Part I. Historical Survey: 1. History of an approximation method of wide importance in various branches of physics Part II. Description of the Phase-Integral Method: 2. Form of the wave function and the q-equation 3. Phase-integral approximation generated from an unspecified base function 4. F-matrix method 5. F-matrix connecting points on opposite sides of a well isolated turning point, and expressions for the wave function in these regions 6. Phase-integral connection formulas for a real, smooth, single-hump potential barrier Part III. Problems With Solutions: 1. Determination of a convenient base function 2. Determination of a phase-integral function satisfying the Schroedinger equation exactly 3. Properties of the phase-integral approximation along certain paths 4. Stokes constants and connection formulas 5. Airy's differential equation 6. Change of phase of the wave function in a classically allowed region due to the change of a boundary condition imposed in an adjacent classically forbidden region 7. Phase shift 8. Nearlying energy levels 9. Quantization conditions 10. Determination of the potential from the energy spectrum 11. Formulas for the normalization integral, not involving the wave function 12. Potential with a strong attractive Coulomb singularity at the origin 13. Formulas for expectation values and matrix elements, not involving the wave function 14. Potential barriers References.

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Physical Problems Solved by the Phase-Integral Method

A first approximation is obtained for the side force or the lift of a body of revolution inclined in a supersonic flow from the linearized equation of motion of compressible fluids. I t is shown that the lift a t any fixed Mach's number is directly proportional to the angle of attack of the body. The case of the cone is calculated in detail and a general method using step-wise doublet distribution is developed for a pointed projectile.

Supersonic Flow over an Inclined Body of Revolution

In some smart weapons, estimation of the impact point of the shell at each computation cycle of the control law is an integral part of the control strategy. In these situations, the impact point predictor is part of the imbedded computing system onboard the projectile. Practical considerations dictate that the impact point predictor yield rapid yet reasonably accurate estimates. Common methods for rapid trajectory prediction are numerical integration of point mass dynamic equations and evaluation of approximate closed-form solutions of the rigid-body projectile dynamic equations. These methods are shown to exhibit poor impact point prediction for long-range shots with high gun elevations characteristic of smart indirect fire munitions. Through modifications of projectile linear theory, a rapid projectile impact point predictor is proposed that eliminates the accuracy problems of the other methods while preserving low computational requirements. Typical results are provided for a short-range trajectory of a direct fire fin-stabilized projectile and a long-range trajectory for an indirect fire spin-stabilized round to substantiate these claims.

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Modified Projectile Linear Theory for Rapid Trajectory Prediction

a) A-> + oo, z on the real interval [z0, zx], b) a (z) twice continuously diff erentiable, and b (z) continuously differentiate on [z0, *J, and c) a(z)>0 on [*o,*J. Convergent expansions in I/A were obtained for a transformed equation. The inadequacies of Liouville's result foreshadow in embryo form the problems that have given birth to the future development of the theory. Permitting a (z) to have one or more zeros in the domain of discussion leads to the immense complications of turning point theory, while replacing the closed interval [z0, zt] by a complex

An historical survey of ordinary linear differential equations with a large parameter and turning points

This paper argues that a tunneling prefactor should appear in expressions for the tunneling probability D relevant to cold field electron emission (CFE) and in Fowler–Nordheim (FN) type equations. Except in the case of the “ideally smooth” parabolic barrier, a prefactor is always present for barriers where D can be found by exact solution of the Schrodinger equation. A review of the Jeffreys–Wentzel–Kramers–Brillouin (JWKB) approach to solving the Schrodinger equation shows that tunneling barriers should be classified according to whether they are weak or strong and ideally smooth or not: there are four different JWKB-type formulas, depending on the nature of the barrier. CFE tunneling barriers are not ideally smooth but since the 1950s have usually been analyzed using the JWKB formulas for ideally smooth barriers. These analyses, and the standard Murphy–Good FN-type equation, seem mathematically and physically incomplete. The FN-type equations currently used to describe CFE should be revised to explicitly include a tunneling prefactor. Some implications are explored.

On the need for a tunneling pre-factor in Fowler-Nordheim tunneling theory

This paper contains the results, theoretical and experimental, of work undertaken, at the request of the Ordnance Committee, by the authors as Technical Officers of the Munitions Inventions Department. Permission to publish such parts as appear to be of general scientific interest has now been granted by the Ordnance Committee and the Director of Artillery. The publication of this paper has received their sanction. The experiments in question were carried out at the firing ground of H. M. S. “Excellent,” Portsmouth; the Experimental Department, H. M. S. “Excellent,” also provided the 3-inch guns used and the material for the construction of the range. The authors’ best thanks are due to the officers of this department, especially Lieut.-Commander R. F. P. Maton, O. B. E., R. N., without whose cordial co-operation these experiments could never have been carried out; also to the other officers of the Munitions Inventions Department who assisted in the heavy work of making and analysing the observations. The aeronautical measurements at low velocities, required for comparison, were made in the wind channels of the National Physical Laboratory, by arrangement with the Director and the Superintendent of the Aero­nautical Department, to whom also we wish to express our thanks.

The Aerodynamics of a Spinning Shell

The paper to which this abstract is meant to serve as a non-technical introduction contains the results of work undertaken at the request of the Ordnance Committee and is published with their permission. (1) The Problems of Experimental External Ballistics .—A spinning shell moving through the air is subjected in general to a complex reaction, which is determined primarily by its motion relative to the air. But a shell has an axis of symmetry, and when this axis of symmetry, its axis of rotation, and the direction of motion of its centre of gravity through the air all coincide the reaction naturally assumes its simplest form. When a shell is moving in such a manner it may be said to be moving “nose-on” through the air, and it is a matter of experience that, by giving a shell a suitable axial spin, it may be made to move practically nose-on through the air over very considerable distances. Under such conditions the symmetry clearly demands that the reaction of the air shall reduce to a single force, acting in the direction of motion reversed, which therefore merely tends to destroy the linear velocity of the shell, and a couple about the axis of symmetry, which tends to destroy the axial spin. The latter is comparatively unimportant, and it appears that, so long as the motion remains practically nose-on, the problem of determining the motion of the shell is reduced to one of particle dynamics, whose solution is now classical.

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Herbert William Richmond

Papers

The Aerodynamics of a Spinning Shell

The Aerodynamics of a Spinning Shell