Changes made in thesis for minor ammendments - P2 typo - P23 Added to abbreviations list - P24 not just elastic composites - P25 Bragg interference of scattered acoustic waves - P25 expanded revolutionary components - P26 Rayleigh line replaced with SAWs - P26 are typo - P27 holes typo - P29 abbreviations - P30 clarification of particle trajectory - P31 Figure 2.1 caption - P35 change of notation to i, positive kx from j, negative kx - P35 I tried putting eqs. 2.9 and 2.11 after 2.10, 2.12 and 2.13 but it didn't work because the 4 solutions mentioned in the paragraph after 2.13 are a result of substitution into 2.11. It just didn't flow very well with this change. - P36 uo and phi0 aren't constant scalars. - P36 the solutions are linearly dispersive rather than non-dispersive - P36 changed wording of the boundary condition to 'no net surface traction' instead of 'no net force'. Also changed the boundary condition to the more neat x3.T = 0 instead of T11 = T22 = T33 = 0. - P36 changed i-j notation - P37 chnaged i-j notation... and so on through subsequent eqautions - P37 clarification of D3 being electric displacement component normal to the surface. - P39 clarification of x axis in the caption - P40 corrected the subheading numbering. - P40 abbreviation definition - P42 Re-wording of Euler angle explanation for clarity. - P42 Added citation in figure caption - P43 simplified free and metallized cases for SAW velocities - P58 PnCs aren't only elastic composites. - P58 eigenfrequencies - P58 corrected the missing vector notations of lattice and reciprocal lattice vectors. - P59 changed transverse subscript from t to T - P62 definition of y axis in caption. - P63 distinction of acoustic and elastic waves. - P65 typo - P66 typo - P68 changed wording of rdiated energy to leaked energy - P68 removed the definition that metamaterials are periodic because they don't strictly have to be. - P69 'the' instead of 'this' - P69 briefly explained Lamb waves. - P70 typo - P71 syntax - P72 explained the Bragg band gap. - P79 improved the section describing the work by Columbi et al. as requested. - P81 defined single/double negative materials for acoustics. - P92 reasoning for using the MUMPS solver - P93 dispersion of SAWs in PnCs, not the PnCs - P94 syntax - P100 typo - P130 SAW bandgap not PnC bandgap and specify the shear BAW soundline. - P138 Typo line - P152 caption explains the arrow in b - P162 rho is the density and v is the group velocity of the SAW modes. - P174 added reasoning for origin choice - P190 Added list of publications - P191 Reviewed the formatting of all references. Lower cased initials were corrected.