Закономерности частотного сканирования в волноводно-щелевой антенне,возбуждаемой замедленной волной

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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.

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.

ȼɫɟ ɩɪɚɜɚ ɡɚɳɢɳɟɧɵ.

ɉɟɪɟɩɟɱɚɬɤɚ ɦɚɬɟɪɢɚɥɨɜ ɠɭɪɧɚɥɚ ɧɟ ɜɨɡɦɨɠɧɚ ɛɟɡ ɩɢɫɶɦɟɧɧɨɝɨ ɪɚɡɪɟɲɟɧɢɹ ɪɟɞɚɤɰɢɢ.

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7 Ⱥɧɧɨɬɚɰɢɹ

Ɋɚɛɨɬɚ ɩɨɫɜɹɳɟɧɚ ɜɨɩɪɨɫɚɦ ɱɚɫɬɨɬɧɨɝɨ ɫɤɚɧɢ- ɪɨɜɚɧɢɹ ɜ ɚɧɬɟɧɧɟ ɛɟɝɭɳɟɣ ɜɨɥɧɵ ɫ ɞɢɫɤɪɟɬɧɨɣ ɫɢɫɬɟɦɨɣ ɢɡɥɭɱɚɬɟɥɟɣ. Ɉɫɧɨɜɧɨɟ ɜɧɢɦɚɧɢɟ ɭɞɟɥɹ- ɟɬɫɹ ɫɥɭɱɚɸ, ɤɨɝɞɚ ɮɚɡɨɜɚɹ ɫɤɨɪɨɫɬɶ ɛɟɝɭɳɟɣ ɜɨɥɧɵ ɦɟɧɶɲɟ ɫɤɨɪɨɫɬɢ ɫɜɟɬɚ. ɉɨɥɭɱɟɧɵ ɭɫɥɨɜɢɹ ɨɞɧɨɥɟ- ɩɟɫɬɤɨɜɨɝɨ ɫɤɚɧɢɪɨɜɚɧɢɹ ɜ ɫɟɤɬɨɪɟ ɞɟɣɫɬɜɢɬɟɥɶɧɵɯ ɭɝɥɨɜ. ɂɫɫɥɟɞɨɜɚɧ ɩɪɨɰɟɫɫ ɫɤɚɧɢɪɨɜɚɧɢɹ ɧɚ ɩɪɢɦɟ- ɪɟ ɜɨɥɧɨɜɨɞɧɨ-ɳɟɥɟɜɨɣ ɚɧɬɟɧɧɵ ɧɚ ɛɚɡɟ ɜɨɥɧɨɜɨɞɚ ɫ ɱɚɫɬɢɱɧɵɦ ɞɢɷɥɟɤɬɪɢɱɟɫɤɢɦ ɡɚɩɨɥɧɟɧɢɟɦ.

Ʉɥɸɱɟɜɵɟ ɫɥɨɜɚ: ɜɨɥɧɨɜɨɞ, ɳɟɥɶ, ɢɡɥɭɱɚɬɟɥɶ, ɥɭɱ, ɫɤɚɧɢɪɨɜɚɧɢɟ, ɱɚɫɬɨɬɚ, ɤɨɷɮɮɢɰɢɟɧɬ ɢɡɥɭ- ɱɟɧɢɹ

© Ȼɝɭɩɫɶ, 2012

Яцук Л.П. — доктор физико-математических наук, профессор, Харьковский национальный университет им. В.Н. Кара- зина, Украина. [email protected]

Блинова Н.К. — кандидат физико-математических наук, старший научный сотрудник, Харьковский национальный университет им. В.Н. Каразина, Украина. [email protected]

Ляховский А.А. — научный сотрудник, Харьковский национальный университет им. В.Н. Каразина, Украина.

[email protected]

Ляховский А.Ф. — кандидат физико-математических наук, старший научный сотрудник, Харьковский националь- ный университет им. В.Н. Каразина, Украина. [email protected]

