Significant effort has recently been directed towards the development of numerically efficient iterative techniques for the solution of boundary integral equation formulations of time harmonic scattering problems. The primary result of this effort has been the development of several advanced numerical techniques which enable the dense matrix-vector products associated with the iterative solution of boundary integral equations to be rapidly computed. However, an important aspect of this problem which has yet to be adequately addressed is the development of rapidly convergent iterative techniques to complement the relatively more mature numerical algorithms which expedite the matrix-vector product operation. To this end, a class of efficient iterative methods for boundary integral equation formulations of two-dimensional scattering problems is presented. This development is based on an attempt to approximately factor (i.e., renormalize) the boundary integral formulation of an arbitrary scattering problem into a product of one-way wave operators and a corresponding coupling operator which accounts for the interactions between oppositely propagating waves on the surface of the scatterer. The original boundary integral formulation of the scattering problem defines the coupling between individual equivalent sources on the surface of the scatterer. The renormalized version of this equation defines the coupling between the forward and backward propagating fields obtained by re-summing the individual equivalent sources present in the original boundary integral formulation of the scattering problem. An important feature of this class of rapidly convergent iterative techniques is that they are based on an attempt to incorporate the important physical aspects of the scattering problem into the iterative procedure. This leads to rapidly convergent iterative series for a number of two-dimensional scattering problems. The iterative series obtained using this renormalization procedure are much more rapidly convergent than the series obtained using Krylov subspace techniques. In fact, for several of the geometries considered the number of iterations required to achieve a specified residual error is independent of the size of the scatterer. This desirable property of the iterative methods presented here is not shared by other iterative schemes for wave scattering problems. Moreover, because the approach used to develop these iterative series depends only on the assumption that the total field can be approximately represented by a summation of independent and oppositely directed waves (and not on the presence of special geometries, etc.), the proposed iterative methods are very general and are thus applicable to a large number of complex scattering problems. / Ph. D.

Ultrasonic wave field is modeled inside non-planar and inhomogeneous structures using a newly developed mesh-free semi-analytical technique called Distributed Point Source Method (DPSM). Wave field inside a corrugated plate which is a non-planar structure is modeled using DPSM when the structure is excited by a bounded acoustic beam generated by a finite-size transducer. The ultrasonic field is computed both inside the plate and in the surrounding fluid medium. It is observed that the reflected beam strength is weaker for the corrugated plate in comparison to that of the flat plate, as expected. Whereas the backward scattering is found to be stronger for the corrugated plate. DPSM generated results in the surrounding fluid medium are compared with the experimental results.Ultrasonic wave field is also modeled inside inhomogeneous structures. Two types of inhomogeneity are considered - a circular hole and a damaged layered half-space. Elastic wave scattering inside a half-space containing a circular hole is first modeled using DPSM when the structure is excited with a bounded acoustic beam. Then the ultrasonic wave field is computed in presence and absence of a defect in a layered half-space. For the layered problem geometry it is shown how the layer material influences the amount of energy that propagates through the layer and that penetrates into the solid half-space when the solid structure is struck by a bounded acoustic beam. It is also shown how the presence of a crack and the material properties of the layer material affect the ultrasonic fields inside the solid and fluid media.After solving the above problems in the frequency domain the DPSM technique is extended to produce the time domain results by the Fast Fourier Transform technique. Time histories are obtained for a bounded beam striking an elastic half-space. Numerical results are generated for normal and inclined incidences, for defect-free and cracked half-spaces. A number of useful information that is hidden in the steady state response can be obtained from the transient results.

The energy wave scattering method (EWS method) makes use of transmission lines and junctions to model the systems, and switches physical quantities to be energy wave variables then simulates the dynamic response of the systems, finally switches the analyzed results from energy wave variables back to physical quantities. Although using EWS method to simulate the dynamic response of structures is still on the initial stage, figuring out the time domain problems as example as transient analysis is suitable for use. Transient analysis is an important segment of dynamic analysis, it needs more extensive mathematics and newer method of calculation. Probably the EWS method is a workable and typical way. The study tries to use the EWS method to simulate the dynamic response of mutildegree of freedom systems, the response are due to different factors such as initial condition factors, damping factors and external force factors else. Let the simulated results display as displacement-time figures and displacement tables, and compared the results from lumped method or the finite element software-ANSYS with system characteristics by the figures and time domain displacements by the tables. On the whole, the simulated results almost matched with the analytical lumped methods. From the results of the study could confirm the feasibility that using the EWS method to simulate the dynamic response of mutildegree of freedom systems, and further tested and verified the applications of the EWS method on the dynamic analysis.

