The main objective is to provide a wide range of approaches that yield useful understanding for the different scenarios of waves in locally stratified media. A classification system is constructed, where the physical problems are put into categories with common mathematical models and problems. This provides a convenient tool for identifying which methods and approaches to employ for a particular physical problem. The bulk of the analysis is centred around two methods or theories - multiple scale homogenisation and generalised ray theory. Multiple scale homogenisation is relevant for long-wave propagation and finding the effective medium of the layered structure. The Backus average for finely layered media is derived in a new way, where multiple scale techniques and the propagator matrix formulation are combined to form a general framework for most linear waves in locally stratified media. With this framework, one can easily derive known formulae for effective media, including the anisotropic effective mass density for acoustic waves. Since the original Backus average was derived for systems close to the static limit, and the anisotropic mass density is a low frequency dynamic effect, acoustic waves are analysed in more detail. The technique is also applied to more complicated media, and it is shown how effective stiffness tensor can be found for piezoelectric media. However, the key feature of this framework is that higher order terms are easily calculated. Nevertheless, the higher order terms are sensitive to the distance propagated, and the effective medium solution is only valid for short distances of propagation. This is in agreement with previous simulations. The validity of Backus averaging or accuracy of effective medium theory is investigated for the one dimensional case. Using Magnus series and manipulation of exponential operators, a new relationship between the effective medium and exact solutions are derived. Based on this relation, the same observations are made as in previous numerical work. Bounds for the factor between the effective and exact solutions are also calculated.
| Identifer | oai:union.ndltd.org:bl.uk/oai:ethos.bl.uk:599757 |
| Date | January 2002 |
| Creators | Grover, B. |
| Publisher | University of Cambridge |
| Source Sets | Ethos UK |
| Detected Language | English |
| Type | Electronic Thesis or Dissertation |
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