УДК 621.396.677

ɂȻɅɉɈɉɇɀɋɈɉɌɍɃ ɒȻɌɍɉɍɈɉȾɉ ɌɅȻɈɃɋɉȽȻɈɃɚ Ƚ ȽɉɆɈɉȽɉȿɈɉ-ɔɀɆɀȽɉɄ ȻɈɍɀɈɈɀ,

ȽɉɂȼɎɁȿȻɀɇɉɄ ɂȻɇɀȿɆɀɈɈɉɄ ȽɉɆɈɉɄ

ɆȺɌȿɆȺɌɂɑȿɋɄɈȿ ɆɈȾȿɅɂɊɈȼȺɇɂȿ ɎɂɁɂɑȿɋɄɂɏ ɉɊɈɐȿɋɋɈȼ

ФИЗИЧЕСКИЕ ОСНОВЫ ПРИБОРОСТРОЕНИЯ. 2012 Том 1, № 1

ȼɜɟɞɟɧɢɟ

ȼɨɥɧɨɜɨɞɧɨ-ɳɟɥɟɜɵɟ ɚɧɬɟɧɧɵ (ȼɓȺ) ɧɚ ɛɚɡɟ ɛɟɫɤɨɧɟɱɧɨɝɨ ɜɨɥɧɨɜɨɞɚ ɨɬɧɨɫɹɬɫɹ ɤ ɚɧɬɟɧɧɚɦ ɛɟ- ɝɭɳɟɣ ɜɨɥɧɵ ɫ ɞɢɫɤɪɟɬɧɨɣ ɫɢɫɬɟɦɨɣ ɢɡɥɭɱɚɬɟɥɟɣ. Ɍɟɨɪɢɹ ɬɚɤɢɯ ɚɧɬɟɧɧ ɩɪɟɞɫɬɚɜɥɟɧɚ ɜ ɪɹɞɟ ɤɥɚɫ- ɫɢɱɟɫɤɢɯ ɢɡɞɚɧɢɣ ɩɨ ɚɧɬɟɧɧɨɣ ɬɟɦɚɬɢɤɟ [1–4]. ɂɯ ɱɚɫɬɨ ɢɫɩɨɥɶɡɭɸɬ ɜ ɫɢɫɬɟɦɚɯ ɫ ɱɚɫɬɨɬɧɵɦ ɫɤɚɧɢ- ɪɨɜɚɧɢɟɦ ɞɢɚɝɪɚɦɦɵ ɧɚɩɪɚɜɥɟɧɧɨɫɬɢ (Ⱦɇ). Ɉɫɨɛɵɣ ɢɧɬɟɪɟɫ ɩɪɟɞɫɬɚɜɥɹɸɬ ɫɨɛɨɣ ɚɧɬɟɧɧɵ ɧɚ ɛɚɡɟ ɜɨɥɧɨɜɨɞɚ ɫ ɡɚɦɟɞɥɟɧɧɨɣ ɜɨɥɧɨɣ, ɩɨɫɤɨɥɶɤɭ ɨɧɢ ɩɨɡɜɨɥɹɸɬ ɩɨɥɭɱɢɬɶ ɛɨɥɟɟ ɜɵɫɨɤɭɸ ɭɝɥɨ-ɱɚɫɬɨɬɧɭɸ ɡɚɜɢɫɢɦɨɫɬɶ ɩɪɢ ɫɤɚɧɢɪɨɜɚɧɢɢ, ɱɟɦ ɚɧɬɟɧɧɵ ɧɚ ɛɚɡɟ ɩɨɥɨɝɨ ɜɨɥɧɨɜɨɞɚ. Ʉɚɤ ɩɪɚɜɢɥɨ, ɬɪɟɛɭɟɬɫɹ, ɱɬɨ- ɛɵ ɩɪɢ ɫɤɚɧɢɪɨɜɚɧɢɢ Ⱦɇ ɢɦɟɥɚ ɨɞɢɧ ɝɥɚɜɧɵɣ ɦɚɤɫɢɦɭɦ (ȽɆ). ɗɬɨɦɭ ɫɨɨɬɜɟɬɫɬɜɭɸɬ ɨɩɪɟɞɟɥɟɧɧɵɟ ɫɨɨɬɧɨɲɟɧɢɹ ɦɟɠɞɭ ɤɨɷɮɮɢɰɢɟɧɬɨɦ ɡɚɦɟɞɥɟɧɢɹ [ O O_g , ɞɥɢɧɨɣ ɜɨɥɧɵ ɜ ɫɜɨɛɨɞɧɨɦ ɩɪɨɫɬɪɚɧɫɬɜɟ Ȝ ɢ ɪɚɫɫɬɨɹɧɢɟɦ ɦɟɠɞɭ ɫɨɫɟɞɧɢɦɢ ɢɡɥɭɱɚɬɟɥɹɦɢ (O_g — ɞɥɢɧɚ ɜɨɥɧɵ ɜ ɜɨɥɧɨɜɨɞɟ). ɗɬɢ ɫɨɨɬɧɨɲɟɧɢɹ ɩɨɞɪɨɛɧɨ ɨɛɫɭɠɞɚɸɬɫɹ ɜ [1] ɢ ɞɪɭɝɢɯ ɢɫɬɨɱɧɢɤɚɯ. ȼ ɧɟɤɨɬɨɪɵɯ ɢɡ ɧɢɯ, ɧɚɩɪɢɦɟɪ [5, 6], ɭɬɜɟɪɠɞɚɟɬɫɹ, ɱɬɨ ɜ ɫɥɭɱɚɟ ɤɨɷɮɮɢɰɢɟɧɬɚ ɡɚɦɟɞɥɟɧɢɹ [, ɛɨɥɶɲɟɝɨ ɟɞɢɧɢɰɵ, «…ɜɨɥɧɚ ɨɬɤɥɨɧɹɬɶɫɹ ɨɬ ɨɫɢ ɜɨɥɧɨɜɨɞɚ ɧɟ ɦɨɠɟɬ». Ɍɟɦ ɧɟ ɦɟɧɟɟ ɜɚɠɧɨ ɞɚɬɶ ɱɟɬɤɨɟ ɩɪɟɞɫɬɚɜɥɟɧɢɟ ɨ ɬɨɦ, ɜ ɱɟɦ ɡɚɤɥɸɱɚɟɬɫɹ ɪɚɡɥɢɱɢɟ ɜ ɩɪɨ- ɰɟɫɫɟ ɮɨɪɦɢɪɨɜɚɧɢɹ Ⱦɇ ɚɧɬɟɧɧɵ ɛɟɝɭɳɟɣ ɜɨɥɧɵ ɫ ɧɟɩɪɟɪɵɜɧɨɣ ɢ ɞɢɫɤɪɟɬɧɨɣ ɫɢɫɬɟɦɨɣ ɢɡɥɭɱɚɬɟɥɟɣ, ɫ ɤɨɷɮɮɢɰɢɟɧɬɨɦ ɡɚɦɟɞɥɟɧɢɹ ɜɨɥɧɵ [, ɛɨɥɶɲɢɦ ɢɥɢ ɦɟɧɶɲɢɦ ɟɞɢɧɢɰɵ, ɜɵɩɨɥɧɟɧɢɟ ɤɚɤɢɯ ɭɫɥɨɜɢɣ ɨɛɟɫɩɟɱɢɜɚɟɬ ɧɚɥɢɱɢɟ ɬɨɥɶɤɨ ɨɞɧɨɝɨ ȽɆ ɜ ɫɟɤɬɨɪɟ ɫɤɚɧɢɪɨɜɚɧɢɹ. ɗɬɢ ɜɨɩɪɨɫɵ ɢɫɫɥɟɞɭɸɬɫɹ ɜ ɧɚɫɬɨ- ɹɳɟɣ ɪɚɛɨɬɟ.