The structure of seismic discontinuities at depths of about 410 km and 660 km provides important constraints on mantle convection as the associated mineral phase transformations in the transition zone are sensitive to thermal perturbations. Teleseismic P-to-S receiver functions have been widely used to map the depths of the two discontinuities. In this study, we investigate the resolution of receiver functions in imaging topographic variations of the 410-km and 660-km discontinuities based on wave propagation simulations using a Spectral Element Method (SEM). We investigate finite-frequency effects of direct P waves as well as P-to-S converted waves by varying the length scale of discontinuity topography in the transition zone. We show that wavefront healing effects are significant in broadband receiver functions. For example, at a period of 10 to 20 seconds, the arrivaltime anomaly in P-to-S converted waves is about 50% of what predicted by ray theory when the topography length scale is in the order of 400 km. The observed arrival anomaly further reduces to 10-20% when the topography length scale reduces to about 200 km. We calculate 2-D boundary sensitivity kernels for direct P waves as well as receiver functions based on surface wave mode summation and confirm that finite frequency-effects can be properly accounted for. Three-dimensional wavespeed structure beneath seismic stations can also introduce significant artifacts in transition zone discontinuity topography if time corrections are not applied, and, the effects are dependent on frequency. / Master of Science

In this thesis we explore several topics with applications to both aero and hydroacoustics. Due to the much larger speed of sound in water compared to in air, several of the approximations used in aeroacoustics are not applicable underwater over the range of frequencies of interest. Specifically, we study the finite-chord effects on two broadband noise mechanisms: the trailing edge noise and the ingested noise problems. We start by investigating the acoustic wave diffraction by a finite rigid plate using three different methods. We compare the behaviour of the different solutions as a function of the reduced acoustic wavenumber. Our results reveal that the Mathieu function expansion is the most appropriate method as long as the reduced acoustic wavenumber is not too large. Finally, we show how the Mathieu functions can be used to build a Green's function tailored to an elliptic cylinder of arbitrary aspect ratio without relying on addition theorems. The results obtained in chapter two motivated the search for an exact solution to the trailing edge noise problem using a Mathieu function expansion. It is shown that the approximate methods used in aeroacoustics are not accurate enough for reduced acoustic wavenumbers less than unity, and for all wavenumbers near cut-off. Furthermore it is shown that, even at low Mach numbers, it is crucial to take into account the effects of convection at low frequency. Finally Lighthill's analogy is used, combined with the tailored Green's function introduced previously, to recover the two asymptotic Mach number scalings of the acoustic power for a flat plate at high frequency and low frequency. In chapter four, we introduce a novel method to solve the ingested noise problem by decomposing the pressure field into a singular part whose functional form can easily be found, and a regular part that we express using a Mathieu function expansion. It was found that finite-chord effects do have a strong impact for reduced acoustic wavenumbers less than unity, and for all wavenumbers near cut-off. The following chapter focuses on the trailing edge noise mechanism and details how the theory for a single stationary aerofoil can be applied to a rotating propeller. Due to the general geometry of a blade, we extended Amiet's model to take into account a mean flow misaligned with the blade chordline. Different semi-analytical models of wall pressure spectra are introduced and compared. We make extensive use of Brooks' data for a NACA 0012 aerofoil to obtain realistic inputs in the semi-analytical models. Finally, we introduce and compare two models of rotating blade trailing edge noise. The effects of both the angle of attack and the number of strips are then investigated. The final chapter is distinct from the rest of the thesis. We propose a model for studying the low Mach number flow noise from a 2D circular cylinder with small roughness. The method is based on using the Green's function tailored to a smooth cylinder in Curle's acoustic analogy. It was found that the main source of noise was the tonal low frequency scattering by the smooth geometry. However, it is suggested that roughness elements might be the dominant source of noise at higher frequency.

