tailieunhanh - Acoustic Waves Part 3

Tham khảo tài liệu 'acoustic waves part 3', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | Rectifying Acoustic Waves 49 Here we consider acoustic waves propagating through an array of triangular holes drilled in an isotropic material. In this case the equations of motion governing the displacement vectors u r t are given by p x ùi r t õịơịị r t i 1 2 3 1 ơij r t cijmn x dnùm r t 2 where r x z x y z and the summation convention over repeated indices is assumed in Eqs. 1 and 2 . p x and cijmn x are the position-dependent mass density and elastic stiffness tensor of the system and ơij r t is the stress tensor. Furthermore we need to impose proper boundary conditions for SAWs. SAWs should satisfy the stress-free boundary condition at the surface z 0 or i 3 z 0 Ci3mn Vm z 0 0 i 1 2 3 . 3 Solving the equations with finite-difference time-domain FDTD method numerically we can obtain the time evolutions of displacement vectors u r t and stress tensors ơij r t at each point in the system. To calculate the transmission rate through the periodic array of triangle holes we define the acoustic Poynting vector Ji r t -Uj r t ơji r t from the continuity of energy flow. In terms of the Fourier components of the displacement u r o and the stress tensor ỡij r o the energy flow at frequency o in the x direction at the position x is expressed by Jx x o -4xỊ Im rnuij r m T x r m dydz. 4 Hence we can determine the transmission rate T O by the ratio of the element of the acoustic Poynting vector in the x direction Jx xD o to that in the absence of scatterers jX xD 0 which is given by T 0 Jx xD o Í0 . jx xD o 5 where xD is the detecting position which is on the right side of the scatterers for case I and on the left side of the scatterers for case II . We introduce an efficiency n 0 to quantify rectification by n o Ti o - Tii o Tj Tii o 6 where TI and TII are the transmission rates for cases I and II respectively. 3. Numerical results Bulk acoustic waves In the section we illustrate the rectifying effects of bulk acoustic waves propagating in the x-direction. Because of the