Σύγκλιση Και Ταλαντώσεις Στη Μέθοδο Βοηθητικών Πηγών (mas) Εφαρμοζόμενη Σε Τρισδιάστατα Ακουστικά Προβλήματα Σκέδασης

Abstract

The Method of Auxiliary Sources (MAS) is an approximate method for the solution of scattering problems. In the case of interest in the present thesis, that of scattering by an acoustically soft sphere, excited externally, one assumes MxN fictitious sources of acoustic field (to be referred to here as ‘MAS currents’) located on an auxiliary spherical surface inside the sphere-scatterer, for finite M,N. The ‘MAS currents’ are such that the boundary condition of the vanishing acoustic field is satisfied on MxΝ collocation points on the soft scatterer. A [(P-1).Q]x[(M-1).N] system of linear algebraic equations thus results. Once the MAS currents are found and calculated, the acoustic field (‘MAS field’) due to them can be easily determined. What is shown in the numerical investigations of this thesis (by means of MATLAB) is that, in the case of 3-D problems, for the placement of the mentioned auxiliary sources in a certain area (which is found in this thesis and determined by a ‘critical’ radius), the auxiliary currents may oscillate, but we obtain a convergent field despite these oscillations; furthermore the oscillations are neither due to round-off nor matrix ill-conditioning. It is also demonstrated that, as M and N go to infinity, it is possible to have a ‘MAS field’ convergent to the true, correct field (for all points outside the sphere) together with divergent ‘MAS currents’. The thesis describes therefore a difficulty (namely oscillations) associated with the implementing of ‘MAS’. The main advantages of illustrating a difficulty via a simple problem are two: (1) if the difficulty occurs in a simple problem, it is also likely to occur in more complicated problems and (2) it is less likely to confuse the said difficulty with other difficulties (namely, effects due to round-off, matrix ill-conditioning or shape elongation).