Abstract

The work is devoted to the questions of scanning in the antenna of traveling wave having a discrete system of radiators. The main attention is paid to the case when the phase velocity of the traveling wave is less then light velocity. The conditions of one-beam scanning in a sector of real angles are obtained. The scanning process is studied on the example of slotted waveguide with partial dielectric ¿ lling.

Key words: waveguide, slot, radiator, beam, scanning, frequency, radiation coef¿ cieent

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8

əɰɭɤ Ʌ.ɉ., Ȼɥɢɧɨɜɚ ɇ.Ʉ., Ʌɹɯɨɜɫɤɢɣ Ⱥ.Ⱥ., Ʌɹɯɨɜɫɤɢɣ Ⱥ.Ɏ.

ɂɫɫɥɟɞɨɜɚɧɢɟ ɜɨɩɪɨɫɚ ɩɪɨɜɟɞɟɦ ɧɚ ɩɪɢɦɟɪɟ ȼɓȺ ɛɟɝɭɳɟɣ ɜɨɥɧɵ ɧɚ ɛɚɡɟ ɩɪɹɦɨɭɝɨɥɶɧɨɝɨ ɜɨɥɧɨɜɨɞɚ, ɱɚɫɬɢɱɧɨ ɡɚɩɨɥɧɟɧɧɨɝɨ ɞɢɷɥɟɤɬɪɢɤɨɦ. ɋɥɨɣ ɞɢɷɥɟɤɬɪɢɤɚ ɦɨɠɟɬ ɛɵɬɶ ɪɚɫɩɨɥɨɠɟɧ ɩɚɪɚɥɥɟɥɶɧɨ ɭɡɤɢɦ ɢɥɢ ɲɢɪɨɤɢɦ ɫɬɟɧɤɚɦ, ɪɚɫɫɦɚɬɪɢɜɚɟɦɵɟ ɳɟɥɢ ɩɪɨɞɨɥɶɧɵɟ ɢɥɢ ɩɨɩɟɪɟɱɧɵɟ, ɜɨɥɧɚ ɛɟɠɢɬ ɜɞɨɥɶ ɩɪɨɞɨɥɶɧɨɣ ɨɫɢ ɜɨɥɧɨɜɨɞɚ z. ȼ ɥɢɬɟɪɚɬɭɪɟ ɞɥɹ ɫɢɫɬɟɦ ɬɚɤɢɯ ɳɟɥɟɣ ɢɧɨɝɞɚ ɢɫɩɨɥɶɡɭɸɬɫɹ ɧɚɡɜɚɧɢɹ ɩɪɹɦɨɮɚɡɧɵɟ (ɞɥɹ ɩɨɩɟɪɟɱɧɵɯ) ɢ ɩɟɪɟɦɟɧɧɨ-ɮɚɡɧɵɟ (ɞɥɹ ɩɪɨɞɨɥɶɧɵɯ).

ɉɪɢɧɰɢɩɢɚɥɶɧɨɟ ɪɚɡɥɢɱɢɟ ɫɬɪɭɤɬɭɪ ɞɢɚɝɪɚɦɦ ɧɚɩɪɚɜɥɟɧɧɨɫɬɢ ɚɧɬɟɧɧ ɛɟɝɭɳɟɣ ɜɨɥɧɵ ɫ ɧɟɩɪɟɪɵɜɧɨɣ ɢ ɞɢɫɤɪɟɬɧɨɣ ɫɢɫɬɟɦɚɦɢ ɢɡɥɭɱɚɬɟɥɟɣ

Ʉɚɤ ɢɡɜɟɫɬɧɨ [1, 3, 4], Ⱦɇ ɚɧɬɟɧɧ ɛɟɝɭɳɟɣ ɜɨɥɧɵ ɫ ɧɟɩɪɟɪɵɜɧɨɣ ɫɢɫɬɟɦɨɣ ɢɡɥɭɱɚɬɟɥɟɣ ɢɦɟɸɬ ɜɫɟɝɨ ɨɞɢɧ ȽɆ ɩɪɢ [d1 ɢɥɢ ɧɢ ɨɞɧɨɝɨ — ɩɪɢ [!1. ɗɬɨɬ ȽɆ ɜ [1, 4] ɧɚɡɜɚɧ ɨɫɧɨɜɧɵɦ. Ɉɧ ɩɨɹɜɥɹɟɬɫɹ ɜ ɫɟɤ- ɬɨɪɟ ɩɨɥɨɠɢɬɟɥɶɧɵɯ ɭɝɥɨɜ, ɨɬɫɱɢɬɵɜɚɟɦɵɯ ɨɬ ɧɨɪɦɚɥɢ ɜ ɫɬɨɪɨɧɭ ɧɚɩɪɚɜɥɟɧɢɹ ɛɟɝɭɳɟɣ ɜɨɥɧɵ. ȼ ɷɬɨɦ ɧɚɩɪɚɜɥɟɧɢɢ ɩɨɥɹ ɩɪɢɯɨɞɹɬ ɜ ɬɨɱɤɭ ɧɚɛɥɸɞɟɧɢɹ ɢɡ ɪɚɡɧɵɯ ɬɨɱɟɤ ɚɧɬɟɧɧɵ ɫ ɧɭɥɟɜɵɦ ɫɞɜɢɝɨɦ ɩɨ ɮɚɡɟ.

ɉɪɢ [d1 ɜɨɥɧɨɜɨɟ ɱɢɫɥɨ J 2S O_g ɜ ɜɨɥɧɨɜɨɞɟ ɦɟɧɶɲɟ (ɢɥɢ ɪɚɜɧɨ) ɜɨɥɧɨɜɨɝɨ ɱɢɫɥɚ k 2S O ɜ ɫɜɨ- ɛɨɞɧɨɦ ɩɪɨɫɬɪɚɧɫɬɜɟ, ɩɨɷɬɨɦɭ ɧɚɛɟɝ ɮɚɡɵ ɦɟɠɞɭ ɞɜɭɦɹ ɢɡɥɭɱɚɬɟɥɹɦɢ ɜɧɭɬɪɢ ɜɨɥɧɨɜɨɞɚ ɦɨɠɟɬ ɛɵɬɶ ɫɤɨɦɩɟɧɫɢɪɨɜɚɧ ɛɨɥɟɟ ɤɨɪɨɬɤɨɣ ɪɚɡɧɨɫɬɶɸ ɯɨɞɚ ɥɭɱɟɣ ɫɧɚɪɭɠɢ. ɗɬɨ ɧɟɜɨɡɦɨɠɧɨ ɩɪɢ [!1. ɂ ɟɫɥɢ ɭɝɨɥ -, ɨɬɫɱɢɬɵɜɚɟɦɵɣ ɨɬ ɧɨɪɦɚɥɢ ɤ ɨɫɢ ɚɧɬɟɧɧɵ (ɪɢɫ. 1), ɩɨɞ ɤɨɬɨɪɵɦ ɜɢɞɟɧ ɷɬɨɬ ɦɚɤɫɢɦɭɦ, ɨɩɪɟɞɟɥɹɟɬɫɹ ɫɨɨɬɧɨɲɟɧɢɟɦ [1–4]

