Kirchhoff Formula Research Articles

Far-field approximations to the Kirchoff-Helmholtz representations of scattered fields

The standard far-field approximation to the Kirchhoff formula for the field scattered by a flat metallic plate <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">S</tex> of arbitrary shape is given by a certain surface (double) integral. This double integral can be reduced to a line integral evaluated around the boundary of S. Moreover, if <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">S</tex> is a polygon, this line integral can be reduced to a closed form expression involving no integrations at all. The use of such line integral representations can easily reduce the costs of numerical calculation by orders of magnitude. If the integrands are to be sampled <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">p</tex> times per wavelength to achieve an acceptable degree of precision, and if <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">A</tex> is the area of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">S</tex> , then the numerical evaluation of the double integral requires <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">p^{2}A/\lambda^{2}</tex> functional evaluations whereas the line integral only requires <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">p\sqrt{A/\lambda^{2}}</tex> . If <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">S</tex> is a polygon with <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N</tex> vertices, then only <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2N</tex> functional evaluations are required to evaluate the closed form expression with no quadrature error at all.

Edge capacitance in the measurement of dielectric properties

The correction for edge capacitance, which is usually calculated by Kirchhoff's formula, can be separated by graphical methods into components at various distances from the edge, and so into components which fall within the area of a disc specimen and those which are entirely outside it. In this way, it can be shown that the correction for edge capacitance is usually negligible when permittivity is measured by the adjustment of electrode separation so as to restore the capacitance, after removal of the specimen, to its previous value. The correction to be applied to the observed value of loss tangent is typically about ?3% of the loss tangent. The paper is a companion to that by Blanco White (p. 939).

Intercoupled thermoelasticity with a finite velocity of heat propagation

General theorems are derived concerning intercoupled thermoelasticity with a finite velocity of heat propagation. The variational principle is applied and the solution to intercoupled problems is given in an integral form analogous to the Kirchhoff formula.

One possible formulation of the thermodynamics of sorption equilibrium

1. To obtain the basic functions of the thermodynamics of Sorption equilibrium there is no need to separate the entire sorbate into gas and sorption phases. 2. The thermodynamics of sorption equilibrium can be represented as the result of replacement of the variables in the thermodynamic functions for a cylinder filled with gas and closed with a piston, within which the sorbent is included. 3. The method of replacement of variables makes the relationship of functions of the type of Clausius-Clapeyron and Kirchhoff formulas, used in the thermodynamics of sorption, to the axioms of thermodynamics more graphic, and also permits a refinement of the thermodynamic meaning of these functions.

A Variation of Hadamard’s Finite Part Integrals

A variation of Hadamard’s finite part integrals is described which can be used to solve a class of problems in a direct manner, eliminating the “ascent-descent” technique which Hadamard used. It is shown that the Green’s formula derived from the standard singular solution for the scalar wave equation yields the Kirchhoff formulas, if, when the space dimension is odd, one retains the logarithmically infinite part of the principal value integral.

A method for computing the transfer of solar radiation through clouds of hexagonal ice crystals

The angular patterns that result from the scattering of solar radiation by hexagonal ice prisms were computed under the assumption that the sizes of the prisms were many times the wavelengths of the incident radiation. Using the results of geometric ray optics, a sufficiently large number of equally spaced, parallel rays were mathematically traced through the prisms (including internal reflections) in order to determine the contribution to the scattering pattern that was due to refraction and reflection. The contribution due to forward diffraction was approximated by the Kirchhoff formula. Results for a distribution of prisms in random orientation show the expected large diffraction peak in the intensity in the forward direction as well as a prominent relative maximum at an angle corresponding to the well-known halo phenomenon occasionally seen around the sun or moon. In comparison, the scattering pattern for a sphere whose size was such as to scatter out of an incident beam the same total amount of radiation as scattered out by the prisms, is also shown. The major difference in the two patterns is the absence of the halo in the case of the sphere. It can be readily shown that the energy of the radiation scattered in the angular region of the halo is comparable in magnitude to that in the forward peak, thereby showing the importance of the halo.

Focal-Region Fields of Annular and Sectoral Microwave Apertures*

This paper concerns an experimental and theoretical study of the amplitude and phase structure of the image of a point source, formed by an ideal microwave-lens focusing system having its circular aperture partially obscured by disks or sectors of various sizes located in the aperture plane. The f/D ratio, the wavelength and the aperture diameter are respectively 2.7, 3.2 cm and 19.5 wavelengths. The exact Helmholtz–Kirchhoff formula is used in the numerical computations and is compared with the Rayleigh formulas.Diagrams are presented, showing measured and calculated fields and their characteristics are discussed. In the case of asymmetrical apertures, only the amplitude distribution is symmetrical in the focal plane, whereas the phase is not and both the amplitude and phase are unsymmetrical in all other regions.

