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Analytical approximations of surface fields induced on convex scatters by exteriorly incident scalar fields
The 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...
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| Format: | Thesis |
| Language: | English |
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Department of Mathematics and Applied Mathematics
2016
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| _version_ | 1867613203410714624 |
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| access_status_str | Open Access |
| author | Nel, W J F |
| author2 | Du Plessis, N M |
| author_browse | Du Plessis, N M Nel, W J F |
| author_facet | Du Plessis, N M Nel, W J F |
| author_sort | Nel, W J F |
| collection | Thesis |
| description | The 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. |
| format | Thesis |
| id | oai:open.uct.ac.za:11427/22216 |
| institution | University of Cape Town (South Africa) |
| language | eng |
| last_indexed | 2026-06-10T12:32:24.523Z |
| license_str | Not specified — see source repository |
| provenance_str_mv | Harvested via OAI-PMH from UCTD — University of Cape Town Open Access Repository |
| publishDate | 2016 |
| publishDateRange | 2016 |
| publishDateSort | 2016 |
| publisher | Department of Mathematics and Applied Mathematics |
| publisherStr | Department of Mathematics and Applied Mathematics |
| record_format | dspace |
| source_str | UCTD — University of Cape Town Open Access Repository |
| spelling | oai:open.uct.ac.za:11427/22216 Analytical approximations of surface fields induced on convex scatters by exteriorly incident scalar fields Nel, W J F Du Plessis, N M Mathematics and Applied Mathematics The 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. 2016-10-20T03:34:40Z 2016-10-20T03:34:40Z 1989 Doctoral Thesis Doctoral PhD http://hdl.handle.net/11427/22216 eng application/pdf Department of Mathematics and Applied Mathematics Faculty of Science University of Cape Town |
| spellingShingle | Mathematics and Applied Mathematics Nel, W J F Analytical approximations of surface fields induced on convex scatters by exteriorly incident scalar fields |
| thesis_degree_str | Doctoral |
| title | Analytical approximations of surface fields induced on convex scatters by exteriorly incident scalar fields |
| title_full | Analytical approximations of surface fields induced on convex scatters by exteriorly incident scalar fields |
| title_fullStr | Analytical approximations of surface fields induced on convex scatters by exteriorly incident scalar fields |
| title_full_unstemmed | Analytical approximations of surface fields induced on convex scatters by exteriorly incident scalar fields |
| title_short | Analytical approximations of surface fields induced on convex scatters by exteriorly incident scalar fields |
| title_sort | analytical approximations of surface fields induced on convex scatters by exteriorly incident scalar fields |
| topic | Mathematics and Applied Mathematics |
| url | http://hdl.handle.net/11427/22216 |
| work_keys_str_mv | AT nelwjf analyticalapproximationsofsurfacefieldsinducedonconvexscattersbyexteriorlyincidentscalarfields |