Reflection and transmission coefficients of a thin bed

The study of thin-bed seismic response is an important part in lithologic and methane reservoir modeling, critical for predicting their physical attributes and/or elastic parameters. The complex propagator matrix for the exact reflections and transmissions of thin beds limits their application in thin-bed inversion. Therefore, approximation formulas with a high accuracy and a relatively simple form are needed for thin-bed seismic analysis and inversion. We have derived thin-bed reflection and transmission coefficients, defined in terms of displacements, and approximated them to be in a quasi-Zoeppritz matrix form under the assumption that the middle layer has a very thin thickness. We have verified the approximation accuracy through numerical calculation and concluded that the errors in PP-wave reflection coefficients RPP are generally smaller than 10% when the thin-bed thicknesses are smaller than one-eighth of the PP-wavelength. The PS-wave reflection coefficients RPS have lower approximation accuracy than RPP for the same ratios of thicknesses to their respective wavelengths, and the RPS approximation is not acceptable for incident angles approaching the critical angles (when they exist) except in the case of extremely strong impedance difference. Errors in phase for the RPP and RPS approximation are less than 10% for the cases of thicknesses less than one-tenth of the wavelengths. As expected, a thinner middle layer and a weaker impedance difference would result in higher approximation accuracy

Anomalous physical properties of underdoped weak-ferromagnetic superconductor RuSr2_2EuCu2_{2}O8_{8}

Similar to the optimal-doped, weak-ferromagnetic (WFM induced by canted antiferromagnetism, T$_{Curie}$ = 131 K) and superconducting (T$_{c}$ = 56 K) RuSr$_{2}$GdCu$_{2}$O$_{8}$, the underdoped RuSr$_{2}$EuCu$_{2}$O$_{8}$ (T$_{Curie}$ = 133 K, T$_{c}$ = 36 K) also exhibited a spontaneous vortex state (SVS) between 16 K and 36 K. The low field ($\pm$20 G) superconducting hysteresis loop indicates a weak and narrow Meissner state region of average lower critical field B$_{c1}^{ave}$(T) = B$_{c1}^{ave}$(0)[1 - (T/T$_{SVS}$)$^{2}$], with B$_{c1}^{ave}$(0) = 7 G and T$_{SVS}$ = 16 K. The vortex melting transition (T$_{melting}$ = 21 K) below T$_{c}$ obtained from the broad resistivity drop and the onset of diamagnetic signal indicates a vortex liquid region due to the coexistence and interplay between superconductivity and WFM order. No visible jump in specific heat was observed near T$_{c}$ for Eu- and Gd-compound. This is not surprising, since the electronic specific heat is easily overshadowed by the large phonon and weak-ferromagnetic contributions. Furthermore, a broad resistivity transition due to low vortex melting temperature would also lead to a correspondingly reduced height of any specific heat jump. Finally, with the baseline from the nonmagnetic Eu-compound, specific heat data analysis confirms the magnetic entropy associated with antiferromagnetic ordering of Gd$^{3+}$ (J = S = 7/2) at 2.5 K to be close to $\it{N_{A}k}$ ln8 as expected.Comment: 7 figure

Attenuation of stress waves in single and multi-layered structures

Analytical and experimental studies were made of the attenuation of the stress waves during passage through single and multilayer structures. The investigation included studies on elastic and plastic stress wave propagation in the composites and those on shock mitigating material characteristics such as dynamic stress-strain relations and energy absorbing properties. The results of the studies are applied to methods for reducing the stresses imposed on a spacecraft during planetary or ocean landings