ДИФРАКЦИЯ ПЛОСКОЙ Е-ПОЛЯРИЗОВАННОЙ ВОЛНЫ НА ЦИЛИНДРИЧЕСКОМ ВКЛЮЧЕНИИ В ПЛОСКОСЛОИСТОЙ СРЕДЕ

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ɎɂɁɂɑȿɋɄɂȿ ɈɋɇɈȼɕ

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ISSN: 2225-4293

Учредитель: Учреждение Российской академии наук

“Научно-технологический центр уникального приборостроения”

Издатель: Учреждение Российской академии наук

“Научно-технологический центр уникального приборостроения”

Журнал зарегистрирован 15 февраля 2000 г.

Министерством Российской Федерации по делам печати, телерадиовещания и средств массовых коммуникаций Свидетельство о регистрации ПИ № 77-1685

РЕДКОЛЛЕГИЯ:

Пустовойт В.И. – гл. редактор, академик РАН, доктор физ.-мат. наук, профессор Кравченко В.Ф. – зам. гл. редактора, д.ф.-м.н., профессор;

Боритко С.В. – д.ф.-м.н., профессор;

Васильев В.П. – д.т.н., профессор;

Виноградов Е.А. – академик РАН, д.ф.-м.н., профессор;

Гуляев Ю.В. – академик РАН. д.ф.-м.н, профессор;

Дианов Е.М. – академик РАН, д.ф.-м.н., профессор;

Жижин Г.Н. – д.ф.-м.н., профессор.;

Компанец О.Н. – д.ф.-м.н., профессор;

Кошкин В.И. – д.ф.-м.н., профессор;

Крохин О.Н. – академик РАН, д.ф.-м.н.. профессор;

Мазур М.М. – д.т.н.;

EDITORIAL BOARD:

Pustovoit V.I. – Editor in Chief, academician RAS, Dr.Sci. (Phys.-Math.), Prof.

Kravchenko V.F. – Deputy Editor in Chief, Dr.Sci. (Phys.-Math.), Prof.

Boritko S.V. – Dr.Sci. (Phys.-Math.), Prof.

Vasiliev V.P. – Dr.Sci. (Techn.), Prof.

Vinogradov E.A. – academician RAS, Dr.Sci. (Phys.-Math.), Prof.

Gulyaev Yu.V. – academician RAS, Dr.Sci. (Phys.-Math.), Prof.

Dianov E.M. – academician RAS, Dr.Sci. (Phys.-Math.), Prof.

Zhizhin G.N. – Dr.Sci. (Phys.-Math.), Prof.

Kompanets O.N. – Dr.Sci. (Phys.-Math.), Prof.

Koshkin V.I. – Dr.Sci. (Phys.-Math.), Prof.

Krohin O.N. – academician RAS, Dr.Sci. (Phys.-Math.), Prof.

Mazur M.M. – Dr.Sci. (Techn.)

ɀɭɪɧɚɥ ɩɟɪɟɢɡɞɚɟɬɫɹ ɧɚ ɚɧɝɥɢɣɫɤɨɦ ɚɡɵɤɟ ɩɨɞ ɧɚɡɜɚɧɢɟɦ «Physical Base of Instrumentation»

Морозов А.Н. – д.ф.-м.н.. профессор;

Отливанчик Е.А. – к.ф.-м.н.;

Пожар В.Э. – д.ф.-м.н.;

Федоров И.Б. – академик РАН, д.т.н., профессор;

Филачев А.М. – чл.-корр. РАН, д.т.н., профессор;

Яковлев В.П. – д.ф.-м.н., профессор

Morozov A.N. – Dr.Sci. (Phys.-Math.), Prof.

Otlivanchik E.A. – Cd.Sci. (Phys.-Math.) Pozhar V.E. – Dr.Sci. (Phys.-Math.)

Fedorov I.B. – academician RAS, Dr.Sci. (Techn.), Prof.

Filachev A.M. – Associate of the Russian Academy of Sciences, Dr.Sci. (Techn.), Prof.

Yakovlev V.P. – Dr.Sci. (Phys.-Math.), Prof.

© ɇɌɐ ɍɉ ɊȺɇ, 201

Ⱥɞɪɟɫ ɪɟɞɚɤɰɢɢ: 117342, Ɇɨɫɤɜɚ, ɭɥ. Ȼɭɬɥɟɪɨɜɚ, ɞ. 15, ɤɨɦɧ. 232.

Ɍɟɥɟɮɨɧ ɪɟɞɚɤɰɢɢ: (495) 334-83-50 E-mail: [email protected]

Ɉɬɜ. ɫɟɤɪɟɬɚɪɶ Ʉɚɥɚɲɧɢɤɨɜɚ ɂ.Ƚ.

Ɂɚɜ. ɪɟɞɚɤɰɢɟɣɋɵɪɨɜɚ Ɇ.ȼ.

ɄɨɪɪɟɤɬɨɪɆɚɤɟɟɜɚ ȿ.ɂ.

ȼɟɪɫɬɤɚɇɟɤɪɚɫɨɜ ɋ.Ƚ., Ɋɨɠɟɤ ɋ.Ʌ.

ɀɭɪɧɚɥ ɩɟɪɟɢɡɞɚɟɬɫɹ ɧɚ ɚɧɝɥɢɣɫɤɨɦ ɹɡɵɤɟ ɩɨɞ ɧɚɡɜɚɧɢɟɦ «Physical Bases of Instumentation»

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СОДЕРЖАНИЕ:

ПАМЯТНЫЕ ДАТЫ Шифрин Я.С.

К 100-летию Якова Наумовича Фельда МАТЕМАТИЧЕСКОЕ МОДЕЛИРОВАНИЕ ФИЗИЧЕСКИХ ПРОЦЕССОВ

Яцук Л.П., Блинова Н.К., Ляховский А.А., Ляховский А.Ф.

Закономерности частотного сканирования в волноводно- щелевой антенне, возбуждаемой замедленной волной Батраков Д.О, Головин Д.В.

Дифракция плоской Е-поляризованной волны на цилиндрическом включении в плоскослоистой среде

Легенький М.Н., Бутрым А.Ю.

Распространение нестационарного электромагнитного поля в диэлектрическом волноводе

Комарь Г.И.

Моделирование энергетических характеристик произвольных связных и радиолокационных систем

ПРИБОРЫ И МЕТОДЫ ФИЗИКИ И ТЕХНИКИ СВЧ-ДИАПАЗОНА

Губский Д.С., Синявский Г.П.

Учет особенности электромагнитного поля при проектировании цилиндрических волноводных структур для СВЧ приборов

ПРОБЛЕМЫ ОБРАБОТКИ СИГНАЛОВ И ИЗОБРАЖЕНИЙ В АКУСТООПТИКЕ И РАДИОФИЗИКЕ Антюфеев В.И., Быков В.Н., Иванченко Д.Д.

Влияние шумовой температуры антенного обтекателя на изображение, формируемое матричными радиометрическими системами

МЕТОДЫ РАДИОЛОКАЦИОННЫХ И РАДИОМЕТРИЧЕСКИХ ИЗМЕРЕНИЙ

Волосюк В.К., Павликов В.В.

Статистический синтез оптимальных и квазиоптимальных одноантенных радиометров модуляционного типа ФИЗИЧЕСКИЕ ОСНОВЫ КОСМИЧЕСКОГО

ПРИБОРОСТРОЕНИЯ Иванов И.И.

Синхронизация бортовых и наземных ионозондов при системном зондировании ионосферы

CONTENT:

MEMORIALS Shifrin Ya.S.

To 100-years anniversary of professor Ya.N.Fel’d MATHEMATICAL MODELING

OF PHYSICAL PROCESSES Yatsuk L.P., Blinova N.K.,

Lyakhovsky A.A., Lyakhovsky A.F.

Regularities of Frequency Scanning in Slotted- Waveguide Antenna excited with slowed-down wave Batrakov D.O., Golovin D.V.

Diffraction of the E-polarized plane wave by a cylindrical inclusion in plane-layered medium Legenkiy M.N., Butrym A.Yu.

Nonstationary electromagnetic field propagation in dielectric waveguide Komar G.I.

Simulation of energy characteristics

of arbitrary communication and radar systems DEVICES AND PHYSICS AND EQUIPMENT METHODS MICROWAVE RANGE

GubskyD.S. , SinyavskyG.P.

The development of circular waveguide structures for microwave devices taking into account the electromagnetic field singularity PROBLEMS OF SIGNAL

AND PROCESSING IN ACUSTO-OPTICS AND RADIOPHYSICS

Antyufeev V.I., Bykov V.N., Ivanchenko D.D.

