Exponential Variation Of Density Research Articles

Impact of inhomogeneity on SH-type wave propagation in an initially stressed composite structure

The present analysis has been made on the influence of distinct form of inhomogeneity in a composite structure comprised of double superficial layers lying over a half-space, on the phase velocity of SH-type wave propagating through it. Propagation of SH-type wave in the said structure has been examined in four distinct cases of inhomogeneity viz. when inhomogeneity in double superficial layer is due to exponential variation in density only (Case I); when inhomogeneity in double superficial layers is due to exponential variation in rigidity only (Case II); when inhomogeneity in double superficial layer is due to exponential variation in rigidity, density and initial stress (Case III) and when inhomogeneity in double superficial layer is due to linear variation in rigidity, density and initial stress (Case IV). Closed-form expression of dispersion relation has been accomplished for all four aforementioned cases through extensive application of Debye asymptotic analysis. Deduced dispersion relations for all the cases are found in well-agreement to the classical Love-wave equation. Numerical computation has been carried out to graphically demonstrate the effect of inhomogeneity parameters, initial stress parameters as well as width ratio associated with double superficial layers in the composite structure for each of the four aforesaid cases on dispersion curve. Meticulous examination of distinct cases of inhomogeneity and initial stress in context of considered problem has been carried out with detailed analysis in a comparative approach.

Influence of density variation on the arbitrarily propagating crack tip fields in functionally graded materials

The stress and displacement fields for an arbitrarily propagating crack tip in functionally graded materials (FGMs) with exponential variation of density and shear modulus are obtained. Nonhomogeneous parameters of density and shear modulus are different from each other. The solutions for higher order terms in the dynamic equilibriums are obtained by transforming the general differential equations to the scaled Laplace’s equations. Using the stress fields, the effects of the nonhomogeneous density on stress components is investigated. In addition, the contours of the constant maximum shear stress at a propagating crack tip are generated and the effects of the nonhomogeneous density on the isochromatics are discussed.

Reflection and transmission of compressional waves: Some exact results

Exact results are derived for the reflection and transmission of acoustic compressional waves by an arbitrary stratification. These results include conservation and reciprocity theorems, low-frequency and high-frequency limiting forms, and analytic solutions for two special stratifications, both having exponential variation of density with depth, and linear or exponential variations of sound speed.

Analytic theory of orbit contraction due to atmospheric drag

The motion of a satellite in orbit, subject to atmospheric force and the motion of a reentry vehicle are governed by the same forces, namely, gravitational and aerodynamic. This suggests the derivation of a uniform set of equations applicable to both cases. For the case of satellite motion, by a proper transformation and by the method of averaging, a technique appropriate for long duration flight, the classical nonlinear differential equation describing the contraction of the major axis is derived. While previous authors, and in particular King-Hele, integrated this equation using various heuristic methods, the present authors present a rigorous analytic solution, with a high degree of accuracy, using Poincaré's method of small parameters. Next, using Lagrange's expansion, the major axis is expressed explicitly as a function of the eccentricity. The solution is uniformly valid for moderate and small eccentricities. This is a major achievement due to the discovery of a certain recurrence formula which facilitates the long and tedious analytic process. For highly eccentric orbits, the asymptotic equation is derived directly from the general equation. To obtain the same equation King-Hele must use an entirely different method. Again, while King-Hele only succeeded in obtaining an approximate solution to this case using a heuristic method of integration, the exact solution to the asymptotic equation has been obtained by the present authors. Numerical solutions have been generated to display the accuracy of the analytic theory. The explicit solution has been derived using a spherically symmetrical atmosphere with exponential variation of density with height but the basic equations developed and the technique for their integration apply to the case of an oblate atmosphere which is locally exponential.

Effect of continuous variation of the density of a liquid on the waves generated by moving surface pressures

This study investigates the plane linear problem of steady-state internal waves in an ideal incompressible liquid with nonuniform density. The waves are generated by surface pressures applied in a bounded region which moves at constant velocity. It is assumed that the density in the unperturbed state varies continuously with depth, remaining constant in the upper and lower layers and varying according to an exponential law in the middle layer. The problem may be regarded, in particular, as a hydrodynamic model for the study of internal waves produced by a cyclone moving over the surface of the ocean. Analogous investigations for a homogeneous liquid were carried out in [1–3]; internal waves for a liquid with the above-mentioned law of density variation but with stationary pressure changes which are periodic with respect to time were studied in [4]. Problems analogous to the one considered here, both for exponential variation of density in the entire layer and for the case of a nonuniform layer near the surface, were investigated in [5, 6]. An analysis of non-linear waves of the steady-state type with arbitrary distribution of vorticity and density with respect to depth was carried out in [7, 8].

Stability of heterogeneous flows to nonaxisymmetric disturbances

The stability of heterogeneous fluids has been extensively investigated by TAYLOl~ (1931), SYI~OE (1933), MILES (196i), HOW~a~D and GuPTA (1962) and others. These results have also been extended to include the stability Of cylindrical masses of fluid but mostly for axisymmetric disturbances. When the fluid is homogeneous and incompressible and is having a 'solid body' rotation the stability for nonaxisymmetric disturbances has been investigated by HOWARD and GO-eTA (1962), LvI)WlEO (1961) and PEDLEY (1968). In the case of a homogeneous fluid, in addition to the solid body rotation, the presence of even ~ small axial shear makes the system unstable as has been shown by LIIDWlEG (1961) for a narrow gap and by PEDLEu (1968) without this restriction. In this paper we consider the stability for non-axisymmetrical disturbances of a cylindrical mass of heterogeneous fluid, with an exponential variation of density in the radial direction and having both axial and azimuthal velocities. Assuming the dependence of the radial perturbation velocity on r ~o be of the form u = ?l-'~H(r), we discuss two cases m = t/2 and 1. The general stability criteria, for both cases, have been derived. In the second case (m-----1), as an illustration the general stability criterion is applied to the Poiseuille type flow and a bound for instability is obtained. The growth rate of the most rapidly growing disturbances is also determined. 2. Formulation ol the Problem