Interaction of a compressional impulse with a slot normal to the surface of an elastic half space – II

Summary. An improved finite difference scheme has been used to simulate the propagation of a plane P -impulse in an elastic half space with a slot normal to its surface. Various angles of incidence and dimensions of slot are considered. The numerical results are presented in several visualizations; each emphasized a different type of wave and all representations help in understanding the scattered and diffracted wave pattern. Experiments were carried out using 0.5—6 MHz ultrasonic pulses on a duralumin semidisc with a surface-breaking slot and the results are compared with those given by the numerical models.
The scattered wavefield includes compressional and Rayleigh pulses whose amplitude increases at the front of the slot and decreases behind it, as the angle of incidence is reduced. A diffracted compressional pulse is generated with a semicircular wavefront centred at the mid-point of the bottom of the slot. Also, two elliptical eddies are excited at the lower corners. These ellipses propagate into the medium and eventually spread out to form arc-shaped shear pulses.
In the shadow zone, behind the slot, the two components of displacement show independent behaviour. The horizontal component decreases either with decreasing angle of incidence or as the slot is made deeper. For acute angles, a reduction of displacement amplitude of about 50 per cent is obtained when the depth of the slot is made a half pulse width. On the other hand there is no diminution of the vertical displacement behind the slot, and, near the upper right corner, it is even amplified.     

B-polarization induction in two thin half-sheets coupled to the mantle by a conducting crust—II. Solution by the mode-matching method, the fields above the conductors, and example results

The mode-matching method is used to obtain an exact analytical solution to the problem of B -polarization induction in two adjacent thin half-sheets, lying on a conducting layer that is terminated by a perfect conductor at finite depth. These components of the model represent, respectively, the Earth's conducting surface layers, crust, and mantle. In dimensionless variables, the model has three independent parameters, these being the two thin-sheet conductances and the layer thickness. The mode-matching solution obtained in this paper is shown to be identical lo that derived via the Wiener-Hopf method in a companion paper (Dawson 1996), and so provides additional verification of that solution. As was shown in the companion paper, the solution for the present model contains, as special limiting cases, those for three models considered earlier by various authors. The second part of the present paper addresses the solutions for the electric fields in the non-conducting half-space above the conductors, which represents the atmosphere. In the final part, sample numerical calculations are presented to illustrate the solution.