Influence of noise temperature of antenna radome on image formed by matrix radiometric systems

METHODS OF RADAR

AND RADIOMETRIC MEASUREMENTS Volosyuk V.K., Pavlikov V.V.

Statistical synthesis optimal and quasi-optimal single-antenna chopper radiometers type PHYSICAL BASES

OF SPACE INSTRUMENTATION Ivanov I.I.

Synchronization of onboard and ground-based ionosondes for the ionospheric sounding system

ɋɞɚɧɨ ɜ ɧɚɛɨɪ 26.06.12. ɉɨɞɩɢɫɚɧɨ ɜ ɩɟɱɚɬɶ 20.06.2012

Ɏɨɪɦɚɬ ɛɭɦɚɝɢ (70×100)1/16. ɉɟɱɚɬɶ ɨɮɫɟɬɧɚɹ. ɉɟɱɚɬɧɵɯ ɥɢɫɬɨɜ 7.

Ɉɬɩɟɱɚɬɚɧɨ: ɈɈɈ «ȻɍȾɈɄȼȺɃ», 105062, ɝ. Ɇɨɫɤɜɚ, ɭɥ. ɉɨɤɪɨɜɤɚ, ɞ. 41, ɫɬɪ. 2 Ɍɢɪɚɠ 500 ɷɤɡ. ɐɟɧɚ ɞɨɝɨɜɨɪɧɚɹ.

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16

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Ⱥɧɧɨɬɚɰɢɹ

ȼ ɪɚɛɨɬɟ ɩɪɟɞɥɨɠɟɧɨ ɪɚɡɜɢɬɢɟ ɱɢɫɥɟɧɧɨɣ ɪɟɚ- ɥɢɡɚɰɢɢ ɦɟɬɨɞɚ ɧɭɥɟɜɨɝɨ ɩɨɥɹ. ɉɪɢɜɟɞɟɧɨ ɪɟɲɟ- ɧɢɟ ɡɚɞɚɱɢ ɞɢɮɪɚɤɰɢɢ ɧɨɪɦɚɥɶɧɨ ɩɚɞɚɸɳɟɣ ɩɥɨ- ɫɤɨɣ ȿ-ɩɨɥɹɪɢɡɨɜɚɧɧɨɣ ɜɨɥɧɵ ɧɚ ɰɢɥɢɧɞɪɢɱɟɫɤɨɦ ɨɞɧɨɪɨɞɧɨɦ ɜɤɥɸɱɟɧɢɢ ɜ ɩɥɨɫɤɨɫɥɨɢɫɬɨɣ ɫɪɟɞɟ.

Ɂɚɞɚɱɚ ɨɩɪɟɞɟɥɟɧɢɹ ɧɟɢɡɜɟɫɬɧɵɯ ɤɨɷɮɮɢɰɢɟɧɬɨɜ ɪɚɡɥɨɠɟɧɢɹ ɩɨɥɹ ɧɚ ɤɨɧɬɭɪɟ ɪɚɫɫɟɢɜɚɬɟɥɹ ɩɨ ɰɢ- ɥɢɧɞɪɢɱɟɫɤɢɦ ɝɚɪɦɨɧɢɤɚɦ ɫɜɟɞɟɧɚ ɤ ɮɪɟɞɝɨɥɶɦɨ- ɜɨɣ ɛɟɫɤɨɧɟɱɧɨɣ ɫɢɫɬɟɦɟ ɥɢɧɟɣɧɵɯ ɚɥɝɟɛɪɚɢɱɟ- ɫɤɢɯ ɭɪɚɜɧɟɧɢɣ ɜɬɨɪɨɝɨ ɪɨɞɚ ɫ ɜɩɨɥɧɟ ɧɟɩɪɟɪɵɜ- ɧɵɦ ɦɚɬɪɢɱɧɵɦ ɨɩɟɪɚɬɨɪɨɦ. Ⱦɨɤɚɡɚɬɟɥɶɫɬɜɨ ɫɯɨ- ɞɢɦɨɫɬɢ ɦɟɬɨɞɚ ɪɟɞɭɤɰɢɢ ɩɨɡɜɨɥɢɥɨ ɨɛɨɫɧɨɜɚɬɶ ɤɪɢɬɟɪɢɢ ɩɨɪɹɞɤɚ ɭɫɟɱɟɧɢɹ ɫɢɫɬɟɦɵ ɢ ɨɩɬɢɦɢɡɢ- ɪɨɜɚɬɶ ɜɵɱɢɫɥɢɬɟɥɶɧɵɣ ɚɥɝɨɪɢɬɦ. ɉɪɢɜɟɞɟɧɧɵɟ ɱɢɫɥɟɧɧɵɟ ɪɟɡɭɥɶɬɚɬɵ ɩɨɞɬɜɟɪɠɞɚɸɬ ɷɮɮɟɤɬɢɜ- ɧɨɫɬɶ ɩɪɟɞɥɨɠɟɧɧɨɝɨ ɩɨɞɯɨɞɚ.

Ʉɥɸɱɟɜɵɟ ɫɥɨɜɚ: ɦɟɬɨɞ ɧɭɥɟɜɨɝɨ ɩɨɥɹ, ɩɥɨɫɤɨ- ɫɥɨɢɫɬɚɹ ɫɪɟɞɚ, ɦɟɬɨɞ ɪɟɞɭɤɰɢɢ

ȼɜɟɞɟɧɢɟ

Ɋɚɫɫɟɹɧɢɟ ɷɥɟɤɬɪɨɦɚɝɧɢɬɧɵɯ ɜɨɥɧ ɧɚ ɰɢɥɢɧɞɪɢɱɟɫɤɢɯ ɜɤɥɸɱɟɧɢɹɯ ɜ ɩɥɨɫɤɨɫɥɨɢɫɬɵɯ ɫɪɟɞɚɯ ɢɫɫɥɟ- ɞɨɜɚɥɨɫɶ ɦɧɨɝɢɦɢ ɚɜɬɨɪɚɦɢ, ɜ ɱɚɫɬɧɨɫɬɢ [1–7]. Ⱦɥɹ ɪɟɲɟɧɢɹ ɡɚɞɚɱɢ ɜ ɨɞɧɢɯ ɫɥɭɱɚɹɯ ɚɜɬɨɪɵ ɢɫɩɨɥɶɡɨɜɚ- ɥɢ ɩɪɟɞɫɬɚɜɥɟɧɢɹ ɩɨɥɟɣ ɜ ɜɢɞɟ ɪɚɡɥɨɠɟɧɢɣ ɩɨ ɚɡɢɦɭɬɚɥɶɧɵɦ ɝɚɪɦɨɧɢɤɚɦ ɫ ɩɨɫɥɟɞɭɸɳɢɦ ɨɩɪɟɞɟɥɟɧɢɟɦ ɧɟɢɡɜɟɫɬɧɵɯ ɤɨɷɮɮɢɰɢɟɧɬɨɜ ɷɬɢɯ ɪɚɡɥɨɠɟɧɢɣ [1–4]. ȼ ɞɪɭɝɢɯ ɫɥɭɱɚɹɯ ɨɫɧɨɜɭ ɩɨɞɯɨɞɚ ɫɨɫɬɚɜɥɹɥ ɩɟɪɟɯɨɞ ɤ ɢɧɬɟɝɪɚɥɶɧɵɦ ɭɪɚɜɧɟɧɢɹɦ. ɗɬɨ ɛɵɥɢ ɢɧɬɟɝɪɚɥɶɧɵɟ ɭɪɚɜɧɟɧɢɹ ɨɬɧɨɫɢɬɟɥɶɧɨ ɞɢɚɝɪɚɦɦɵ ɪɚɫɫɟɹɧɢɹ [1]

ɥɢɛɨ ɨɛɴɟɦɧɵɟ [6] ɢ ɩɨɜɟɪɯɧɨɫɬɧɵɟ [7] ɢɧɬɟɝɪɚɥɶɧɵɟ ɭɪɚɜɧɟɧɢɹ. ɗɬɢ ɭɪɚɜɧɟɧɢɹ, ɢɫɩɨɥɶɡɭɸɳɢɟ ɮɭɧɤ- ɰɢɸ Ƚɪɢɧɚ ɢ ɜɤɥɸɱɚɸɳɢɟ ɜ ɤɨɦɩɚɤɬɧɨɣ ɮɨɪɦɟ ɜɫɟ ɞɨɩɨɥɧɢɬɟɥɶɧɵɟ ɭɫɥɨɜɢɹ, ɨɛɥɚɞɚɸɬ ɪɹɞɨɦ ɩɪɟɢɦɭ- ɳɟɫɬɜ. Ɉɞɧɚɤɨ ɚɧɚɥɢɬɢɱɟɫɤɢ ɱɢɫɥɟɧɧɨ ɪɟɲɢɬɶ ɢɯ ɜ ɛɨɥɶɲɢɧɫɬɜɟ ɫɥɭɱɚɟɜ ɧɟ ɭɞɚɟɬɫɹ, ɢ ɡɚɞɚɱɚ ɫɜɨɞɢɬɫɹ ɤ ɨɬɵɫɤɚɧɢɸ ɪɟɲɟɧɢɹ ɤɨɧɟɱɧɵɯ ɥɢɛɨ ɛɟɫɤɨɧɟɱɧɵɯ ɫɢɫɬɟɦ ɥɢɧɟɣɧɵɯ ɚɥɝɟɛɪɚɢɱɟɫɤɢɯ ɭɪɚɜɧɟɧɢɣ (ȻɋɅȺɍ).

