CE 5310 Lecture Notes - Lecture 9: Wavenumber, Applied Physics Letters, Waveguide
Document Summary
Assuming one-dimensional (1-d) motion, the strain-stress relationship of a linear elastic material follows hooke"s law as: where (cid:2252) , (cid:2161) , (cid:2239) are stress, elastic young"s modulus and strain, respectively. Assuming propagating waves have small amplitude (low intensity), the governing (1) wave equation (pressure waves for a simple example) along x-direction is given by: (cid:2252)(cid:3404)(cid:2161)(cid:2239) (cid:2251)(cid:2260)(cid:2779)(cid:2203)(cid:2260)(cid:2202)(cid:2779)(cid:3404)(cid:2161)(cid:2260)(cid:2779)(cid:2203)(cid:2260)(cid:2206)(cid:2779) (2) transmission and reflection coefficients are of relevance in this case. For nonlinear elastic materials, the strain-stress relationship includes higher where (cid:2251) is density of material. Such parameters as wave velocity, attenuation, order strain terms as: (cid:2252)(cid:3404)(cid:2161)(cid:2239)(cid:4666)(cid:2778)(cid:3397)(cid:2236)(cid:2239)(cid:3397)(cid:2238)(cid:2239)(cid:2779)(cid:3397) (cid:4667) where (cid:2236) and (cid:2238) are a second order (quadratic in strain) and a third order (cubic in strain) nonlinear parameters, respectively. Here, the amplitude of propagating waves is no longer assumed infinitesimally small, so called finite amplitude waves. Due to the finite amplitude and nonlinear terms in equation (3), the governing wave equation becomes as: (cid:2251)(cid:2260)(cid:2779)(cid:2203)(cid:2260)(cid:2202)(cid:2779)(cid:3404)(cid:2161)(cid:2260)(cid:2779)(cid:2203)(cid:2260)(cid:2206)(cid:2779)(cid:3397)(cid:2779)(cid:2161)(cid:2236)(cid:2260)(cid:2203)(cid:2260)(cid:2206)(cid:2260)(cid:2779)(cid:2203)(cid:2260)(cid:2206)(cid:2779)(cid:3397)(cid:2780)(cid:2161)(cid:2238)(cid:3436)(cid:2260)(cid:2203)(cid:2260)(cid:2206)(cid:3440)(cid:2779)(cid:2260)(cid:2779)(cid:2203)(cid:2260)(cid:2206)(cid:2779)(cid:3397) .