In the western hemisphere, the piano is one of the most important instruments. While its evolution lasted for more than three centuries, and the most important physical aspects have already been investigated, some parts in the characterization of the piano remain not well understood. Considering the pivotal piano soundboard, the effect of ribs mounted on the board exerted on the sound radiation and propagation in particular, is mostly neglected in the literature. The present investigation deals exactly with the sound wave propagation effects that emerge in the presence of an array of equally-distant mounted ribs at a soundboard. Solid-state theory proposes particular eigenmodes and -frequencies for such arrangements, which are comparable to single units in a crystal. Following this 'linear chain model' (LCM), differences in the frequency spectrum are observable as a distinct band structure. Also, the amplitudes of the modes are changed, due to differences of the damping factor. These scattering effects were not only investigated for a well-understood conceptional rectangular soundboard (multichord), but also for a genuine piano resonance board manufactured by the piano maker company 'C. Bechstein Pianofortefabrik'. To obtain the possibility to distinguish between the characterizing spectra both with and without mounted ribs, the typical assembly plan for the Bechstein instrument was specially customized. Spectral similarities and differences between both boards are found in terms of damping and tone. Furthermore, specially prepared minimal-invasive piezoelectric polymer sensors made from polyvinylidene fluoride (PVDF) were used to record solid-state vibrations of the investigated system. The essential calibration and characterization of these polymer sensors was performed by determining the electromechanical conversion, which is represented by the piezoelectric coefficient. Therefore, the robust 'sinusoidally varying external force' method was applied, where a dynamic force perpendicular to the sensor's surface, generates movable charge carriers. Crucial parameters were monitored, with the frequency response function as the most important one for acousticians. Along with conventional condenser microphones, the sound was measured as solid-state vibration as well as airborne wave. On this basis, statements can be made about emergence, propagation, and also the overall radiation of the generated modes of the vibrating system. Ultimately, these results acoustically characterize the entire system. / Betrachtet man den westlichen Kulturkreis, ist der Flügel bzw. das Klavier wohl eines der bedeutendsten Instrumente. Trotz einer stetigen, empirischen Weiterentwicklung dieses Instrumentes in den letzten drei Jahrhunderten und des Wissens um die wichtigsten physikalischen Effekte, sind viele Teile der Charakterisierung des Klaviers (sowohl akustisch als auch physikalisch) immer noch nicht vollständig verstanden. Nehmen wir nur den Resonanzboden - das entscheidende Bauteil für die Akustik eines Klaviers - und betrachten die Auswirkung, den die Berippung auf die Schallausbreitung des Instruments hat. Bis auf wenige Ausnahmen wird dieser Struktur-Aspekt in der Literatur weitestgehend übergangen. Die vorliegende Arbeit untersucht genau diese Ausbreitungscharakteristiken und Streueffekte, welche dadurch entstehen, dass Rippen, die denselben Abstand zueinander haben, auf dem Resonanzboden angebracht werden. Die Festkörperphysik stellt ein einfaches Modell über die Eigenfrequenzen für solche Anordnungen bereit. Dafür werden die Rippen und deren Abstände wie Einheitszellen eines Kristalls betrachtet. Ausgehend vom sogenannten 'Modell der linearen Ketten', werden gemessene Frequenzbänder im Spektrum erklärbar. Zusätzlich ändern sich auch die spektralen Amplituden des Resonanzbodens durch das Anbringen der Rippen. Diese Streueffekte wurden nicht nur an einem konzeptionellen rechteckigen Resonanzboden untersucht, sondern auch an einem originalen Klavier-Resonanzboden, welcher von dem Klavierbauer 'C. Bechstein Pianofortefabrik' hergestellt wurde und auch später in einem fertigen Klavier zum Einsatz kommen wird. Der traditionelle Zusammenbau des Klaviers wurde speziell für diese Untersuchung abgeändert, um die Möglichkeit zu haben, die Berippung des Resonanzbodens spektral zu charakterisieren. Alle gefundenen Eigenschaften des konzeptionellen und des originalen Bodens wurden verglichen. Für die Dämpfung und für die Brillianz des Tons wurden Übereinstimmungen, aber auch Unterschiede gefunden. Ein großer Teil dieser Untersuchung erforderte den Einsatz von speziell angefertigten piezoelektrischen Polymer-Beschleunigungsaufnehmern aus Polyvinylidenfluorid. Direkt fest eingeklebt im Instrument, wurden diese eingesetzt, um die Körperschwingungen des vibrierenden Systems aufzunehmen. Die essentielle Kalibrierung und Charakterisierung dieser Sensoren wurde durchgeführt, indem die elektromechanische Umwandlung bestimmt wurde, die durch den piezoelektrischen Koeffizienten gegeben ist. Durch eine sinusförmig variierende, externe Kraft und die dadurch entstehenden Ladungsträger an den Oberflächen des Sensormaterials kann dieser Koeffizient sehr genau bestimmt werden. In Abhängigkeit entscheidender physikalischer Größen, z.B. der Frequenz-Antwort-Funktion, wurde das Verhalten des piezoelektrischen Koeffizienten untersucht. Die erzeugten Vibrationen als Körperschall (aufgenommen durch die Piezopolymere) und als Luftschallwelle (aufgenommen durch konventionelle Kondensator-Mikrophone) wurden simultan gemessen und dann untersucht. Daraus kann man Aussagen über Entstehung, Ausbreitung und Abstrahlung der erzeugten Moden in das umgebende Medium ableiten. Letztlich charakterisieren diese Ergebnisse das gesamte vibrierende System akustisch.

