tailieunhanh - Wave Propagation 2011 Part 17

Tham khảo tài liệu 'wave propagation 2011 part 17', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | Wave Velocity Dispersion and Attenuation in Media Exhibiting Internal Oscillations 457 and frequency-dependent attenuation . However such oscillatory phenomena are not included in current effective medium theories such as the Biot theory Biot 1962 . Oscillations are usually described on the microscale . the scale of one single oscillator Dvorkin et al. 1990 Graham Higdon 2000 Hilpert et al. 2000 Beresnev et al. 2005 not taking into account the macroscale . continuous scale. Frehner et al. 2009 presented a model to combine the dynamics of wave propagation and rock-internal oscillations caused by capillarity effects using an effective medium theory. However the model was presented for internal undamped oscillations with only one particular resonance frequency. In this chapter we extend the effective medium model of Frehner et al. 2009 by introducing a viscous damping term and generalize the model for an arbitrary number of resonance frequencies described . by a probability density function of possible resonance frequencies in the limit case . We describe the simplified case of an acoustic medium exhibiting internal oscillations . a linear elastic medium with an elastic shear modulus equal to zero. In such a material only one type of wave . a compressional wave or P-wave can propagate. The internal oscillations will be described by the equation of motion for a damped harmonic oscillator. Different to Frehner et al. 2009 we use as an example for such a medium water containing gas bubbles where the dispersed gas bubbles can oscillate with their natural frequency. For that specific example the elastic and viscous interaction forces between the phases can be derived analytically and a comparison with literature and experimentally obtained results can be performed. 2. Mathematical model of acoustic medium exhibiting internal oscillations For simplicity reasons and because we want to understand first-order effects we choose a basic model. By choosing an .