 0, is attached to a circular cylinder of radius a, represented by the circle. A line source parallel to the cylinder is at Q, which has coordinates r 0 and θ 0. Two rays from Q are tangent to the cylinder at Q 1 and Q 2. Their extensions bound the two parts of the shadow, A below the screen and B above it. The tangent ray at Q 1 produces a diffracted surface ray which proceeds along the cylinder in the counterclockwise direction. The two diffracted rays which it sheds at P 1 and P′1 are shown passing through P, the second ray having been reflected from the screen at P′1. If a is large compared to the wavelength λ of the incident radiation, then the resulting field can be determined by means of the geometrical theory of diffraction [2, 3]. This theory provides a simple geometrical procedure for constructing the field. The construction is carried out in Section 2. In making this construction, it is assumed that the field u is a scalar which satisfies the reduced wave equation and either it or its normal derivative vanishes on the screen. Thus u may represent either the component of electric or magnetic field parallel to the generators of the cylinder, and the screen may represent a perfect conductor. In the second case, u may also represent acoustic pressure and the screen may represent a rigid body.