Nonlinear Acoustics in Underwater and Biomedical Applications: Array Performance Degradation and Time Reversal Invariance

This dissertation describes a model for acoustic propagation in inhomogeneous flu- ids, and explores the focusing by arrays onto targets under various conditions. The work explores the use of arrays, in particular the time reversal array, for underwater and biomedical applications. Aspects of propagation and phasing which can lead to reduced focusing effectiveness are described. An acoustic wave equation was derived for the propagation of finite-amplitude waves in lossy time-varying inhomogeneous fluid media. The equation was solved numerically in both Cartesian and cylindrical geometries using the finite-difference time-domain (FDTD) method. It was found that time reversal arrays are sensitive to several debilitating factors. Focusing ability was determined to be adequate in the presence of temporal jitter in the time reversed signal only up to about one-sixth of a period. Thermoviscous absorption also had a debilitating effect on focal pressure for both linear and nonlinear propagation. It was also found that nonlinearity leads to degradation of focal pressure through amplification of the received signal at the array, and enhanced absorption in the shocked waveforms. This dissertation also examined the heating effects of focused ultrasound in a tissue-like medium. The application considered is therapeutic heating for hyperther- mia. The acoustic model and a thermal model for tissue were coupled to solve for transient and steady temperature profiles in tissue-like media. The Pennes bioheat equation was solved using the FDTD method to calculate the temperature fields in tissue-like media from focused acoustic sources. It was found that the temperature-dependence of the medium's background prop- erties can play an important role in the temperature predictions. Finite-amplitude effects contributed excess heat when source conditions were provided for nonlinear ef- fects to manifest themselves. The effect of medium heterogeneity was also found to be important in redistributing the acoustic and temperature fields, creating regions with hotter and colder temperatures than the mean by local scattering and lensing action. These temperature excursions from the mean were found to increase monotonically with increasing contrast in the medium's properties.

Identifying the damage mechanisms in paper using the acoustic emission monitoring technique

This paper investigates the use of the acoustic emission (AE) monitoring technique for use in identifying the damage mechanisms present in paper associated with its production process. The microscopic structure of paper consists of a random mesh of paper fibres connected by hydrogen bonds. This implies the existence of two damage mechanisms, the failure of a fibre-fibre bond and the failure of a fibre. This paper describes a hybrid mathematical model which couples the mechanics of the mass-spring model to the acoustic wave propagation model for use in generating the acoustic signal emitted by complex structures of paper fibres under strain. The derivation of the mass-spring model can be found in [1,2], with details of the acoustic wave equation found in [3,4]. The numerical implementation of the vibro-acoustic model is discussed in detail with particular emphasis on the damping present in the numerical model. The hybrid model uses an implicit solver which intrinsically introduces artificial damping to the solution. The artificial damping is shown to affect the frequency response of the mass-spring model, therefore certain restrictions on the simulation time step must be enforced so that the model produces physically accurate results. The hybrid mathematical model is used to simulate small fibre networks to provide information on the acoustic response of each damage mechanism. The simulated AEs are then analysed using a continuous wavelet transform (CWT), described in [5], which provides a two dimensional time-frequency representation of the signal. The AEs from the two damage mechanisms show different characteristics in the CWT so that it is possible to define a fibre-fibre bond failure by the criteria listed below. The dominant frequency components of the AE must be at approximately 250 kHz or 750 kHz. The strongest frequency component may be at either approximately 250 kHz or 750 kHz. The duration of the frequency component at approximately 250 kHz is longer than that of the frequency component at approximately 750 kHz. Similarly, the criteria for identifying a fibre failure are given below. The dominant frequency component of the AE must be greater than 800 kHz. The duration of the dominant frequency component must be less than 5.00E-06 seconds. The dominant frequency component must be present at the front of the AE. Essentially, the failure of a fibre-fibre bond produces a low frequency wave and the failure of a fibre produces a high frequency pulse. Using this theoretical criteria, it is now possible to train an intelligent classifier such as the Self-Organising Map (SOM) [6] using the experimental data. First certain features must be extracted from the CWTs of the AEs for use in training the SOM. For this work, each CWT is divided into 200 windows of 5E-06s in duration covering a 100 kHz frequency range. The power ratio for each windows is then calculated and used as a feature. Having extracted the features from the AEs, the SOM can now be trained, but care is required so that the both damage mechanisms are adequately represented in the training set. This is an issue with paper as the failure of the fibre-fibre bonds is the prevalent damage mechanism. Once a suitable training set is found, the SOM can be trained and its performance analysed. For the SOM described in this work, there is a good chance that it will correctly classify the experimental AEs.

