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## The Optimal Shape of the Reflex Tube of a Bass Loudspeaker

• In the thesis the author presents a mathematical model which describes the behaviour of the acoustical pressure (sound), produced by a bass loudspeaker. The underlying physical propagation of sound is described by the non--linear isentropic Euler system in a Lagrangian description. This system is expanded via asymptotical analysis up to third order in the displacement of the membrane of the loudspeaker. The differential equations which describe the behaviour of the key note and the first order harmonic are compared to classical results. The boundary conditions, which are derived up to third order, are based on the principle that the small control volume sticks to the boundary and is allowed to move only along it. Using classical results of the theory of elliptic partial differential equations, the author shows that under appropriate conditions on the input data the appropriate mathematical problems admit, by the Fredholm alternative, unique solutions. Moreover, certain regularity results are shown. Further, a novel Wave Based Method is applied to solve appropriate mathematical problems. However, the known theory of the Wave Based Method, which can be found in the literature, so far, allowed to apply WBM only in the cases of convex domains. The author finds the criterion which allows to apply the WBM in the cases of non--convex domains. In the case of 2D problems we represent this criterion as a small proposition. With the aid of this proposition one is able to subdivide arbitrary 2D domains such that the number of subdomains is minimal, WBM may be applied in each subdomain and the geometry is not altered, e.g. via polygonal approximation. Further, the same principles are used in the case of 3D problem. However, the formulation of a similar proposition in cases of 3D problems has still to be done. Next, we show a simple procedure to solve an inhomogeneous Helmholtz equation using WBM. This procedure, however, is rather computationally expensive and can probably be improved. Several examples are also presented. We present the possibility to apply the Wave Based Technique to solve steady--state acoustic problems in the case of an unbounded 3D domain. The main principle of the classical WBM is extended to the case of an external domain. Two numerical examples are also presented. In order to apply the WBM to our problems we subdivide the computational domain into three subdomains. Therefore, on the interfaces certain coupling conditions are defined. The description of the optimization procedure, based on the principles of the shape gradient method and level set method, and the results of the optimization finalize the thesis.

• Dokument_1.pdf • Dokument_1.djvu Author: Jevgenijs Jegorovs urn:nbn:de:hbz:386-kluedo-20925 Axel Klar Doctoral Thesis English 2007 2007 Technische Universität Kaiserslautern Technische Universität Kaiserslautern 2007/04/25 2007/05/07 Asympotic Analysis; Helmholtz Type Boundary Value Problems; Optimization; Wave Based Method Fachbereich Mathematik 5 Naturwissenschaften und Mathematik / 510 Mathematik 30-XX FUNCTIONS OF A COMPLEX VARIABLE (For analysis on manifolds, see 58-XX) / 30Exx Miscellaneous topics of analysis in the complex domain / 30E25 Boundary value problems [See also 45Exx] 49-XX CALCULUS OF VARIATIONS AND OPTIMAL CONTROL; OPTIMIZATION [See also 34H05, 34K35, 65Kxx, 90Cxx, 93-XX] / 49Qxx Manifolds [See also 58Exx] / 49Q12 Sensitivity analysis 76-XX FLUID MECHANICS (For general continuum mechanics, see 74Axx, or other parts of 74-XX) / 76Mxx Basic methods in fluid mechanics [See also 65-XX] / 76M25 Other numerical methods 78-XX OPTICS, ELECTROMAGNETIC THEORY (For quantum optics, see 81V80) / 78Mxx Basic methods / 78M35 Asymptotic analysis Standard gemäß KLUEDO-Leitlinien vor dem 27.05.2011