Wave Number Approximation Research Articles

Acoustic wave propagation in a radial duct

The results of an experimental investigation of acoustic wave propagation through a radial duct are reported. The investigation was limited to large wavenumbers in the absence of flow, and experimental results are compared to a correspondng analytical model. The apparatus consisted of an anechoic termination, one rigid duct wall, and one variable impedance duct wall. Preliminary experiments revealed that the measurement techniques were appropriate and that the apparatus was behaving as expected. Experimental plots of pressure versus radial position were found to be in agreement with analytical predictions for cases involving two rigid duct walls. A variable backing depth Helmholtz resonator array was substituted for one of the rigid duct walls, and measurements of pressure versus radial position were made for three backing depths. Values of radial attenuation were estimated from the pressure measurements using a large wavenumber approximation. Experimental radial attenuation estimates are in good qualitative agreement with analytical predictions.

Analytic approximations to distorted wave integrals, for inelastic molecular collision cross sections

Several methods-for approximating distorted wave integrals using analytic formulae are presented and compared. The various methods are applied to the calculation of rotationally and vibrationally inelastic HeH 2 collisions and the resulting integrals compared with their exact counterparts. Some of the methods involve the use of the modified wave number approximation. This approximation is shown to break down seriously at large values of the orbital angular momentum. A further important point which is demonstrated is the importance of the region around the outer classical turning point of the two channels involved in the calculation. Among all the methods examined, a recently proposed one based on a generalisation of the well-known Mies formula is found to be the most reliable. A slightly improved version of this method is also developed and tested. Comparison of this improved method with exact calculations establishes its reliability over a large range of energies and total angular momentum quantum numbers.

Improved modified wavenumber approximation applied to elastic collision

A method to improve the usual modified wavenumber result is applied to an elastic collision problem. For an exponential repulsive potential, simple expressions for the phase shift are obtained. The partial cross sections obtained by this improved modified wavenumber approximation agree qualitatively with those of the exact numerical solutions. The total cross sections agree very well with the exact results.

Vibrational Excitation in a Morse Oscillator

Vibrational excitation in the system He+H2 is studied theoretically by considering H2 as a rotating Morse oscillator. The anharmonicity of molecular vibration is found to have only a small effect on the 0 → 1 transitions. Vibrational transition probabilities are also found to be insensitive to the rotational energy of the diatom. Following the approach of Mies and Shuler, based on the modified wavenumber approximation, three-dimensional transition probabilities are calculated from one-dimensional results and the steric factor is obtained as a function of the collision energy.

Semiclassical Treatment of the Optical Model

Semiclassical scattering theory is applied to the optical model for two different complex potentials. For the case of a Lennard-Jones interaction, the results for the imaginary part of the phase shift are in excellent agreement with the exact calculation of Marriott and Micha. The use of a modified wavenumber approximation yields useful estimates. For a complex exponential potential, the semiclassical and quantum expressions for the opacity function at zero impact parameter are identical.

Instability of Three-Layer Viscous Stratified Flow

Instability of plane Couette flow of three superposed layers of fluids of different viscosity between two horizontal planes is investigated. It is found that there are two modes of disturbance dominated, respectively, by the two interfaces. The flow is found to be unstable in the zero-order approximation of wavenumber α for certain values of the depth and viscosity ratios. This is owing to a sort of resonance which prevails when a free wave at the lower interface forces a free wave at the upper interface. As is known from results previously obtained, the existence of a single surface of viscosity discontinuity will cause instability. The presence of an additional surface of discontinuity may or may not be stabilizing, depending upon the values of the depth and viscosity ratios at the additional interface.

Rotational relaxation in mixtures of hydrogen isotopes and noble gases: Part III. Comparison with theory

In this article we will make a comparison between theory and the experimental results on the rotational relaxation times of mixtures of pH 2 and oD 2 with the noble gases. It is found that inelastic scattering cross sections obtained with the modified wave number approximation in the distorted wave method do not give satisfactory results.

Excitation of Molecular Vibration on Collision: Role of the High-Order Angular Momenta

Excitation of molecular vibration in the collision between two diatomic molecules is studied. The specific aims are to calculate the partial cross sections for general values of the collision angular momentum l and to show the important role of the high-order partial waves in the calculation of total cross sections. It is shown that the partial cross section becomes largest at lmax=12[(2/L)12−1], where L is dependent on the collision parameters and is of the order of 10−3−10−4; in H2–H2 values of l up to about 30, 55, 90, and 125 must be considered for the relative energy of 0.5, 1, 2, and 3 eV, respectively, for the calculation of total cross sections. In O2–O2 the corresponding l values are 125, 200, 340, and 450. It is also shown that the modified wavenumber approximation underestimates the average total cross section by a factor of 1.7 at 300°K and 1.3 at 4000°K for H2–H2 and for O2–O2 by 1.8 and 1.4 at the corresponding temperatures. The connection between the cross section for inverse power potentials and those for exponential potentials is discussed analytically.

Quantum-Mechanical Calculation of Harmonic Oscillator Transition Probabilities. II. Three-Dimensional Impulsive Collisions

The previous one-dimensional calculations of Shuler and Zwanzig for the vibrational transition probabilities for harmonic oscillators undergoing impulsive hard-sphere collisions with an incident atom have been extended to three dimensions. By the use of the modified wave number approximation, the three-dimensional scattering problem is related to the one-dimensional solution and the three-dimensional transition probabilities are obtained through numerical integration over the previously calculated one-dimensional probabilities. It is shown that vibrational transitions with | Δv | >1 are strongly allowed (in addition to the | Δv | = 1 transitions) in good agreement with the one dimensional impulsive collision results. The implication of this result on the mode and rate of the dissociation of diatomic molecules is discussed briefly.

Excitation of the Molecular Vibration by Collision

Excitation of the molecular vibration in H 2 + H 2 collision is studied using the so-called modified wave number method. The same problem has been investigated quite recently by Salkoff and Bauer who applied the distorted wave method thoroughly without making the modified wave number approximation. Thus the present work serves as a test of the accuracy of the modified wave number method which has been used by many authors to study this sort of problem. The result obtained here is in good agreement with the result by Salkoff and Bauer.