Bivectors and Waves in Mechanics and Optics
Bivectors occur naturally in the description of elliptically polarized homogeneous and inhomogeneous plane waves. The description of a homogeneous plane wave generally involves a vector (the unit vector along the propagation direction) and a bivbector (the complex amplitude of the wave). Inhomogeneous plane waves are described in terms of two bivectors - the complex amplitude and the complex slowness. The use of bivectors and their associated ellipses is essential for the presentation of the 'directional ellipse' method given in this book, in deriving all possible inhomogeneous plane wave solutions in a given context.
The purpose of this book is to give an extensive treatment of the properties of bivectors and to show how these may be applied to the theory of homogeneous and inhomogeneous plane waves. For each chapter there are exercises with answers, many of which present further useful properties which are referred to afterwards. The material in this book is suitable for senior undergraduate and first year graduate students. It will also prove useful for researchers interested in homogeneous and inhomogeneous plane waves.
Complex symmetric matrices
Complex orthogonal matrices and complex
Homogeneous and inhomogeneous plane waves
Description of elliptical polarization
Electromagnetic plane waves
Plane waves in linearized elasticity theory
Plane waves in viscous fluids
Appendix Spherical trigonometry
amplitude bivector angle arbitrary associated assumed bivector C₁ called central Chapter circle circularly polarized complex components conjugate directions conjugate radii consider corresponding crystals defined definite described determined direction double root eigenbivectors eigenvalue elastic ellipse ellipsoid energy density equation equivalently Exercises expression factor field Figure Find follows given gives Hence homogeneous waves inhomogeneous plane waves Introducing isotropic linear linearly independent material mean energy flux obtain optic axis orthogonal matrix parallel phase planes of constant polarization form polarized waves positive possible prescribed principal axes Proof propagation condition real direction reciprocal relation respectively root satisfy scalar secular equation similar and similarly similarly situated simple slowness solutions symmetric symmetric matrix tensor Theorem transformation triple unit vectors write written yields zero