УДК 537.874

ȿɃɏɋȻɅɑɃɚ ɊɆɉɌɅɉɄ ɀ-ɊɉɆɚɋɃɂɉȽȻɈɈɉɄ ȽɉɆɈɖ ɈȻ ɑɃɆɃɈȿɋɃɒɀɌɅɉɇ ȽɅɆəɒɀɈɃɃ Ƚ ɊɆɉɌɅɉɌɆɉɃɌɍɉɄ Ɍɋɀȿɀ

Abstract

Development of the numerical implementation of null-¿ eld technique is proposed. Solution of the diffraction problem for normally incident plane E- polarized wave on a homogeneous cylindrical inclusion in a plane-layered medium is presented. The problem of determining the unknown coef¿ cients of expansion of the full ¿ eld on the contour of the inclusion in cylindrical harmonics basis is reduced to a Fredholm in¿ nite system of linear algebraic equations of the second kind with completely continuous matrix operator. Proof of the reduction method con- vergence allowed to obtain the criteria for the system truncation order determination and optimize the computational algorithm. Numerical results con¿ rm the effectiveness of the proposed approach.

Key words: null-¿ eld technique, plane-layered medium, reduction method

ɆȺɌȿɆȺɌɂɑȿɋɄɈȿ ɆɈȾȿɅɂɊɈȼȺɇɂȿ ɎɂɁɂɑȿɋɄɂɏ ɉɊɈɐȿɋɋɈȼ

© Ȼɝɭɩɫɶ, 201

Батраков Д.О., Головин Д.В. — Харьковский национальный университет им. В.Н. Каразина радиофизический фа- культет, кафедра теоретической радиофизики

61077, Украина, г. Харьков, пл. Свободы.

E-mail: [email protected], [email protected]

ФИЗИЧЕСКИЕ ОСНОВЫ ПРИБОРОСТРОЕНИЯ. 2012 Том 1, № 1

(4)

Ⱦɢɮɪɚɤɰɢɹ ɩɥɨɫɤɨɣ ȿ-ɩɨɥɹɪɢɡɨɜɚɧɧɨɣ ɜɨɥɧɵ ɧɚ ɰɢɥɢɧɞɪɢɱɟɫɤɨɦ ɜɤɥɸɱɟɧɢɢ ɜ ɩɥɨɫɤɨɫɥɨɢɫɬɨɣ ɫɪɟɞɟ

ȼ ɫɥɭɱɚɟ ɨɛɴɟɦɧɵɯ ɢɧɬɟɝɪɚɥɶɧɵɯ ɭɪɚɜɧɟɧɢɣ (ɂɍ) ɪɚɡɦɟɪɧɨɫɬɶ ɫɨɨɬɜɟɬɫɬɜɭɸɳɟɣ ɫɢɫɬɟɦɵ ɛɨɥɶɲɟ, ɱɟɦ ɜ ɫɥɭɱɚɟ ɩɨɜɟɪɯɧɨɫɬɧɵɯ ɂɍ, ɱɬɨ ɞɟɥɚɟɬ ɢɯ ɛɨɥɟɟ ɩɪɢɜɥɟɤɚɬɟɥɶɧɵɦɢ. ȼ ɫɜɨɸ ɨɱɟɪɟɞɶ, ɫɪɟɞɢ ɦɧɨɝɨɨɛɪɚɡɢɹ ɬɚɤɢɯ ɦɟɬɨɞɨɜ ɦɨɠɧɨ ɜɵɞɟɥɢɬɶ ɦɟɬɨɞ ɧɭɥɟɜɨɝɨ ɩɨɥɹ, ɨɩɢɪɚɸɳɢɣɫɹ ɧɚ ɨɞɧɭ ɢɡ ɷɮɮɟɤɬɢɜɧɵɯ ɩɪɨɟɤɰɢɨɧ- ɧɵɯ ɫɯɟɦ [8, 9]. Ɉɞɧɢɦ ɢɡ ɧɟɞɨɫɬɚɬɤɨɜ ɷɬɨɝɨ ɦɟɬɨɞɚ ɞɨ ɫɢɯ ɩɨɪ ɛɵɥɨ ɢɫɩɨɥɶɡɨɜɚɧɢɟ ɩɪɢ ɱɢɫɥɟɧɧɵɯ ɪɚɫ- ɱɟɬɚɯ ɛɟɫɤɨɧɟɱɧɨɣ ɫɢɫɬɟɦɵ ɥɢɧɟɣɧɵɯ ɚɥɝɟɛɪɚɢɱɟɫɤɢɯ ɭɪɚɜɧɟɧɢɣ ɩɟɪɜɨɝɨ ɪɨɞɚ (ȻɋɅȺɍ 1), ɞɥɹ ɤɨɬɨɪɨɣ ɞɨɤɚɡɚɬɟɥɶɫɬɜɨ ɫɯɨɞɢɦɨɫɬɢ ɦɟɬɨɞɚ ɪɟɞɭɤɰɢɢ ɨɬɫɭɬɫɬɜɨɜɚɥɨ. ɉɨɷɬɨɦɭ ɜ ɞɚɧɧɨɣ ɪɚɛɨɬɟ ɩɪɟɞɥɨɠɟɧɵ ɩɪɨ- ɰɟɞɭɪɚ ɜɵɞɟɥɟɧɢɹ ɢɡ ɦɚɬɪɢɱɧɨɝɨ ɨɩɟɪɚɬɨɪɚ ȻɋɅȺɍ 1 ɝɥɚɜɧɨɣ ɱɚɫɬɢ ɢ ɫɜɟɞɟɧɢɟ ɡɚɞɚɱɢ ɤ ɮɪɟɞɝɨɥɶɦɨɜɨɣ ɛɟɫɤɨɧɟɱɧɨɣ ɫɢɫɬɟɦɟ ɥɢɧɟɣɧɵɯ ɚɥɝɟɛɪɚɢɱɟɫɤɢɯ ɭɪɚɜɧɟɧɢɣ ɫɨ ɜɩɨɥɧɟ ɧɟɩɪɟɪɵɜɧɨɣ ɮɨɪɦɨɣ. Ⱦɥɹ ɪɟɲɟɧɢɹ ɬɚɤɨɣ ȻɋɅȺɍ 2 ɨɛɨɫɧɨɜɚɧɨ ɩɪɢɦɟɧɟɧɢɟ ɦɟɬɨɞɚ ɪɟɞɭɤɰɢɢ, ɱɬɨ ɩɨɡɜɨɥɹɟɬ ɩɨɥɭɱɚɬɶ ɭɫɬɨɣɱɢɜɵɟ ɪɟɲɟɧɢɹ.

