Analytical approximations of surface fields induced on convex scatters by exteriorly incident scalar fields

 

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dc.contributor.advisor Du Plessis, N M en_ZA
dc.contributor.author Nel, W J F en_ZA
dc.date.accessioned 2016-10-20T03:34:40Z
dc.date.available 2016-10-20T03:34:40Z
dc.date.issued 1989 en_ZA
dc.identifier.citation Nel, W. 1989. Analytical approximations of surface fields induced on convex scatters by exteriorly incident scalar fields. University of Cape Town. en_ZA
dc.identifier.uri http://hdl.handle.net/11427/22216
dc.description.abstract 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. en_ZA
dc.language.iso eng en_ZA
dc.subject.other Mathematics and Applied Mathematics en_ZA
dc.title Analytical approximations of surface fields induced on convex scatters by exteriorly incident scalar fields en_ZA
dc.type Doctoral Thesis
uct.type.publication Research en_ZA
uct.type.resource Thesis en_ZA
dc.publisher.institution University of Cape Town
dc.publisher.faculty Faculty of Science en_ZA
dc.publisher.department Department of Mathematics and Applied Mathematics en_ZA
dc.type.qualificationlevel Doctoral
dc.type.qualificationname PhD en_ZA
uct.type.filetype Text
uct.type.filetype Image
dc.identifier.apacitation Nel, W. J. F. (1989). <i>Analytical approximations of surface fields induced on convex scatters by exteriorly incident scalar fields</i>. (Thesis). University of Cape Town ,Faculty of Science ,Department of Mathematics and Applied Mathematics. Retrieved from http://hdl.handle.net/11427/22216 en_ZA
dc.identifier.chicagocitation Nel, W J F. <i>"Analytical approximations of surface fields induced on convex scatters by exteriorly incident scalar fields."</i> Thesis., University of Cape Town ,Faculty of Science ,Department of Mathematics and Applied Mathematics, 1989. http://hdl.handle.net/11427/22216 en_ZA
dc.identifier.vancouvercitation Nel WJF. Analytical approximations of surface fields induced on convex scatters by exteriorly incident scalar fields. [Thesis]. University of Cape Town ,Faculty of Science ,Department of Mathematics and Applied Mathematics, 1989 [cited yyyy month dd]. Available from: http://hdl.handle.net/11427/22216 en_ZA
dc.identifier.ris TY - Thesis / Dissertation AU - Nel, W J F AB - 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. DA - 1989 DB - OpenUCT DP - University of Cape Town LK - https://open.uct.ac.za PB - University of Cape Town PY - 1989 T1 - Analytical approximations of surface fields induced on convex scatters by exteriorly incident scalar fields TI - Analytical approximations of surface fields induced on convex scatters by exteriorly incident scalar fields UR - http://hdl.handle.net/11427/22216 ER - en_ZA


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