This paper studies is considered the problem of coherent re ection of an acoustic plane wave from a rough seabed with a randomly inhomogeneous sediment layer overlying a uniform elastic basement. The randomness of the sound eld is attributable to the rough- ness of the seabed and the sound-speed perturbation in the sediment layer, resulting in a joint rough surface and volume scattering problem. An approach based upon perturbation theory, combined with a derived Green's function for a slab bounded above and below by a uid and an elastic half space, respectively, is employed to obtain an analytic solution for the coherent eld in the sediment layer. Furthermore, a boundary perturbation the- ory developed by Kuperman and Schmidt [22] is applied to treat the problem of rough surface scattering. A linear system is then established to facilitate the computation of the coherent re ection eld. The coherent re ection coe cients for various surface roughness, sediment randomness, frequency, sediment thickness, and basement elasticity have been generated numerically and analyzed. It was found that the higher/larger size of surface and/or medium randomness, frequency, thickness, and shear-wave speed, the lower the coherent re ection. Physical interpretations of the various results are provided.

<p>This thesis concerns the numerical simulation of creeping rays and their contribution to high frequency scattering problems.</p><p>Creeping rays are a type of diffracted rays which are generated at the shadow line of the scatterer and propagate along geodesic paths on the scatterer surface. On a perfectly conducting convex body, they attenuate along their propagation path by tangentially shedding diffracted rays and losing energy. On a concave scatterer, they propagate on the surface and importantly, in the absence of dissipation, experience no attenuation. The study of creeping rays is important in many high frequency problems, such as design of sophisticated and conformal antennas, antenna coupling problems, radar cross section (RCS) computations and control of scattering properties of metallic structures coated with dielectric materials.</p><p>First, assuming the scatterer surface can be represented by a single parameterization, we propose a new Eulerian formulation for the ray propagation problem by deriving a set of <i>escape </i>partial differential equations in a three-dimensional phase space. The equations are solved on a fixed computational grid using a version of fast marching algorithm. The solution to the equations contain information about all possible creeping rays. This information includes the phase and amplitude of the ray field, which are extracted by a fast post-processing. The advantage of this formulation over the standard Eulerian formulation is that we can compute multivalued solutions corresponding to crossing rays. Moreover, we are able to control the accuracy everywhere on the scatterer surface and suppress the problems with the traditional Lagrangian formulation. To compute all possible creeping rays corresponding to all shadow lines, the algorithm is of computational order O(<i>N</i>³ log <i>N</i>), with<i> N</i>³ being the total number of grid points in the computational phase space domain. This is expensive for computing the wave field for only one shadow line, but if the solutions are sought for many shadow lines (for many illumination angles), the phase space method is more efficient than the standard methods such as ray tracing and methods based on the eikonal equation.</p><p>Next, we present a modification of the single-patch phase space method to a multiple-patch scheme in order to handle realistic problems containing scatterers with complicated geometries. In such problems, the surface is split into multiple patches where each patch has a well-defined parameterization. The escape equations are solved in each patch, individually. The creeping rays on the scatterer are then computed by connecting all individual solutions through a fast post-processing.</p><p>We consider an application to mono-static radar cross section problems where creeping rays from all illumination angles must be computed. The numerical results of the fast phase space method are presented.</p>

