Wave propagation in stratified media may be explained by ray theory, effective medium theory, or scattering theory depending on the scales of wavelength and layer spacing. To effectively integrate and use seismic data at different frequencies and widely varying scales, it is essential to understand the domain of applicability of long and short wavelength behavior and the transition between them. A joint experimental and theoretical study was conducted to investigate velocity behavior at the transition from ray theory to effective medium theory in stratified media. Velocity measurements were performed at 50 and 500 kHz on periodic media composed of steel and plastic discs. The ratio of wavelength to layer spacing, λ/d, spanned more than two orders of magnitude between 0.1 and 50, and the volume fraction of steel ranged from 9 to 89 percent by volume. Our results confirm that velocities in stratified media depend on composition and are controlled by the ratio of wavelength to layer spacing. Velocities in the short wavelength limit are generally faster than velocities in the long wavelength limit. We find that transition from ray to effective medium approximations occurs over a narrow range of λ/d at a value of approximately 10. The amount of velocity change increases with impedance contrast, but the value of λ/d at the transition is generally independent of the composition of the stratified medium. Our numerically simulated waveforms are in close agreement with the experimentally observed delayed first arrival in the long wavelength limit and with the reduced amplitudes at the transition from short to long wavelength regime.
Effective medium theory for magnetodielectric composites: Beyond the long-wavelength limit