ɉɨɫɬɚɧɨɜɤɚ ɡɚɞɚɱɢ

Ɋɚɫɫɦɨɬɪɢɦ ɦɧɨɝɨɫɥɨɣɧɭɸ ɩɥɨɫɤɨɫɥɨɢɫɬɭɸ ɫɬɪɭɤɬɭɪɭ, ɜ ɨɞɧɨɦ ɢɡ ɫɥɨɟɜ ɤɨɬɨɪɨɣ ɪɚɫɩɨɥɨɠɟɧɨ ɨɞ- ɧɨɪɨɞɧɨɟ ɰɢɥɢɧɞɪɢɱɟɫɤɨɟ ɜɤɥɸɱɟɧɢɟ ɩɪɨɢɡɜɨɥɶɧɨɝɨ ɩɨɩɟɪɟɱɧɨɝɨ ɫɟɱɟɧɢɹ S_p. ȼɜɟɞɟɦ ɫɢɫɬɟɦɭ ɞɟɤɚɪɬɨ- ɜɵɯ ɤɨɨɪɞɢɧɚɬ XYZ (ɫɦ. ɪɢɫ. 1). ȼɧɟɲɧɸɸ ɨɬɧɨɫɢɬɟɥɶɧɨ ɜɤɥɸɱɟɧɢɹ ɨɛɥɚɫɬɶ ɩɪɨɫɬɪɚɧɫɬɜɚ ɨɛɨɡɧɚɱɢɦ S_e. Ɂɚ ɩɪɟɞɟɥɚɦɢ ɫɥɨɢɫɬɨɣ ɫɪɟɞɵ ɪɚɫɩɨɥɨɠɟɧ ɢɫɬɨɱɧɢɤ ɩɚɞɚɸɳɟɝɨ ɩɨɥɹ. ȼ ɧɚɫɬɨɹɳɟɣ ɫɬɚɬɶɟ ɛɭɞɟɦ ɪɚɫɫɦɚ- ɬɪɢɜɚɬɶ ɫɥɭɱɚɣ ɧɨɪɦɚɥɶɧɨɝɨ ɩɚɞɟɧɢɹ ɦɨɧɨɯɪɨɦɚɬɢɱɟɫɤɨɣ ɩɥɨɫɤɨɣ ȿ-ɩɨɥɹɪɢɡɨɜɚɧɧɨɣ ɜɨɥɧɵ (ɤɨɝɞɚ ɜɟɤ- ɬɨɪɵ ɧɚɩɪɹɠɟɧɧɨɫɬɢ ɷɥɟɤɬɪɢɱɟɫɤɨɝɨ ɢ ɦɚɝɧɢɬɧɨɝɨ ɩɨɥɹ ɦɨɝɭɬ ɛɵɬɶ ɩɪɟɞɫɬɚɜɥɟɧɵ ɜ ɜɢɞɟ ,

). Ɍɨɝɞɚ ɪɟɲɟɧɢɟ ɫɢɫɬɟɦɵ ɜɟɤɬɨɪɧɵɯ ɭɪɚɜɧɟɧɢɣ Ɇɚɤɫɜɟɥɥɚ ɦɨɠɟɬ ɛɵɬɶ ɫɜɟɞɟɧɨ ɤ ɪɟɲɟɧɢɸ ɜɨɥɧɨɜɨɝɨ ɭɪɚɜɧɟɧɢɹ ɨɬɧɨɫɢɬɟɥɶɧɨ ɤɨɦɩɨɧɟɧɬɵ E_z ɜɟɤɬɨɪɚ ɧɚɩɪɹɠɟɧɧɨɫɬɢ ɷɥɟɤɬɪɢɱɟɫɤɨɝɨ ɩɨɥɹ (ɜ ɞɚɥɶ- ɧɟɣɲɟɦ ɢɡɥɨɠɟɧɢɢ ɢɧɞɟɤɫ z ɛɭɞɟɦ ɨɩɭɫɤɚɬɶ).

Ɋɢɫ. 1. Ƚɟɨɦɟɬɪɢɹ ɢɫɫɥɟɞɭɟɦɨɣ ɫɬɪɭɤɬɭɪɵ

Ɇɟɬɨɞ ɪɟɲɟɧɢɹ

ɉɪɟɞɫɬɚɜɢɦ ɩɨɥɧɨɟ ɩɨɥɟ, ɫɭɳɟɫɬɜɭɸɳɟɟ ɜ ɨɛɥɚɫɬɢ S_e(E_e) ɜ ɜɢɞɟ ɫɭɦɦɵ E_e = E_in + E_sc.

ɋɥɚɝɚɟɦɨɟ E_in ɫɨɨɬɜɟɬɫɬɜɭɟɬ ɩɨɥɸ, ɫɭɳɟɫɬɜɭɸɳɟɦɭ ɜ ɫɪɟɞɟ ɛɟɡ ɜɤɥɸɱɟɧɢɹ, ɚ E_sc — ɩɨɥɸ, ɪɚɫɫɟɹɧɧɨ- ɦɭ ɜɤɥɸɱɟɧɢɟɦ. Ɍɚɤɠɟ ɜɜɟɞɟɦ ɜ ɪɚɫɫɦɨɬɪɟɧɢɟ ɩɨɥɟ ɜɧɭɬɪɢ ɜɤɥɸɱɟɧɢɹ — E_p ɢ ɝɪɚɧɢɱɧɵɟ ɭɫɥɨɜɢɹ ɧɚ ɤɨɧ- ɬɭɪɟ ɜɤɥɸɱɟɧɢɹ L

, (1) ɝɞɟ — ɜɟɤɬɨɪ ɜɧɟɲɧɟɣ ɧɨɪɦɚɥɢ ɤ ɤɨɧɬɭɪɭ L.

(5)

18

Ȼɚɬɪɚɤɨɜ Ⱦ.Ɉ., Ƚɨɥɨɜɢɧ Ⱦ.ȼ.

ȼ ɫɨɨɬɜɟɬɫɬɜɢɢ ɫ ɩɪɨɰɟɞɭɪɨɣ, ɩɨɞɪɨɛɧɨ ɨɩɢɫɚɧɧɨɣ ɜ [11], ɢɡ ɩɪɢɜɟɞɟɧɧɵɯ ɭɪɚɜɧɟɧɢɣ ɦɨɠɟɬ ɛɵɬɶ ɩɨ- ɥɭɱɟɧɨ ɢɧɬɟɝɪɨɞɢɮɮɟɪɟɧɰɢɚɥɶɧɨɟ ɭɪɚɜɧɟɧɢɟ ɞɥɹ ɨɩɪɟɞɟɥɟɧɢɹ ɧɟɢɡɜɟɫɬɧɵɯ ɚɦɩɥɢɬɭɞ ɩɨɥɧɨɝɨ ɩɨɥɹ E_e ɧɚ ɤɨɧɬɭɪɟ L

, (2)

ɝɞɟ μ — ɦɚɝɧɢɬɧɚɹ ɩɪɨɧɢɰɚɟɦɨɫɬɶ; L_p — ɤɪɭɝɨɜɨɣ ɤɨɧɬɭɪ, ɰɟɥɢɤɨɦ ɥɟɠɚɳɢɣ ɜɧɭɬɪɢ ɨɛɥɚɫɬɢ S_p; — ɮɭɧɤɰɢɹ Ƚɪɢɧɚ ɩɥɨɫɤɨɫɥɨɢɫɬɨɣ ɫɪɟɞɵ, ɭɞɨɜɥɟɬɜɨɪɹɸɳɚɹ ɭɪɚɜɧɟɧɢɸ

. (3)

Ⱦɥɹ ɪɟɲɟɧɢɹ ɭɪɚɜɧɟɧɢɹ (2) ɩɪɟɞɫɬɚɜɢɦ ɧɟɢɡɜɟɫɬɧɵɟ ɚɦɩɥɢɬɭɞɵ ɩɨɥɧɨɝɨ ɩɨɥɹ E_e ɢ ɟɝɨ ɧɨɪɦɚɥɶɧɨɣ ɩɪɨ- ɢɡɜɨɞɧɨɣ ɧɚ ɤɨɧɬɭɪɟ ɜɤɥɸɱɟɧɢɹ L ɜ ɜɢɞɟ ɪɚɡɥɨɠɟɧɢɣ

, (4) ɝɞɟ — ɱɚɫɬɧɨɟ ɪɟɲɟɧɢɟ ɨɞɧɨɪɨɞɧɨɝɨ ɜɨɥɧɨɜɨɝɨ ɭɪɚɜɧɟɧɢɹ ɜ ɨɛɥɚɫɬɢ S_p

. (5) ɉɨɞɫɬɚɜɢɜ ɪɚɡɥɨɠɟɧɢɟ (4) ɜ ɭɪɚɜɧɟɧɢɟ (2), ɩɨɥɭɱɢɦ

. (6) Ɏɭɧɤɰɢɹ ɩɟɪɜɢɱɧɨɝɨ ɩɨɥɹ E_in ɜ ɩɪɚɜɨɣ ɱɚɫɬɢ ɭɪɚɜɧɟɧɢɹ ɩɪɟɞɫɬɚɜɢɦɚ (ɟɞɢɧɫɬɜɟɧɧɵɦ ɨɛɪɚɡɨɦ) ɪɚɡɥɨ- ɠɟɧɢɟɦ ɜɢɞɚ

, ɝɞɟ τ — ɞɜɭɦɟɪɧɵɣ ɢɧɞɟɤɫ, ɚɧɚɥɨɝɢɱɧɵɣ ɢɧɞɟɤɫɭ σ.