An analysis of the changes to the mechanical and acoustical properties of paper caused by the damage mechanisms present in the fibre structure

This paper will analyse two of the likely damage mechanisms present in a paper fibre matrix when placed under controlled stress conditions: fibre/fibre bond failure and fibre failure. The failure process associated with each damage mechanism will be presented in detail focusing on the change in mechanical and acoustic properties of the surrounding fibre structure before and after failure. To present this complex process mathematically, geometrically simple fibre arrangements will be chosen based on certain assumptions regarding the structure and strength of paper, to model the damage mechanisms. The fibre structures are then formulated in terms of a hybrid vibro-acoustic model based on a coupled mass/spring system and the pressure wave equation. The model will be presented in detail in the paper. The simulation of the simple fibre structures serves two purposes; it highlights the physical and acoustic differences of each damage mechanism before and after failure, and also shows the differences in the two damage mechanisms when compared with one another. The results of the simulations are given in the form of pressure wave contours, time-frequency graphs and the Continuous Wavelet Transform (CWT) diagrams. The analysis of the results leads to criteria by which the two damage mechanisms can be identified. Using these criteria it was possible to verify the results of the simulations against experimental acoustic data. The models developed in this study are of specific practical interest in the paper-making industry, where acoustic sensors may be used to monitor continuous paper production. The same techniques may be adopted more generally to correlate acoustic signals to damage mechanisms in other fibre-based structures.

Boundary conditions in the simplest model of linear and second harmonic magneto-optical effects

This paper is concerned with linear and nonlinear magneto- optical effects in multilayered magnetic systems when treated by the simplest phenomenological model that allows their response to be represented in terms of electric polarization, The problem is addressed by formulating a set of boundary conditions at infinitely thin interfaces, taking into account the existence of surface polarizations. Essential details are given that describe how the formalism of distributions (generalized functions) allows these conditions to be derived directly from the differential form of Maxwell's equations. Using the same formalism we show the origin of alternative boundary conditions that exist in the literature. The boundary value problem for the wave equation is formulated, with an emphasis on the analysis of second harmonic magneto-optical effects in ferromagnetically ordered multilayers. An associated problem of conventions in setting up relationships between the nonlinear surface polarization and the fundamental electric field at the interfaces separating anisotropic layers through surface susceptibility tensors is discussed. A problem of self- consistency of the model is highlighted, relating to the existence of resealing procedures connecting the different conventions. The linear approximation with respect to magnetization is pursued, allowing rotational anisotropy of magneto-optical effects to be easily analyzed owing to the invariance of the corresponding polar and axial tensors under ordinary point groups. Required representations of the tensors are given for the groups infinitym, 4mm, mm2, and 3m, With regard to centrosymmetric multilayers, nonlinear volume polarization is also considered. A concise expression is given for its magnetic part, governed by an axial fifth-rank susceptibility tensor being invariant under the Curie group infinityinfinitym.

The Steady-State and Transient Characteristic Analysis of Thermal Fixing a Photorefractive Hologram Grating in Lithium Niobate

The transient interaction between a refraction index grating and light beams during simultaneous writing and thermal fixing of a photorefractive hologram is investigated. With a diffusion- and photovoltaic-dominated carrier transport mechanism and carrier thermal activation (temperature dependent) considered in Fe:LiNbO3 crystal, from the standpoint of field-material coupling, the theoretical thermal fixing time and the space-charge field buildup, spatial distribution, and temperature dependence are given numerically by combining the band transport model with mobile ions with the coupled-wave equation

Continuum bound states as surface states of a finite periodic system

We discuss the relation between continuum bound states (CBSs) localized on a defect, and surface states of a finite periodic system. We model an experiment of Capasso et al. [F. Capasso, C. Sirtori, J. Faist, D. L. Sivco, S-N. G. Chu, and A. Y. Cho, Nature (London) 358, 565 (1992)] using the transfer-matrix method. We compute the rate for intrasubband transitions from the ground state to the CBS and derive a sum rule. Finally we show how to improve the confinement of a CBS while keeping the energy fixed.

