### Browsing by Author "Du Plessis, N M"

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- ItemOpen AccessAnalytical approximations of surface fields induced on convex scatters by exteriorly incident scalar fields(1989) Nel, W J F; Du Plessis, N MThe boundary value problems for the Helmholtz equation give rise to boundary integral equations for the unknown surface field or its normal derivative. These integral equations involve the Helmholtz surface potentials in the form of weakly singular surface integrals. This thesis is based on a method of parameterisation of the surface integrals which removes the weak singularities provided that the surface satisfies certain convexity conditions. Firstly this method of parameterisation is applied to investigate the properties of the Helmholtz surface potentials on convex surface elements, and some new proofs are given. The theory is then applied to the boundary integral equations which arise when a scalar field is incident on a bounded scatterer. The surface integrals in these integral equations are Helmholtz potentials and can be regularised by suitable parameterisation. It is assumed that the unknoWn density function is an analytical function on the boundary of the scatterer, and can therefore be expanded as a Taylor series at any point of the surface. If this expansion is substituted into the regularised integral equation and if the operations of integration and summation are formally interchanged, then the end result is a partial differential equation of infinite order involving only the field coordinates and having analytical coefficients. However, if the Taylor expansions are truncated then partial differential equations of finite orders result. The view is taken that analytical solutions of such differential equations of finite orders can serve as _approximations for the surface field or its normal derivative provided that suitable initial conditions are imposed to ensure uniqueness. On the other hand the general solution of such a differential equation can serve as a local approximation at any point on the surface. Some basic properties of the differential equations and their solutions, called analytical approximations, are discussed and the theory is then applied to the problem of acoustic scattering from a sound hard sphere.
- ItemOpen AccessAn investigation into the use of the method of spherical currents for the determination of scattered wave fields(1976) Du Plessis, N M; Brundrit, GeoffIn this thesis an approximation method, the method of spherical currents, is advanced for the determination of scattered fields (acoustic and electromagnetic) from convex bodies for plane harmonic incident waves. The method is worked out for the scalar (acoustic) case and applied to sound soft and sound hard obstacles of a given shape. The surface fields and backscattering cross sections of these obstacles are calculated according to the method and the results are compared with those of other methods. For the vector (electromagnetic) case only the basic theory is worked out for a perfectly conducting convex body.