Theoretical investigation of cylindrical dielectric-rod antennas

Present-day theory of uniform cylindrical dielectric-rod antennas rests on the three theoretical treatments of Horton, Bouix and Fradin (leaky-guide theory), and on Brown&apos;s end-fire theory. A digression on the general relationship between Schelkunoff&apos;s equivalence principle and the vector Kirchhoff formula reveals that these radiation formulas may only be applied to regions in which the vector potentials are known to satisfy the inhomogeneous wave equation. On applying this axiom to the theories, all but Brown&apos;s are shown to be fallacious. A detailed application of the vector Kirchhoff formula to uniform cylindrical dielectric rods supporting the HE11 mode produces an expression for the far-field radiation pattern, which, when computed, gives good agreement with existing experimental data. The analysis allows for dielectric losses and can also be extended to nonuniform dielectric antennas with gradual tapers. Fresnel interaction of the apertures and lens effects are considered. Finally, some design recommendations for dielectric rod antennas are made.

Backscattering of an Isotropic, Spherical, Acoustic Pulse of Arbitrary Shape from a Rough-Sea Surface

When an isotropic spherical pulse generated by an underwater source at A reflects from a rough-sea surface S, the pressure p(A) subsequently measured at the backscatter point depends on the shape of the incident pulse and the displacement-slope properties of S. If A is at a large distance D from S and approximate boundary conditions appropriate to surfaces of small slope are assumed, then Kirchhoff's general integral formula can be transformed into the sum of two surface integrals over modified source distributions on the zero-level surface So. The first integral po reduces to an “image pulse” reflected from a mirror surface at D + ζo, where ζo is the vertical displacement of S from the specular point on So. The second integral ps is the scattered pulse that interferes with po and persists after the latter disappears. At time t, ps is composed of a superposition of pulselet contributions that have propagated to A from a limited portion So(t) of So. The functional form of the pulselet source at dSo(t) is generally quite different from that of the incident pulse since the “amplitude scattering coefficient” σ that determines the pulselet-source strength is actually a differential operator. Pulselet-source strength also depends directly on the slope of S near dSo(t). This approximate formulation gives physical insight into the causes of surface scattering and is convenient for treating certain problems that arise in underwater sound and model experiments. [Hudson Laboratories, Columbia University Informal Documentation No. 31. This work was supported by the U. S. Office of Naval Research.]

Computation of Far-Field Characteristics of a Transducer from Near-Field Measurements Made in a Reflective Tank

The directivity pattern and source level of a large, cylindrical transducer were computed by using Kirchhoff's formula with a simple approximation for the normal pressure gradient. The integration was performed numerically over a finite cylinder. Near-field pressure amplitude and phase measurements were made over the surface of integration by rotating the transducer and measuring the output of a line hydrophone less than 1 wavelength from the active face. The measurements were made on the leading edge of the received pulses—before steady-state conditions were reached. This was done to simulate this transducer operating in a tank of diameter only about twice that of the transducer. Under such conditions, interference due to tank-wall reflections would begin when the amplitude of the pulse envelope was as much as 2–3 dB below the steady-state level. Computed directivity patterns agreed well with equivalent measured ones, but, naturally, computed source levels were 2–3 dB low. A correction term was derived from the first portion of the pulse risetime curves, and was applied to the source-level values to give good agreement with measured ones. The present results indicate that accurate transducer calibration can be obtained in reflective tanks considerably smaller than those used in previous work.

Determination of Far-Field Characteristics of Large Underwater Sound Transducers from Near-Field Measurements

It has recently become necessary to find a way to determine far-field characteristics of large underwater sound transducers from measurements made in the near field of the transducer (at ranges small compared to the transducer dimensions). It is possible to compute far-field directivity patterns and source levels from pressure amplitude and phase measurements made in the near field. This can be done by using Kirchhoff's formula with a simple approximation for the normal component of the pressure gradient. Pattern computations were made on line, plane-array, dipole, and line-and-cone transducers from measurements made in open water. Extensive pattern and source level computations were made on a large, multistave cylindrical transducer from measurements made in open water and in a highly reflective tank. Agreement with measured far-field patterns and source levels was good throughout and was within ±1 dB in the more recent work with the cylindrical transducer. Particularly significant is the fact that successful measurements were made in a tank of diameter only 2.9 times that of the transducer. These measurements were made near the leading edge of the received pulses in order to avoid interference due to reflections. Preliminary work on a simplified near-field test for a cylindrical transducer has also been done. Such a test would enable one to use near-field data to make certain judgments about the far-field performance of a transducer without computations.

Computation of Far-Field Characteristics of a Transducer from Measurements Made in a Reflective Tank

The directivity pattern and source level of a large cylindrical transducer were computed using the Kirchhoff formula with a simple approximation for the normal pressure gradient. The integration was performed numerically over a finite cylinder. Nearfield pressure amplitude and phase data were made over the surface of integration by rotating the transducer and measuring the output of a line hydrophone less than one wavelength from the active face. These measurements were made in a highly reflective sheet-steel tank of diameter 2.9 times that of the transducer and wall thickness much less than a wavelength. The amplitude and phase were measured on the leading portion of the received pulses. Directivity patterns were computed in a plane normal to the axis of the transducer. They were compared with the equivalent patterns measured in open-sea conditions. Agreement was good both in pattern shape and source level. It is felt that this work shows that it is possible to calibrate such a transducer in a tank that is relatively small and highly reflective.