Exercise sheet 4

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Exam 1998

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MATH2038 Notes, Lectures and handouts

The MATH2038 (Partial Differential Equations) course, as given in semester 2 2008/9. Syllabus has changed slightly from previous years, as has coursework weighting.

Acoustic trapped modes in a three-dimensional waveguide of slowly varying cross section

In this paper we develop an asymptotic scheme to approximate the trapped mode solutions to the time harmonic wave equation in a three-dimensional waveguide with a smooth but otherwise arbitrarily shaped cross section and a single, slowly varying `bulge', symmetric in the longitudinal direction. Extending the work in Biggs (2012), we first employ a WKBJ-type ansatz to identify the possible quasi-mode solutions which propagate only in the thicker region, and hence find a finite cut-on region of oscillatory behaviour and asymptotic decay elsewhere. The WKBJ expansions are used to identify a turning point between the cut-on and cut-on regions. We note that the expansions are nonuniform in an interior layer centred on this point, and we use the method of matched asymptotic expansions to connect the cut-on and cut-on regions within this layer. The behaviour of the expansions within the interior layer then motivates the construction of a uniformly valid asymptotic expansion. Finally, we use this expansion and the symmetry of the waveguide around the longitudinal centre, x = 0, to extract trapped mode wavenumbers, which are compared with those found using a numerical scheme and seen to be extremely accurate, even to relatively large values of the small parameter.

Acoustic scattering by an inhomogeneous layer on a rigid plate

The problem of scattering of time-harmonic acoustic waves by an inhomogeneous fluid layer on a rigid plate in R2 is considered. The density is assumed to be unity in the media: within the layer the sound speed is assumed to be an arbitrary bounded measurable function. The problem is modelled by the reduced wave equation with variable wavenumber in the layer and a Neumann condition on the plate. To formulate the problem and prove uniqueness of solution a radiation condition appropriate for scattering by infinite rough surfaces is introduced, a generalization of the Rayleigh expansion condition for diffraction gratings. With the help of the radiation condition the problem is reformulated as a system of two second kind integral equations over the layer and the plate. Under additional assumptions on the wavenumber in the layer, uniqueness of solution is proved and the nonexistence of guided wave solutions of the homogeneous problem established. General results on the solvability of systems of integral equations on unbounded domains are used to establish existence and continuous dependence in a weighted norm of the solution on the given data.

A GRADIENT-LIKE NONAUTONOMOUS EVOLUTION PROCESS

In this paper we consider a dissipative damped wave equation with nonautonomous damping of the form u(tt) + beta(t)u(t) - Delta u + f(u) (1) in a bounded smooth domain Omega subset of R(n) with Dirichlet boundary conditions, where f is a dissipative smooth nonlinearity and the damping beta : R -> (0, infinity) is a suitable function. We prove, if (1) has finitely many equilibria, that all global bounded solutions of (1) are backwards and forwards asymptotic to equilibria. Thus, we give a class of examples of nonautonomous evolution processes for which the structure of the pullback attractors is well understood. That complements the results of [Carvalho & Langa, 2009] on characterization of attractors, where it was shown that a small nonautonomous perturbation of an autonomous gradient-like evolution process is also gradient-like. Note that the evolution process associated to (1) is not a small nonautonomous perturbation of any autonomous gradient-like evolution processes. Moreover, we are also able to prove that the pullback attractor for (1) is also a forwards attractor and that the rate of attraction is exponential.

A thermoelastic system of memory type in noncylindrical domains

In this paper, we consider a hyperbolic thermoelastic system of memory type in domains with moving boundary. The problem models vibrations of an elastic bar under thermal effects according to the heat conduction law of Gurtin and Pipkin. Global existence is proved by using the penalty method of Lions. (c) 2007 Elsevier Inc. All rights reserved.