The scattering of waves by a finite thin plate in a two-dimensional wave guide and an array of finite thin plates, in the presence of subsonic mean flow, are formulated using a mode matching technique. The influence of mean flow on trapped modes in the vicinity of a finite thin plate in a two-dimensional wave guide is then investigated by putting the amplitude of the forcing term to zero in the scattering problem. The conditions for complex resonances are found, and numerical results are computed. The influence of mean flow on Rayleigh-Bloch modes is investigated by using a similar methodology. The condition for embedded trapped modes to exist is introduced next, and then numerical results for embedded trapped modes without mean flow are presented. Complex resonances without mean flow are then found by fixing the geometry of the waveguide. The influence of mean flow on complex resonances and embedded trapped modes is investigated subsequently. In addition, the investigation of scattering coefficients is discussed when the frequency of an incident wave is near the real part of the frequency of complex resonances or embedded trapped modes. Embedded trapped modes near an indentation in a strip wave guide, which may correspond to a two-dimensional acoustic wave guide or a channel of uniform water depth in water waves, are also found. Modes are sought which are either symmetric or anti-symmetric about the centreline of the guide and the centre of the indentation. In each case, a simple approximate solution is found numerically. Full solutions are then found by using a Galerkin approach in which the singularity near the indentation edge is modelled by choosing proper special functions. The final part of the thesis is devoted to spinning modes (Rayleigh-Bloch modes) in a cylindrical waveguide in the presence of radial fins. A mode matching technique is used to obtain the potential, and the coefficients in the expansion are found numerically by using an efficient Galerkin procedure. In addition, an existence proof for modes symmetric about the centre of the guide and the centre of the section with radial fins is given by applying a variational approach. The connection between Rayleigh-Bloch modes and trapped modes is discussed thereafter, and numerical results for a number of geometric configurations are presented.

Biot's theory when applied to homogeneous media (involving the macroscopic flow mechanism) cannot explain the high level of attenuation observed in natural porous media over the seismic frequency range. However, several successful mesocopic inhomogeneity models have been developed to account for P wave attenuation. In this thesis I further develop the approaches to tackle S wave velocity and attenuation, to simulate transient wave propagation in poroelastic media, and to construct new models for determining the effective parameters of porous media containing mesoscopic inhomogeneities. As an important application of the double-porosity dual-permeability (DPDP) model, I have reformulated the effective Biot model using the total-field variables. This gives rise to new and more general governing equations than the previous approach based on the host phase field variables (which become a special case of the more general treatment). The analytical transient solution and dispersion characteristics for the double-porosity model and also for a poro-viscoacoustic model are derived over the entire frequency range for a homogeneous medium. The comparison between the results of the two models shows that dissipation by local mesoscopic flow of the double porosity model is very hard to fit by a single Zener element over a broad band. I chose the relaxation function to approximate the dispersion behaviour of the double porosity model just around the source centre frequency. It is shown that for most water-filled sandstones having a double porosity structure, wave propagation can be well described by the poro-viscoaoustic model with a single Zener element in the seismic frequency range. The transient solution for heterogeneous double porosity media is obtained by a numerical pseudospectral time splitting technique. This method is extended to 2.5-D poro-viscoelastic media to capture both P and S wave behaviour. I also demonstrate that if the frequency is below several Hz, then a single Kelvin-Voigt element gives an even better result than a single Zener element. I propose a two-phase permeability spherical inclusion model and obtain the dispersion curves of phase velocity and dissipation factor for the composite. I then determine the effective dynamic permeability of porous media with mesoscopic heterogeneities over the whole frequency range. This result is used to check the validity of other measures of effective dynamic permeability, deduced from the effective hydraulic permeability by replacing the permeability of the components with their dynamic values as determined from the Johnson, Koplik and Dashen (JKD) model. I also investigate the scattering of plane transverse waves by a spherical porous inclusion embedded in an infinite poroelastic medium. The vector displacement wave equations of Biot’s theory are solved as an infinite series of vector spherical harmonics for the case of a plane S-wave incidence. Then, the non-self-consistent theory is used to derive the dispersion characteristics of shear wave velocity and attenuation for a porous rock having mesoscopic spherical inclusions which are designed to represent either the patchy saturation model or the double porosity model with dilute concentrations of identical inclusions. / http://proxy.library.adelaide.edu.au/login?url= http://library.adelaide.edu.au/cgi-bin/Pwebrecon.cgi?BBID=1457632 / Thesis (Ph.D.) -- University of Adelaide, School of Chemistry and Physics, 2009