ɉɨɫɤɨɥɶɤɭ ɮɭɧɤɰɢɹ E_in ɡɚɜɢɫɢɬ ɨɬ , ɞɥɹ ɩɟɪɟɯɨɞɚ ɨɬ ɭɪɚɜɧɟɧɢɹ (6) ɤ ȻɋɅȺɍ ɧɭɠɧɨ ɢ ɜ ɥɟɜɨɣ ɱɚɫɬɢ (6) ɩɨɥɭɱɢɬɶ ɪɚɡɥɨɠɟɧɢɟ ɩɨ ɮɭɧɤɰɢɹɦ , ɡɚɜɢɫɹɳɢɦ ɢɦɟɧɧɨ ɨɬ . ɉɨɥɭɱɢɬɶ ɬɚɤɨɟ ɪɚɡɥɨɠɟɧɢɟ ɦɨɠɧɨ, ɩɪɟɞɫɬɚɜɢɜ ɮɭɧɤɰɢɸ Ƚɪɢɧɚ ɜ ɜɢɞɟ

.

Ʉɪɨɦɟ ɬɨɝɨ, ɭɱɬɟɦ, ɱɬɨ ɜ ɪɚɫɫɦɚɬɪɢɜɚɟɦɨɦ ɫɥɭɱɚɟ, ɤɨɝɞɚ ɫɪɟɞɚ, ɜ ɤɨɬɨɪɨɣ ɪɚɫɩɨɥɨɠɟɧɨ ɜɤɥɸɱɟɧɢɟ, ɹɜ- ɥɹɟɬɫɹ ɫɥɨɢɫɬɨɧɟɨɞɧɨɪɨɞɧɨɣ, ɮɭɧɤɰɢɹ Ƚɪɢɧɚ ɦɨɠɟɬ ɛɵɬɶ ɩɪɟɞɫɬɚɜɥɟɧɚ ɜ ɜɢɞɟ

, (7) ɝɞɟ G⁰ — ɮɭɧɤɰɢɹ Ƚɪɢɧɚ ɨɞɧɨɪɨɞɧɨɝɨ ɩɪɨɫɬɪɚɧɫɬɜɚ ɫ ɩɚɪɚɦɟɬɪɚɦɢ ɫɥɨɹ, ɫɨɞɟɪɠɚɳɟɝɨ ɜɤɥɸɱɟɧɢɟ; G^r — ɪɟɲɟɧɢɟ ɨɞɧɨɪɨɞɧɨɝɨ ɜɨɥɧɨɜɨɝɨ ɭɪɚɜɧɟɧɢɹ ɜ ɨɛɥɚɫɬɢ S_e

. ɉɨɞɫɬɚɜɢɦ ɩɪɟɞɫɬɚɜɥɟɧɢɹ ɞɥɹ ɢ ɜ ɭɪɚɜɧɟɧɢɟ (6)

(6)

(8)

ɇɚ ɞɚɧɧɨɦ ɷɬɚɩɟ ɦɨɠɧɨ ɩɟɪɟɣɬɢ ɤ ȻɋɅȺɍ. ɉɨɫɤɨɥɶɤɭ ɫɢɫɬɟɦɚ ɮɭɧɤɰɢɣ ɩɨɥɧɚ ɧɚ ɤɨɧɬɭɪɟ L_p, ɪɚ- ɜɟɧɫɬɜɨ ɞɜɭɯ ɪɚɡɥɨɠɟɧɢɣ ɜ ɛɚɡɢɫɟ ɜɨɡɦɨɠɧɨ ɬɨɥɶɤɨ ɩɪɢ ɪɚɜɟɧɫɬɜɟ ɤɨɷɮɮɢɰɢɟɧɬɨɜ ɷɬɢɯ ɪɚɡɥɨɠɟɧɢɣ.

Ɍɚɤɢɦ ɨɛɪɚɡɨɦ, ɢɡ ɢɧɬɟɝɪɚɥɶɧɨɝɨ ɭɪɚɜɧɟɧɢɹ (8) ɦɨɠɧɨ ɩɨɥɭɱɢɬɶ ȻɋɅȺɍ 1

, (9) ɝɞɟ — ɦɚɬɪɢɰɚ ɫɢɫɬɟɦɵ, ɜɵɪɚɠɟɧɢɟ ɞɥɹ ɷɥɟɦɟɧɬɚ ɤɨɬɨɪɨɣ ɢɦɟɟɬ ɜɢɞ

, ɝɞɟ — ɜɟɤɬɨɪ ɩɪɚɜɨɣ ɱɚɫɬɢ, ɚ — ɜɟɤɬɨɪ ɧɟɢɡɜɟɫɬɧɵɯ ɤɨɷɮɮɢɰɢɟɧɬɨɜ.

Ⱦɥɹ ɪɟɲɟɧɢɹ ɩɨɥɭɱɟɧɧɨɣ ɫɢɫɬɟɦɵ ɧɟɨɛɯɨɞɢɦɨ ɩɪɢɦɟɧɢɬɶ ɦɟɬɨɞ ɪɟɞɭɤɰɢɢ (ɭɫɟɱɟɧɢɹ). Ɉɞɧɚɤɨ ɬɚɤɢɟ ɫɢɫɬɟɦɵ, ɤɚɤ ɩɪɚɜɢɥɨ, ɩɥɨɯɨ ɨɛɭɫɥɨɜɥɟɧɵ, ɢ ɨɛɨɫɧɨɜɚɬɶ ɜɨɡɦɨɠɧɨɫɬɶ ɩɪɢɦɟɧɟɧɢɹ ɞɥɹ ɢɯ ɪɟɲɟɧɢɹ ɦɟɬɨ- ɞɚ ɪɟɞɭɤɰɢɢ ɡɚɬɪɭɞɧɢɬɟɥɶɧɨ. ȼ ɬɨ ɠɟ ɜɪɟɦɹ ɞɥɹ ȻɋɅȺɍ 2 ɫ ɜɩɨɥɧɟ ɧɟɩɪɟɪɵɜɧɵɦ ɦɚɬɪɢɱɧɵɦ ɨɩɟɪɚɬɨɪɨɦ (ɫ ɜɩɨɥɧɟ ɧɟɩɪɟɪɵɜɧɨɣ ɮɨɪɦɨɣ [11]) ɩɪɢɦɟɧɟɧɢɟ ɦɟɬɨɞɚ ɪɟɞɭɤɰɢɢ ɢ ɫɯɨɞɢɦɨɫɬɶ ɩɨɥɭɱɚɟɦɵɯ ɪɟɲɟɧɢɣ ɤ ɬɨɱɧɨɦɭ ɨɛɨɫɧɨɜɚɬɶ ɦɨɠɧɨ.

Ȼɥɚɝɨɞɚɪɹ ɩɪɟɞɥɨɠɟɧɧɨɦɭ ɩɪɟɞɫɬɚɜɥɟɧɢɸ (7) ɞɥɹ ɮɭɧɤɰɢɢ Ƚɪɢɧɚ ɨɛɴɟɦɥɸɳɟɣ ɫɪɟɞɵ, ɝɥɚɜɧɚɹ ɱɚɫɬɶ ɦɚɬɪɢɱɧɨɝɨ ɨɩɟɪɚɬɨɪɚ ɢɫɯɨɞɧɨɣ ȻɋɅȺɍ 1 ɢɦɟɟɬ ɨɬɧɨɫɢɬɟɥɶɧɨ ɩɪɨɫɬɨɣ ɜɢɞ. ɋɥɟɞɨɜɚɬɟɥɶɧɨ, ɫɨɝɥɚɫɧɨ [11], ɬɚɤɭɸ ɫɢɫɬɟɦɭ ɦɨɠɧɨ ɫɜɟɫɬɢ ɤ ɮɪɟɞɝɨɥɶɦɨɜɨɣ ȻɋɅȺɍ 2 ɫ ɜɩɨɥɧɟ ɧɟɩɪɟɪɵɜɧɵɦ ɦɚɬɪɢɱɧɵɦ ɨɩɟɪɚɬɨɪɨɦ.

ȼɨɫɩɨɥɶɡɨɜɚɜɲɢɫɶ ɩɪɟɞɫɬɚɜɥɟɧɢɟɦ (7), ɦɨɠɧɨ ɩɪɟɞɫɬɚɜɢɬɶ ɷɥɟɦɟɧɬɵ ɢɫɯɨɞɧɨɣ ɦɚɬɪɢɰɵ ɜ ɜɢɞɟ ɫɭɦɦɵ: , ɝɞɟ ɩɟɪɜɨɟ ɫɥɚɝɚɟɦɨɟ ɫɨɞɟɪɠɢɬ ɜ ɩɨɞɵɧɬɟɝɪɚɥɶɧɨɦ ɜɵɪɚɠɟɧɢɢ ɮɭɧɤɰɢɸ , ɚ ɜɬɨɪɨɟ — . ȼ ɯɨɞɟ ɩɨɫɥɟɞɭɸɳɢɯ ɨɱɟɜɢɞɧɵɯ ɩɪɟɨɛɪɚɡɨɜɚɧɢɣ ɦɨɠɧɨ ɩɨɥɭɱɢɬɶ ɫɢɫɬɟɦɭ ɜɢɞɚ

, (10)

. ɉɨɤɚɠɟɦ, ɱɬɨ ɦɚɬɪɢɰɚ ɷɬɨɣ ɫɢɫɬɟɦɵ ɨɛɥɚɞɚɟɬ ɫɥɟɞɭɸɳɢɦ ɫɜɨɣɫɬɜɨɦ

.