Retardation and Diffraction Aspects of the Conduction of Heat in Solids

The concept of retardation of phase and energy propagation, borrowed from analyses used in physical optics, is applied to the analysis of heat conduction in solids. Analogies between physical optics and heat conduction in solids are pointed out and these are given a general interpretation in terms of a superposition of traveling waves. It is shown that interference effects may occur in the propagation of heat in a solid. Some of the notions of physical optics are then applied to the analysis of heat conduction in solids. A simple experiment demonstrating interference of traveling waves in heat conduction is analyzed. An analog to the Kirchhoff formula, exhibiting the role of diffraction effects in the conduction of heat in solids, is derived.

Microwave Aperture Antennas and Diffraction Theory*

Various studies of diffraction of electromagnetic waves motivated by and applied to work on microwave aperture antennas are reviewed in relation to their significance to the study of optics. A critique of the Kirchhoff formula is presented to show the essentially heuristic nature of the formula and its range of applicability. Other approaches to high-frequency approximations for diffraction problems are indicated together with various experimental results showing the structure of the electromagnetic field in the neighborhood of the aperture or scatterer. The results illustrate how studies in the microwave region have enlarged the scope of optics and provide a rich body of material for teaching the subject of diffraction in physical optics.

Computation of the Directivity Pattern and Source Level of a Transducer from Near-Field Measurements

Kirchhoff's formula is used to compute the directivity pattern and the source level of a large cylindrical transducer from measurements made near the transducer. Directivity patterns are computed for a plane normal to the axis of the cylinder and a plane which contains the axis of the cylinder. The normal pressure gradient required in the formula is obtained by a simple approximation, and the computations are carried out by means of a numerical integration over the surface of a circular cylinder of finite length. Data have been taken by making various numbers of passes around the transducer at different positions along the axis with a probe hydrophone, or by making one pass with a line hydrophone. There is good agreement between measured and computed patterns and source levels. (This work is supported by the Bureau of Ships.)

Ratios of specific heat and high-frequency viscosities in organic liquids under pressure, derived from ultrasonic propagation

Abstract Measurements of the velocity and attenuation of plane sound waves in a number of organic liquids at pressures up to 10 000 lb/in.2 have been made. From the velocity, the adiabatic compressibility is derived and from this, by comparison with the measured isothermal compressibility, the ratio of specific heats as a function of pressure is calculated. By introducing the measured attenuation into the Kirchhoff formula an ‘ultrasonic’ viscosity as a function of pressure is also derived.

High energy inelastic diffraction phenomena

The results surveyed above are, of course, only examples of the method developed. This method may be extended on different other phenomena (this, extension is in fact now being performed). In particular, the formation of pairs-accompanying the diffraction of a particle by a nucleus and some other effects. have been considered. In some cases it has turned out reasonable to consider the nucleus as being partly transparent. The behaviour of a diffracted particle may be represented in the essential region in a more general way than that offered by the Kirchhoff formula etc. All these and similar other possibilities deserve further investigation.

Attenuation of Sound in a Tube

An experimental method of measuring the attenuation of sound in a circular tube for gases at reduced pressures has been developed. Since the tube effect is inversely proportional to the square root of the pressure, a reduction in pressure increases the attenuation and thus increases the experimental accuracy. The tube attenuation was measured for the frequency over pressure ratio of 8 to 2000 kc/atm. The measured attenuation was found to be 4.5 percent greater than that predicted by the Kirchhoff formula.

Diffraction of Sound Waves by a Circular Aperture

Diffraction of an acoustic wave by an aperture in a plane reflecting screen has been investigated experimentally at a wavelength of about 3 cm. The pressure amplitude distribution within a circular aperture illuminated by a plane, progressive sound wave exhibits oscillations which do not conform to theoretical requirements. A possible explanation for the disagreement offered. Diffraction measurements made in the near-field of the circular hole were found to be in fair agreement with the Fresnel-Kirchhoff theory. A modified theoretical expression, valid for apertures large as compared to the wavelength, is also presented. It is concluded that this theory is capable of predicting the diffraction effects to a greater degree of accuracy and over a larger region of space than Kirchhoff's classical formula.

An Experimental Investigation of Formulas for the Prediction of Horn Radiator Patterns

The E-plane radiation pattern has been measured at short distances and both E and H-plane patterns at relatively great distances from the mouths of several electromagnetic horns. Measured values are compared with values calculated by the Kirchhoff formula and in addition the distant patterns are compared with values calculated by the corrected formula due to Stratton and Chu. For the range of horn length and aperture considered, the Kirchhoff formula gives satisfactory agreement in the E-plane for angles as great as twenty degrees and at distances as short as one foot from the mouth of the horn; the H-plane predictions are not as satisfactory.