Second-order Quantities Research Articles

Are acoustic intensity and potential energy density first- or second-order quantities?

A reviewer’s comment to our manuscript ‘‘Acoustic radiation produced by a beam of sound,’’ [J. Acoust. Soc. Am. 72, 1673 (1982) has prompted us to consider the question posed by the title of this manuscript, which is not an uncommon one when dealing with nonlinear problems. As might be expected, we demonstrate that the problems reside mostly with semantics. To set the record straight we define and distinguish among acoustic potential energy density, potential energy density relative to a reference state, and internal energy density, as well as intensity and acoustic intensity.

Application of the Mohr-circle construction to problems of inhomogeneous deformation

Off-axis Mohr circles are used to represent asymmetric second-order tensor quantities that describe deformation with homogeneous position gradients or flow with homogeneous velocity gradients. The validity of this representation is demonstrated. Several applications to problems of inhomogeneous deformation are explored by considering neighboring domains A and B that are homogeneously but differently deformed or flowing. The compatibility condition in Mohr space is that the circles for A and B should intersect in a point that represents the material line of the A/B boundary. In the first application it is assumed that the stretch tensor is known for each domain, and a Mohr construction is used to find the orientation of a compatible A/B boundary and the rotation of one domain with respect to the other. The second application is similar but involves the instantaneous state of a system. Non-parallel simple shearing at known rates is assumed in A and B, and a construction is used to find the instantaneous orientation of a compatible boundary and the spin of one domain with respect to the other. The third and fourth applications apply Mohr circles to problems of deformation interpolation and deformation averaging. In the former a deformation and shape is found for a domain C that lies between A and B, themselves incompatible, such that the A/C and C/B boundaries are both compatible.

Evaluation of the optimal path in ATDHF theory

The formal resemblance between the adiabatic time-dependent Hartree-Fock (ATDHF) theory of Villars and of Baranger and Vénéroni (BV) has been demonstrated. While the BV approach mixes the zeroth- and second-order quantities. Villars neglects the second-order equation and suggests a self-consistent method of solving the zeroth- and first-order equations. It is shown here that in order to obtain the time evolution of the TDHF solution one has to include the second-order equation along with zeroth- and first-order equations. It is then shown by an analytic method that the consistent solutions of the three equations follow the bottom of the valley of the collective potential.

Similarity behaviour of momentumless turbulent wakes

Similarity solutions are determined for the turbulent wake of a self-propelled body (thrust = drag). The momentumless wake is shown to behave in a manner intermediate to homogeneous grid turbulence and more familiar free-shear flows such as the drag wake or jet. In essence the decay of momentumless-wake turbulence is similar to that of grid turbulence, but proceeds at a somewhat greater rate owing to lateral diffusion. The mean velocity difference is coupled to the difference$\overline{u^2}-\overline{v^2}$between the axial and radial components of the mean-square fluctuating velocity. It is necessary to consider governing relations for various second-order turbulence quantities. Previously developed closure approximations yield far-wake decay rates that agree well with available measurements. Production of turbulent energy is negligible asymptotically; thus there is no balance between production and dissipation, and the far-wake behaviour does not become independent of the initial (near-wake) conditions. Even the radial profiles depend on the initial conditions, and there is no natural length scale with which to characterize the far wake.

Second-order semiclassical calculations for diatomic molecules

The second-order WKB quantization condition is used to derive expressions for the second-order contributions to the vibrational energy levels and the rotational and centrifugal constants of a diatomic molecule. The second-order corrections to the RKR (Rydberg-Klein-Rees) turning points are also calculated. These second-order quantities are related to the quantities calculated from Kaiser's modified first-order quantization condition. Numerical calculations for the ground electronic state of carbon monoxide are compared with quantum-mechanical values obtained by numerical solution of the Schrödinger equation. In general, the second-order semiclassical values are in good agreement with the quantum-mechanical values, and can be computed by the present methods in a small fraction of the time.

On the Accuracy of Whitham's Method

The steady flow of an ideal gas past a conical body is studied by the method of matched asymptotic expansions and by Whitham's method in order to assess the accuracy of the latter. It is found that while Whitham's method does not yield a correct asymptotic representation of the perturbation field to second order in regions where the flow ahead of the Mach cone of the apex is disturbed, it does correctly predict the changes of the second-order perturbation quantities across a shock (the first-order shock strength). The results of the analysis are illustrated by a special case of a flat, rectangular plate at incidence.

Second-order effects in the propagation of elastic waves

Second-order effects in the propagation of elastic waves through homogeneous isotropic media are considered. The jump conditions governing second-order quantities at wave fronts where the first-order displacements have discontinuous derivatives are derived, and it is seen that a consistent theory requires the secondorder displacements to be discontinuous at such wave fronts, in general. The results are illustrated by some one-dimensional examples involving plane and spherical waves.

Two-Dimensional Supercavitating Plate Oscillating Under a Free Surface

The problem of a supercavitating flat plate at zero and nonzero cavitation numberoscillating under a free surface is analyzed by a linearized method using the accelerationpotential. The analysis is based on the concept of small velocity perturbations where in all second-order quantities are neglected. The flow is assumed two-dimensional, irrotational, incompressible, and gravitation-free. The potential-flow region is mapped on to an upper half-plane and the solution is expressed in an integral form using Cheng andRott's method. Special attention is given to the effect of approximate wake boundary conditions on the computed force and moment. It was estimated that the effect is of secondorder when the cavitation number is a first-order small quantity.

Second-order boundary-layer effects in hypersonic flow past axisymmetric blunt bodies

First- and second-order boundary-layer theory are examined in detail for some specific flow cases of practical interest. These cases are for flows over blunt axisymmetric bodies in hypersonic high-altitude (or low density) flow where second-order boundary-layer quantities may become important. These cases consist of flow over a hyperboloid and a paraboloid both with free-stream Mach number infinity and flow over a sphere at free-stream Mach number 10. The method employed in finding the solutions is an implicit finite-difference scheme. It is found to exhibit both stability and accuracy in the examples computed. The method consists of starting near the stagnation-point of a blunt body and marching downstream along the body surface. Several interesting properties of the boundary layer are pointed out, such as the nature of some second-order boundary-layer quantities far downstream in the flow past a sphere and the effect of strong vorticity interaction on the second-order boundary layer in the flow past a hyperboloid. In several of the flow cases, results are compared with other theories and experiments.

The Total Mass of a Particle in General Relativity

Without specifying otherwise the stress-energy distribution inside of a material particle, the assumption is made that the scalar quantity $T$, known in electromagnetism as the "scalar of Laue," shall vanish, in harmony with the Maxwellian stress tensor, and confirmed also by other considerations. It is shown that under this condition a theoretical explanation can be given of the long established empirical fact that the mass of any particle is necessarily positive. All metrical contributions to the total mass come out as necessarily positive, second-order quantities, with the only exception of an eventual spin, which has a decreasing instead of increasing effect on the mass. The strict proportionality of gravitational and and inertial mass can still be established, at least for the spherically symmetric case. The factor of proportionality, however, is numerically different from the value ordinarily assumed.