Ⱦɥɹ ɷɬɨɝɨ ɩɨɥɭɱɢɦ ɚɫɢɦɩɬɨɬɢɱɟɫɤɭɸ ɨɰɟɧɤɭ ɞɥɹ ɦɨɞɭɥɹ ɦɚɬɪɢɱɧɨɝɨ ɷɥɟɦɟɧɬɚ . ȼɨɫɩɨɥɶɡɭɟɦɫɹ ɢɡ- ɜɟɫɬɧɵɦɢ [11] ɚɫɢɦɩɬɨɬɢɱɟɫɤɢɦɢ ɩɪɟɞɫɬɚɜɥɟɧɢɹɦɢ ɞɥɹ ɰɢɥɢɧɞɪɢɱɟɫɤɢɯ ɮɭɧɤɰɢɣ (ɞɥɹ ɫɥɭɱɚɹ )

.

(7)

20

Ⱦɥɹ ɭɩɪɨɳɟɧɢɹ ɞɚɥɶɧɟɣɲɟɝɨ ɢɡɥɨɠɟɧɢɹ (ɛɟɡ ɨɝɪɚɧɢɱɟɧɢɹ ɨɛɳɧɨɫɬɢ) ɪɚɫɫɦɨɬɪɢɦ ɱɚɫɬɧɵɣ ɫɥɭɱɚɣ ɫɢɦ- ɦɟɬɪɢɱɧɨɝɨ ɜɤɥɸɱɟɧɢɹ, ɤɨɷɮɮɢɰɢɟɧɬɵ ɩɪɢ ɧɟɱɟɬɧɵɯ ɮɭɧɤɰɢɹɯ (sin) ɪɚɜɧɵ ɧɭɥɸ. Ɍɨɝɞɚ ɚɫɢɦɩɬɨɬɢɱɟɫɤɚɹ ɨɰɟɧɤɚ ɞɥɹ ɞɢɚɝɨɧɚɥɶɧɨɝɨ ɷɥɟɦɟɧɬɚ ɦɚɬɪɢɰɵ ɫɥɟɞɭɸɳɚɹ:

.

ɇɟɞɢɚɝɨɧɚɥɶɧɵɣ ɷɥɟɦɟɧɬ

Ⱦɚɥɟɟ ɜ ɫɨɨɬɜɟɬɫɬɜɢɢ ɫ ɩɨɞɯɨɞɨɦ, ɩɪɟɞɥɨɠɟɧɧɵɦ ɜ [13], ɭɱɬɟɦ, ɱɬɨ ɞɢɚɝɨɧɚɥɶɧɵɟ ɷɥɟɦɟɧɬɵ ɦɚɬɪɢɰɵ ɫɢɫɬɟɦɵ (10) ɭɛɵɜɚɸɬ ɫ ɪɨɫɬɨɦ l ɧɟ ɯɭɠɟ, ɱɟɦ 1/l. Ʉɪɨɦɟ ɬɨɝɨ, ɷɥɟɦɟɧɬɵ ɷɬɨɣ ɦɚɬɪɢɰɵ, ɪɚɫɩɨɥɨɠɟɧ- ɧɵɟ ɧɚ ɫɨɫɟɞɧɢɯ ɞɢɚɝɨɧɚɥɹɯ, ɜɟɞɭɬ ɫɟɛɹ ɤɚɤ ɤɨɷɮɮɢɰɢɟɧɬɵ ɪɹɞɚ Ɏɭɪɶɟ ɨɬɧɨɫɢɬɟɥɶɧɨ ɪɚɡɧɨɫɬɢ ɢɧɞɟɤɫɨɜ

. ȼ ɬɚɤɨɦ ɫɥɭɱɚɟ ɦɨɠɧɨ ɡɚɩɢɫɚɬɶ ɜɵɪɚɠɟɧɢɟ ɞɥɹ ɫɭɦɦɵ ɤɜɚɞɪɚɬɨɜ ɦɨɞɭɥɟɣ ɦɚɬɪɢɱɧɵɯ ɷɥɟɦɟɧɬɨɜ ɜ ɷɤɜɢɜɚɥɟɧɬɧɨɦ ɜɢɞɟ, ɨɬɤɭɞɚ ɢ ɫɥɟɞɭɟɬ ɧɟɨɛɯɨɞɢɦɚɹ ɨɰɟɧɤɚ

.

Ⱥɧɚɥɨɝɢɱɧɨ ɫ ɩɨɦɨɳɶɸ ɚɫɢɦɩɬɨɬɢɱɟɫɤɢɯ ɨɰɟɧɨɤ ɥɟɝɤɨ ɩɨɤɚɡɚɬɶ, ɱɬɨ ɞɥɹ ɷɥɟɦɟɧɬɨɜ ɩɪɚɜɵɯ ɱɚɫɬɟɣ ɫɢ- ɫɬɟɦɵ (10) ɜɵɩɨɥɧɹɟɬɫɹ ɭɫɥɨɜɢɟ

.

