A contribution to the theoretical study and numerical calculation of edge diffraction

A contribution to the theoretical study and numerical calculation of edge diffraction

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Νικολάου, Πέτρος

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The subject of the present thesis is the sound field around solid obstacles and specifically, the diffraction field generated as sound reaches the edges of the obstacles. Consider, for example, the edge of a half-plane or of a wedge. Diffraction is important in many practical problems, such as noise barriers, sonic boom propagation around buildings, room acoustics, volcanic explosions or ancient theater acoustics. The present work aims to derive new analytical solutions and/or to extend existing models in both time and frequency domain. The purpose of the work is to provide new physical insight in the study of diffraction and to accelerate its computation.
The analysis begins in the frequency domain, where an existing analytical model, the Directive Line Source Model (DLSM), is extended in areas where it was not valid before. The new model has a unified form for all types of incident radiation, being exact for plane incident waves and approximate for cylindrical and spherical incident waves. It is shown that the diffraction field can be interpreted as radiation from a directional line source (as stipulated by the original DLSM) irrespective of the receiver proximity to a shadow boundary, provided that the directivity of the virtual line source is appropriately modified. The properties of the new directivity function are investigated, its parameters recast and appropriate simpler asymptotic forms presented. Based on the new directivity function a new, precisely defined separation of the diffraction field around the shadow boundaries is proposed, which also provides a computational advantage for large scale simulations. Further, the proposed reformulation of the diffraction solution and of its parameters enables the application of the model to two cases of practical interest: (i) directional sound sources and (ii) diffraction by wedges, where the proposed formulation is considerably faster to compute than well-established solutions.
In time domain a new solution in the form of impulse response is also derived as the Fourier transform of the proposed frequency domain solution. The proposed impulse response, as its frequency counterpart, has a unified form for all types of incident signals, is exact for plane incident signals, and approximate for cylindrical and spherical incident signals. The investigation of the derived formulation leads to the derivation of a generator curve that embodies the impulse response at any source–receiver configuration. The generator curve is function of a single variable, namely the diffraction number. The diffraction number is a universal diffraction parameter, which translates the generator curve into impulse response at all times and all source-receiver locations, according to a condition termed the similarity condition. The employment of the generator curve can provide considerable computational benefit compared to direct computations. A separation of the diffracted signal into precisely determined time stages is also proposed. Further, for the case of the spherical incident signal the concept of the generator curve is also extended for the case of the exact solution. The exact solution is embodied in the same generator curve as the approximate solution but with a different diffraction number and similarity condition. A separation of the diffracted signal based on the exact impulse response is also proposed.
The work continues in the time domain with the study of the convolution between the impulse response and the incident signal, which is used to obtain the diffracted signal or diffraction response. The primitive functions of the proposed impulse response are employed to: (i) prove that the convolution of the impulse response with any bounded signal is bounded for all times, (ii) obtain analytically the diffraction response, as combination of elementary functions, for any incident signal approximated piecewise by fitting polynomials, (iii) improve the performance of the numerical convolution by orders of magnitude, and (iv) handle the convolution of very sparsely sampled incident signals.
The analysis also handles the diffraction of spherical signals incident on edges of finite length. Existing finite length diffraction theory and the proposed impulse response for the infinite edge are combined to form a new impulse response for the finite edge. The new impulse response has a simple analytical form and as opposed to other analytical solutions does not require integration along the edge length to compute the diffracted signal. Because it is based on the impulse response for infinite edges, it inherits all afore mentioned benefits associated with its primitive functions. Furthermore, it offers substantial computational benefit compared to traditional integration formulas along the edge.
In the frequency domain the Fourier transform of the new finite length impulse response can be approximated analytically. The resulting frequency formula does not require integration along the edge and offers physical insight on exactly how much different portions of the edge contribute to the total diffraction field.

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Κυματική διάδοση, Περίθλαση, Ακμή, Ημι-επίπεδο, Σφήνα, Πεδίο συχνοτήτων, Πεδίο χρόνου, Μετασχηματισμός Fourier, Συνέλιξη, Απόκριση περίθλασης, Άπειρη ακμή, Πεπερασμένη ακμή