Search

In this thesis, we use asymptotic methods to solve problems of wave propagation through infinite and finite (only consider those that are finite in one direction) arrays of scatterers. Both two- and three-dimensional arrays are considered. We always assume the scatterer size is much smaller than both the wavelength and array periodicity. Therefore a small parameter is involved and then the method of matched asymptotic expansions is applicable. When the array is infinite, the elastic wave scattering in doubly-periodic arrays of cavity cylinders and acoustic wave scattering in triply-periodic arrays of arbitrary shape rigid scatterers are considered. In both cases, eigenvalue problems are obtained to give perturbed dispersion approximations explicitly. With the help of the computer-algebra package Mathematica, examples of explicit approximations to the dispersion relation for perturbed waves are given. In the case of finite arrays, we consider the multiple resonant wave scattering problems for both acoustic and elastic waves. We use the methods of multiple scales and matched asymptotic expansions to obtain envelope equations for infinite arrays and then apply them to a strip of doubly or triply periodic arrays of scatterers. Numerical results are given to compare the transmission wave intensity for different shape scatterers for acoustic case. For elastic case, where the strip is an elastic medium with arrays of cavity cylinders bounded by acoustic media on both sides, we first give numerical results when there is one dilatational and one shear wave in the array and then compare the transmission coefficients when one dilatational wave is resonated in the array for normal incidence. Key words: matched asymptotic expansions, multiple scales, acoustic scattering, elastic scattering, periodic structures, dispersion relation.

Hyperbolic metamaterials are artificial nano-fabricated structures, which can be made by periodic metallo-dielectric layers. Hyperbolic metamaterials show great potential in a wide variety of applications, and hence optical properties of such materials, i.e., the behavior of beams propagating and their transmission characteristics, are of great significance. In this work, first, the propagation of profiled beams is analyzed inside an anisotropic bulk medium using a transfer function approach in one transverse dimension. Then, a transfer matrix method is developed to study arbitrary optical field propagation through the hyperbolic metamaterials, including both incident and emergent media. The feasibility and accuracy of the transfer matrix method are verified numerically. The unique optical properties of the hyperbolic metamaterials, such as negative refractive index, are revealed analytically by the techniques mentioned above and are supported experimentally. Moreover, the dispersion relation of periodic metallo-dielectric structures is derived. It is shown by the dispersion relation that the transmission characteristics of hyperbolic metamaterials are intrinsically affected by many factors, viz., the complex refractive index of the materials, and the geometric parameters of the structures. With proper design, the transmittance spectra of such structures can be manipulated, leading to applications such as transmission filters.By providing alternative methods to study beam propagation and wave transmission across anisotropic metamaterial slabs, this work should be helpful to reveal the optical properties of hyperbolic metamaterials, enabling metamaterial-based devices to be designed and optimized with higher efficiency and greater flexibility.

Abstract: The health monitoring of critical structures plays a crucial role in locating damage positions timely and preventing catastrophic failures. Much attention has been devoted to exploiting piezoelectric sensors/actuators to develop techniques of recording elastic wave signals to realize structural health monitoring (SHM). This thesis is to conduct a systematic investigation of the dynamic behaviour of piezoelectric sensors and their application in quantitative crack identification in SHM systems. A typical SHM system contains piezoelectric sensors bonded to a host structure to be monitored. This structure is subjected to a dynamic excitation which will induce elastic wave propagation in it. When the wave encounters cracks, it will be scattered and the scattered wave will be recorded by the piezoelectric sensors. The recorded signals contain the information of the cracks thus can be used to identify the parameters of the cracks. In this study, theoretical modelling and simulation are conducted to investigate the load transfer between the sensor and the host structure, the dispersion relation of wave propagation, and the multiple scattering of elastic waves. In addition, a crack identification technique is investigated using the voltage signals based on optimization method. Four aspects of the work were accordingly studied and examined. Firstly, a new model is developed for surface-bonded piezoelectric thin-sheets with bending effect. The coupled electromechanical behaviour and the effect of bending upon load transfer and local stress field are studied. Secondly, a new analytical treatment is provided for wave propagation in layered piezoelectric structures, including dispersion characteristics and harmonic wave propagation. The two lowest wave modes of the guided wave in such structures are analyzed. Thirdly, a new semi-analytical solution is determined for the complicated dynamic interaction between piezoelectric sensors and cracks using pseudo incident wave method and superposition. This method has the advantages of the reliability of analytical solutions and the flexibility of typical numerical methods, and finds explicit relations between the voltage output of the piezoelectric sensor and the crack parameters. Inversely, by integrating this relation and known voltage data into an optimization process, a novel crack identification technique is established. This technique quantitatively identifies the position, the length and the orientation of typically embedded cracks effectively. The methods proposed in this thesis can be used to understand the dynamic behaviour of piezoelectric based SHM systems, multiple scattering of elastic waves and provide insights into developing new methods for quantitative crack identification.

The display of an Airy function via an optical image processing technique is demonstrated. Using a two-dimensional object with a cubic amplitude transmittance, one can observe a Fourier transformed image of which intensity variation is identical to that of the Airy function. A lens system is used to achieve the Fourier transformation of the object which is a two-dimensional binary filter. A set of computer simulations are performed prior to the optical experiments. Using the same method of optical display, the transient wave propagation in a dispersive medium is optically displayed based on the analogous relationship found between two equations: One is the Fourier form of a transient wave field propagating in a dispersive medium and the other is an equation of the diffraction pattern of an object constructed with two juxtaposed two-dimensional filters. A theoretical analysis is provided along with computer simulations. The use of a Spatial Light Modulator is proposed as the source of the input object in the optical experiment.