Ʉɪɨɦɟ ɬɨɝɨ, ɩɪɢ ɱɢɫɥɟɧɧɨɣ ɪɟɚɥɢɡɚɰɢɢ ɩɪɟɞɥɨɠɟɧɧɨɝɨ ɚɥɝɨɪɢɬɦɚ ɜɨɡɧɢɤɚɸɬ ɜɚɠɧɵɟ ɩɪɚɤɬɢɱɟɫɤɢɟ ɜɨ- ɩɪɨɫɵ, ɫɜɹɡɚɧɧɵɟ ɫ ɜɵɛɨɪɨɦ ɩɨɪɹɞɤɚ ɪɟɞɭɤɰɢɢ. ȼɨ ɦɧɨɝɢɯ ɪɚɛɨɬɚɯ ɚɜɬɨɪɵ ɜ ɤɚɱɟɫɬɜɟ ɤɪɢɬɟɪɢɹ ɩɪɟɤɪɚɳɟ- ɧɢɹ ɜɵɱɢɫɥɟɧɢɣ ɢɫɩɨɥɶɡɨɜɚɥɢ ɭɫɥɨɜɢɟ ɩɪɟɤɪɚɳɟɧɢɹ ɫɭɳɟɫɬɜɟɧɧɨɝɨ ɢɡɦɟɧɟɧɢɹ ɧɟɜɹɡɤɢ ɩɪɢ ɭɜɟɥɢɱɟɧɢɢ ɩɨɪɹɞɤɚ ɪɟɞɭɤɰɢɢ. Ɉɞɧɚɤɨ ɡɚɤɨɧɧɨɫɬɶ ɩɪɢɦɟɧɟɧɢɹ ɬɚɤɨɝɨ ɤɪɢɬɟɪɢɹ ɧɟɨɛɯɨɞɢɦɨ ɨɛɨɫɧɨɜɚɬɶ ɞɥɹ ɭɫɥɨɜɢɣ ɞɚɧɧɨɣ ɡɚɞɚɱɢ. Ⱦɥɹ ɷɬɨɝɨ ɩɪɟɠɞɟ ɜɫɟɝɨ ɨɬɦɟɬɢɦ, ɱɬɨ ɛɟɫɫɟɥɟɜɵ ɮɭɧɤɰɢɢ ɩɪɢ ɛɨɥɶɲɢɯ ɨɬɧɨɫɢɬɟɥɶɧɨ ɚɪɝɭ- ɦɟɧɬɚ ɡɧɚɱɟɧɢɹɯ ɢɧɞɟɤɫɚ ɫɯɨɞɹɬɫɹ ɤ ɚɫɢɦɩɬɨɬɢɱɟɫɤɢɦ ɩɪɟɞɫɬɚɜɥɟɧɢɹɦ ɪɚɜɧɨɦɟɪɧɨ [14]. ɂɧɵɦɢ ɫɥɨɜɚɦɢ, ɟɫɥɢ ɞɥɹ ɧɟɤɨɬɨɪɨɝɨ ɡɧɚɱɟɧɢɹ (N) ɢɧɞɟɤɫɚ ɮɭɧɤɰɢɢ Ȼɟɫɫɟɥɹ ɧɟɜɹɡɤɚ (ɦɨɞɭɥɶ ɪɚɡɧɨɫɬɢ) ɦɟɠɞɭ ɬɨɱɧɵɦ ɡɧɚ- ɱɟɧɢɟɦ ɮɭɧɤɰɢɢ ɢ ɟɟ ɚɫɢɦɩɬɨɬɢɱɟɫɤɢɦ ɜɵɪɚɠɟɧɢɟɦ ɦɟɧɶɲɟ ɧɟɤɨɬɨɪɨɝɨ ɩɨɫɬɨɹɧɧɨɝɨ ɱɢɫɥɚ ε, ɬɨ ɩɪɢ ɜɫɟɯ ɡɧɚɱɟɧɢɹɯ ɢɧɞɟɤɫɚ, ɛɨɥɶɲɢɯ, ɱɟɦ N, ɷɬɚ ɧɟɜɹɡɤɚ ɛɭɞɟɬ ɦɟɧɶɲɟ (ɩɪɢ ɭɫɥɨɜɢɢ, ɱɬɨ ɚɪɝɭɦɟɧɬ — ɞɟɣɫɬɜɢɬɟɥɶ- ɧɨɟ ɩɨɫɬɨɹɧɧɨɟ ɱɢɫɥɨ, ɚ ɢɧɞɟɤɫ — ɰɟɥɨɟ ɩɨɥɨɠɢɬɟɥɶɧɨɟ ɱɢɫɥɨ). Ɉɬɫɸɞɚ ɫɥɟɞɭɟɬ, ɱɬɨ ɟɫɥɢ ɩɪɢ ɤɚɤɨɦ-ɥɢɛɨ ɩɨɪɹɞɤɟ ɭɫɟɱɟɧɢɹ ɛɭɞɟɬ ɭɫɬɚɧɨɜɥɟɧɨ ɞɨɫɬɢɠɟɧɢɟ ɧɟɜɹɡɤɨɣ ɡɚɞɚɧɧɨɝɨ ɡɧɚɱɟɧɢɹ, ɬɨ ɢ ɩɪɢ ɛɨɥɟɟ ɜɵɫɨɤɢɯ ɩɨ- ɪɹɞɤɚɯ ɪɟɞɭɤɰɢɢ ɷɬɨ ɭɫɥɨɜɢɟ ɬɚɤɠɟ ɛɭɞɟɬ ɜɵɩɨɥɧɟɧɨ. ɋ ɞɪɭɝɨɣ ɫɬɨɪɨɧɵ, ɚɫɢɦɩɬɨɬɢɱɟɫɤɨɟ ɩɨɜɟɞɟɧɢɟ ɦɚ- ɬɪɢɱɧɵɯ ɷɥɟɦɟɧɬɨɜ ɫ ɪɨɫɬɨɦ ɢɧɞɟɤɫɨɜ, ɚ ɢɦɟɧɧɨ ɢɯ ɭɛɵɜɚɧɢɟ ɩɨ ɦɨɞɭɥɸ ɩɪɢ ɭɞɚɥɟɧɢɢ ɨɬ ɝɥɚɜɧɨɣ ɞɢɚɝɨ- ɧɚɥɢ, ɧɟ ɯɭɠɟ, ɱɟɦ 1/l ɢɥɢ 1/j, ɢ ɫɨɜɦɟɫɬɧɨ ɫɨ ɫɤɨɪɨɫɬɶɸ ɭɛɵɜɚɧɢɹ ɞɢɚɝɨɧɚɥɶɧɵɯ ɷɥɟɦɟɧɬɨɜ ɨɡɧɚɱɚɟɬ, ɱɬɨ ɫɭɳɟɫɬɜɭɟɬ ɪɚɜɧɨɦɟɪɧɚɹ ɫɯɨɞɢɦɨɫɬɶ ɩɪɢɛɥɢɠɟɧɧɵɯ ɪɟɲɟɧɢɣ ɤ ɬɨɱɧɨɦɭ. ɋ ɬɨɱɤɢ ɡɪɟɧɢɹ ɜɵɱɢɫɥɢɬɟɥɶɧɨɣ ɪɟɚɥɢɡɚɰɢɢ ɷɬɨ ɩɨɡɜɨɥɹɟɬ ɭɫɬɚɧɨɜɢɬɶ ɤɪɢɬɟɪɢɢ ɩɪɟɤɪɚɳɟɧɢɹ ɜɵɱɢɫɥɟɧɢɣ ɩɨ ɭɫɥɨɜɢɸ ɞɨɫɬɢɠɟɧɢɹ ɧɟɜɹɡ- ɤɨɣ χ_N ɡɚɞɚɧɧɨɝɨ ɩɪɟɞɟɥɶɧɨɝɨ ɡɧɚɱɟɧɢɹ. ɇɟɜɹɡɤɭ χ_N ɩɪɢ ɷɬɨɦ ɦɨɠɧɨ ɨɩɪɟɞɟɥɢɬɶ ɤɚɤ

,

(8)

.

ɝɞɟ ɢ — ɪɟɲɟɧɢɹ, ɩɨɥɭɱɟɧɧɵɟ ɫ ɦɟɬɨɞɨɦ ɪɟɞɭɤɰɢɢ ɞɨ N ɢ N–1 ɩɨɪɹɞɤɨɜ ɫɨɨɬɜɟɬɫɬɜɟɧɧɨ.

Ɋɟɡɭɥɶɬɚɬɵ ɜɵɱɢɫɥɢɬɟɥɶɧɵɯ ɷɤɫɩɟɪɢɦɟɧɬɨɜ ɢ ɢɯ ɚɧɚɥɢɡ

ȼ ɫɨɨɬɜɟɬɫɬɜɢɢ ɫ ɨɩɢɫɚɧɧɵɦ ɚɥɝɨɪɢɬɦɨɦ ɛɵɥɚ ɧɚɩɢɫɚɧɚ ɩɪɨɝɪɚɦɦɚ ɞɥɹ ɪɚɫɱɟɬɚ ɞɢɚɝɪɚɦɦ ɪɚɫɫɟɹɧɢɹ ɢ ɩɪɨɜɟɞɟɧɚ ɫɟɪɢɹ ɫɨɨɬɜɟɬɫɬɜɭɸɳɢɯ ɱɢɫɥɟɧɧɵɯ ɷɤɫɩɟɪɢɦɟɧɬɨɜ.

ɂɡ ɪɢɫ. 2ɚ ɜɢɞɧɨ, ɱɬɨ ɪɚɡɪɚɛɨɬɚɧɧɵɣ ɱɢɫɥɟɧɧɵɣ ɚɥɝɨɪɢɬɦ ɩɨɡɜɨɥɹɟɬ ɩɨɥɭɱɢɬɶ ɭɫɬɨɣɱɢɜɨɟ ɪɟɲɟɧɢɟ ɡɚ- ɞɚɱɢ ɞɢɮɪɚɤɰɢɢ ɩɪɢ ɦɟɧɶɲɟɦ ɩɨɪɹɞɤɟ ɭɫɟɱɟɧɢɹ ɦɚɬɪɢɰɵ ɫɢɫɬɟɦɵ (ɤɪɢɜɚɹ 1), ɱɟɦ ɩɪɟɞɩɨɥɚɝɚɟɬɫɹ ɩɪɢ ɪɟ- ɲɟɧɢɢ ɬɪɚɞɢɰɢɨɧɧɵɦɢ ɦɟɬɨɞɚɦɢ — ɤɪɢɜɚɹ 2 (ɢɦɟɟɬɫɹ ɜ ɜɢɞɭ ɜɟɥɢɱɢɧɚ N = ka + 5, ɝɞɟ k — ɷɬɨ ɜɨɥɧɨɜɨɟ ɱɢɫɥɨ ɜ ɦɚɬɟɪɢɚɥɟ ɜɤɥɸɱɟɧɢɹ, a — ɪɚɞɢɭɫ ɰɢɥɢɧɞɪɚ).

Ɉɫɧɨɜɵɜɚɹɫɶ ɧɚ ɞɚɧɧɵɯ ɨ ɩɨɪɹɞɤɟ ɭɫɟɱɟɧɢɹ ɦɚɬɪɢɰɵ, ɞɨɫɬɚɬɨɱɧɨɦ ɞɥɹ ɩɨɥɭɱɟɧɢɹ ɭɫɬɨɣɱɢɜɨɝɨ ɪɟ- ɲɟɧɢɹ, ɛɵɥɢ ɩɪɨɜɟɞɟɧɵ ɪɚɫɱɟɬɵ ɞɢɚɝɪɚɦɦ ɪɚɫɫɟɹɧɢɹ ɞɥɹ ɪɚɡɥɢɱɧɵɯ ɤɨɧɮɢɝɭɪɚɰɢɣ ɪɚɫɫɟɢɜɚɸɳɟɣ ɫɬɪɭɤɬɭɪɵ.

