Kirchhoff Formula Research Articles

小変歪翼理論に於ける基礎問題に就いて

Abstract This paper forms a fundamental part of the investigations on the theory of the aerodynamical lift, which have been devoloped by the author during the last years of the Pacific War. He takes the point of view of the acceleration potential propounded by L. Prandtl in 1936, as the starting point, which presents a special covenience to the treatment of the high speed problems and oscillating airfoils. We obtain, by application of Huygens' principle and the idea of Lorentz transformation concerning the classical wave equation, which governs the potential function of the field of small disturbances caused by the motion of the lifting system at high speed, the fundamental formulas of the acceleration and velocity field. The expression of the former obtained from Kirchhoff's formula concerning the field of pressure around the airfoil, linearly and automatically splits into two parts, one of which is the effect of the camber form of the wing section, while the other is the effect of the thickness. We also obtain the fundamental formula of the oscillaling lifting surface which is proved to be equivalenl to Kuu89ssner's.

Calculation of the Radiation Properties of Hollow Pipes and Horns

The radiation properties of hollow pipes and horns are analyzed. The vector Kirchhoff formula is employed. The radiation fields from the transverse electric wave in hollow pipes of circular and rectangular cross sections are derived. For the TE1, 1 wave in circular pipe and the TE0, 1 wave in the rectangular pipe, the directivities are analytically expressed in terms of beam angle and power gain. It is found that the two waves have substantially equal power gains on the basis of equal areas of openings. The formulae for the radiation fields of TE0, 1 and TE1, 0 waves in a sectoral horn are given, of which, curves illustrating the radiation properties have been published previously.

On Diffraction and Radiation of Electromagnetic Waves

Inasmuch as it is rarely possible to treat diffraction of electromagnetic waves exactly, the Kirchhoff formulation of Huygens' Principle has been frequently used in approximate calculations. If the Kirchhoff formula is applied directly to the field intensities of the incident wave over the aperture, the diffracted field is found to be inconsistent with Maxwell's equations. If, on the other hand, this formula is applied to some auxiliary vector potential from which the diffracted field is subsequently deduced by differentiation, the result (although consistent with Maxwell's equations) depends on the particular choice of the auxiliary vector and in some instances, at least, is obviously unreasonable (Appendix III). The calculations of diffracted fields and radiation fields, based either on the Equivalence Principle or on the more general Induction Theorem, depend upon a priori verifiable approximations to the actual fields in the neighborhoods of the sources of the diffracted and radiated waves. For this reason we feel that these methods are preferable to those based on the Kirchhoff formula.

Diffraction Theory of Electromagnetic Waves

It has been shown by Larmor, Kottler and others that the classical method of calculating diffraction from the Kirchhoff formula in terms of a scalar light function cannot be applied directly to the electromagnetic field since it takes into account neither the vector character of the field nor the effect of charges along the contour of the opening. The field equations are integrated directly by means of a vector analog of Green's theorem. The results are applied to the calculation of diffraction of electromagnetic waves from a rectangular slit in a screen of infinite conductivity. The results are compared with an exact solution obtained recently by Morse and Rubenstein.

The tube effect in sound-velocity measurements

The modifications of Kirchhoff's formula required to take account of the finite thermal conductivity of the tube, slip between the gas and the walls, temperature-discontinuity between the gas and the walls, and absorption of energy by the walls are calculated and are found to be negligible. The effect of roughness of the walls is discussed, and the conclusion is drawn that the large tube-effects often found in practice are due to irregular motion of the gas. It is argued that the methods of correcting for the tube-effect used hitherto are unreliable. The effects of oscillation of the piston and of the gas behind the piston are investigated and it is shown that these may become appreciable in some forms of apparatus. The yielding of a tube of elliptical cross-section is calculated and is shown to have a negligible effect.

XIV.The kirchhoff formula extended to a moving surface

(1930). XIV. The kirchhoff formula extended to a moving surface. The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science: Vol. 9, No. 55, pp. 141-161.

Some problems in the theory of optical diffraction

The diffraction of waves by a perfectly reflecting wedge is now a problem of considerable celebrity. A semi-infinite plane may be looked upon as a wedge whose internal angle is zero. The application of Kirchhoff's formula in the latter case is considerably extended, it being shown that, through a region of not too small dimensions comprising the whole of the boundary of the geometrical shadow, the solution thus obtained is identical with the accurate solution. An expression is found for a plane wave incident at any angle upon a semi-infinite screen. The diffraction of waves by a straight slit in a plane screen is a problem not yet exactly solved; it is one which occupied Lord Rayleigh's mind for many years. An approximate expression is found for the disturbance distributed over the slit due to a wave incident normally, and by an application of Kirchhoff's formula an important result is obtained very readily, which is in agreement with Rayleigh's investigation by very long analysis. This appears to be the first time that an attempt has been made to apply Kirchhoff's formula to an area of disturbance which is not considered to be uniform over the area. The paper gives a brief treatment of the foregoing points of novelty and interest in these two problems which have long been and still are of some importance in mathematical physics.