ɇɚ ɪɢɫ. 2ɛ ɩɪɢɜɟɞɟɧ ɧɚɛɨɪ ɞɢɚɝɪɚɦɦ ɪɚɫɫɟɹɧɢɹ ɞɥɹ ɞɢɷɥɟɤɬɪɢɱɟɫɤɨɝɨ ɰɢɥɢɧɞɪɚ (ka~10, k = 2π/λ, a — ɪɚɞɢɭɫ ɰɢɥɢɧɞɪɚ), ɪɚɫɩɨɥɨɠɟɧɧɨɝɨ ɜ ɩɟɪɜɨɦ ɢɡ ɬɪɟɯ ɫɥɨɟɜ, ɪɚɡɞɟɥɹɸɳɢɯ ɞɜɚ ɨɞɧɨɪɨɞɧɵɯ ɩɨɥɭɩɪɨɫɬɪɚɧ- ɫɬɜɚ. Ƚɥɭɛɢɧɚ ɩɨɝɪɭɠɟɧɢɹ ɜɤɥɸɱɟɧɢɹ 1,5λ (ɤɪɢɜɚɹ 1) ɢ 2,5λ (ɤɪɢɜɚɹ 2).

ɚ) ɛ)

Ɋɢɫ. 2. Ɋɟɡɭɥɶɬɚɬɵ ɱɢɫɥɟɧɧɵɯ ɷɤɫɩɟɪɢɦɟɧɬɨɜ:

ɚ — ɡɚɜɢɫɢɦɨɫɬɶ ɨɩɬɢɦɚɥɶɧɨɝɨ ɩɨɪɹɞɤɚ ɭɫɟɱɟɧɢɹ ɦɚɬɪɢɰɵ ɫɢɫɬɟɦɵ ɨɬ ɜɨɥɧɨɜɨɝɨ ɪɚɡɦɟɪɚ ɪɚɫɫɟɢɜɚɬɟɥɹ (1 – N = Nmax, 2 – N = ka + 5);

ɛ — ɞɢɚɝɪɚɦɦɵ ɪɚɫɫɟɹɧɢɹ ȿ-ɩɨɥɹɪɢɡɨɜɚɧɧɨɣ ɩɥɨɫɤɨɣ ɜɨɥɧɵ (ɫɥɭɱɚɣ ɧɨɪɦɚɥɶɧɨɝɨ ɩɚɞɟɧɢɹ) ɞɥɹ ɤɪɭɝɨɜɨɝɨ ɰɢɥɢɧɞɪɚ (a/λ = 1,5), ɪɚɫɩɨɥɨɠɟɧɧɨɝɨ ɜ ɩɟɪɜɨɦ ɫɥɨɟ ɬɪɟɯɫɥɨɣɧɨɣ ɫɬɪɭɤɬɭɪɵ Ɂɚɤɥɸɱɟɧɢɟ

Ɍɚɤɢɦ ɨɛɪɚɡɨɦ, ɜ ɪɚɛɨɬɟ ɩɪɟɞɥɨɠɟɧɨ ɪɚɡɜɢɬɢɟ ɱɢɫɥɟɧɧɨɣ ɪɟɚɥɢɡɚɰɢɢ ɦɟɬɨɞɚ ɧɭɥɟɜɨɝɨ ɩɨɥɹ ɞɥɹ ɪɟɲɟ- ɧɢɹ ɡɚɞɚɱɢ ɞɢɮɪɚɤɰɢɢ ɧɨɪɦɚɥɶɧɨ ɩɚɞɚɸɳɟɣ ɩɥɨɫɤɨɣ ȿ-ɩɨɥɹɪɢɡɨɜɚɧɧɨɣ ɜɨɥɧɵ ɧɚ ɰɢɥɢɧɞɪɢɱɟɫɤɨɦ ɨɞɧɨ- ɪɨɞɧɨɦ ɜɤɥɸɱɟɧɢɢ ɜ ɩɥɨɫɤɨɫɥɨɢɫɬɨɣ ɫɪɟɞɟ. ȼ ɱɚɫɬɧɨɫɬɢ, ɫ ɩɨɦɨɳɶɸ ɜɵɞɟɥɟɧɢɹ ɝɥɚɜɧɨɣ ɱɚɫɬɢ ɦɚɬɪɢɱɧɨ- ɝɨ ɨɩɟɪɚɬɨɪɚ ȻɋɅȺɍ 1 ɡɚɞɚɱɚ ɨɩɪɟɞɟɥɟɧɢɹ ɧɟɢɡɜɟɫɬɧɵɯ ɤɨɷɮɮɢɰɢɟɧɬɨɜ ɪɚɡɥɨɠɟɧɢɹ ɪɚɫɫɟɹɧɧɨɝɨ ɩɨɥɹ ɩɨ ɰɢɥɢɧɞɪɢɱɟɫɤɢɦ ɝɚɪɦɨɧɢɤɚɦ ɫɜɟɞɟɧɚ ɤ ɮɪɟɞɝɨɥɶɦɨɜɨɣ ȻɋɅȺɍ 2 ɫ ɜɩɨɥɧɟ ɧɟɩɪɟɪɵɜɧɵɦ ɦɚɬɪɢɱɧɵɦ ɨɩɟ- ɪɚɬɨɪɨɦ. Ɍɚɤɢɟ ɡɚɞɚɱɢ ɤɨɪɪɟɤɬɧɵ, ɩɨɡɜɨɥɹɸɬ ɢɫɩɨɥɶɡɨɜɚɬɶ ɞɥɹ ɢɯ ɪɟɲɟɧɢɹ ɦɟɬɨɞ ɪɟɞɭɤɰɢɢ. Ⱦɨɤɚɡɚɬɟɥɶ- ɫɬɜɨ ɫɯɨɞɢɦɨɫɬɢ ɦɟɬɨɞɚ ɪɟɞɭɤɰɢɢ ɩɨɡɜɨɥɢɥɨ ɬɚɤɠɟ ɨɛɨɫɧɨɜɚɬɶ ɤɪɢɬɟɪɢɢ ɩɨɪɹɞɤɚ ɭɫɟɱɟɧɢɹ ɫɢɫɬɟɦɵ, ɱɬɨ ɜ ɫɜɨɸ ɨɱɟɪɟɞɶ ɩɨɡɜɨɥɢɥɨ ɨɩɬɢɦɢɡɢɪɨɜɚɬɶ ɜɵɱɢɫɥɢɬɟɥɶɧɵɣ ɚɥɝɨɪɢɬɦ. ɗɬɨ ɨɱɟɧɶ ɜɚɠɧɨ ɞɥɹ ɨɬɵɫɤɚɧɢɹ ɩɨɥɟɣ, ɪɚɫɫɟɹɧɧɵɯ ɜɤɥɸɱɟɧɢɹɦɢ ɛɨɥɶɲɢɯ ɜɨɥɧɨɜɵɯ ɪɚɡɦɟɪɨɜ. ɉɪɢɜɟɞɟɧɧɵɟ ɱɢɫɥɟɧɧɵɟ ɪɟɡɭɥɶɬɚɬɵ ɩɨɞ- ɬɜɟɪɠɞɚɸɬ ɷɮɮɟɤɬɢɜɧɨɫɬɶ ɩɪɟɞɥɨɠɟɧɧɨɝɨ ɩɨɞɯɨɞɚ.

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DIFFRACTION OF THE E-POLARIZED PLANE WAVE

BY A CYLINDRICAL INCLUSION IN PLANE-LAYERED MEDIUM

Batrakov D.O., Golovin D.V.

Development of the numerical implementation of null-¿ eld technique is proposed. Solution of the diffraction problem for normally incident plane E-polarized wave on a homogeneous cylindrical inclusion in a plane-layered medium is presented. The problem of determining the unknown coef¿ cients of expansion of the full ¿ eld on the contour of the inclusion in cylindrical harmonics basis is reduced to a Fredholm in¿ nite system of linear algebraic equations of the second kind with completely continuous matrix operator. Proof of the reduction method conver- gence allowed to obtain the criteria for the system truncation order determination and to optimize the computational algorithm. Numerical results con¿ rm the effectiveness of the proposed approach.

ɋɩɢɫɨɤ ɥɢɬɟɪɚɬɭɪɵ