sin- O O_g, (1)

ɬɨ ɩɪɢ [!1 ȽɆ ɬɚɤɨɝɨ ɩɪɨɢɫɯɨɠɞɟɧɢɹ (ɨɫɧɨɜɧɨɣ), ɭɪɨɜɟɧɶ ɤɨɬɨɪɨɝɨ ɪɚɜɟɧ ɟɞɢɧɢɰɟ, ɜ ɫɟɤɬɨɪɟ ɞɟɣɫɬɜɢ- ɬɟɥɶɧɵɯ ɭɝɥɨɜ ɜɨɨɛɳɟ ɨɬɫɭɬɫɬɜɭɟɬ.

ɂɧɚɹ ɤɚɪɬɢɧɚ ɧɚɛɥɸɞɚɟɬɫɹ ɜ ɫɥɭɱɚɟ ɞɢɫɤɪɟɬɧɨɣ ɫɢɫɬɟɦɵ ɢɡɥɭɱɚɬɟɥɟɣ. Ɂɞɟɫɶ ɧɟɡɚɜɢɫɢɦɨ ɨɬ ɪɚɫɫɬɨɹɧɢɹ ɦɟɠɞɭ ɫɨɫɟɞɧɢɦɢ ɢɡɥɭɱɚɬɟɥɹɦɢ ɩɪɢ [d1 ɬɨɠɟ ɫɭɳɟɫɬɜɭɟɬ ɨɫɧɨɜɧɨɣ ȽɆ, ɧɚɩɪɚɜɥɟɧɢɟ ɤɨɬɨɪɨɝɨ ɨɩɪɟɞɟ- ɥɹɟɬɫɹ ɮɨɪɦɭɥɨɣ (1). Ʉɚɤ ɢ ɜ ɫɥɭɱɚɟ ɧɟɩɪɟɪɵɜɧɨɣ ɫɢɫɬɟɦɵ ɢɡɥɭɱɚɬɟɥɟɣ, ɨɧ ɦɨɠɟɬ ɫɭɳɟɫɬɜɨɜɚɬɶ ɬɨɥɶɤɨ ɜ ɫɟɤɬɨɪɟ ɩɨɥɨɠɢɬɟɥɶɧɵɯ ɭɝɥɨɜ -.

Ɋɢɫ. 1. ɂɡɥɭɱɚɬɟɥɢ ɧɚ ɜɨɥɧɨɜɨɞɟ ɫ ɡɚɦɟɞɥɹɸɳɟɣ ɫɢɫɬɟɦɨɣ

ɇɨ ɤɪɨɦɟ ɨɫɧɨɜɧɨɝɨ ɜ Ⱦɇ ɞɢɫɤɪɟɬɧɨɣ ɫɢɫɬɟɦɵ ɢɡɥɭɱɚɬɟɥɟɣ, ɜ ɨɬɥɢɱɢɟ ɨɬ ɧɟɩɪɟɪɵɜɧɨɣ ɫɢɫɬɟɦɵ ɢɡ- ɥɭɱɚɬɟɥɟɣ, ɦɨɝɭɬ ɫɭɳɟɫɬɜɨɜɚɬɶ ɢ ɞɪɭɝɢɟ ȽɆ. Ɉɧɢ ɜɨɡɧɢɤɚɸɬ ɜ ɬɟɯ ɧɚɩɪɚɜɥɟɧɢɹɯ, ɜ ɤɨɬɨɪɵɯ ɪɚɡɧɨɫɬɶ ɮɚɡ ɩɨɥɟɣ, ɩɪɢɲɟɞɲɢɯ ɜ ɬɨɱɤɭ ɧɚɛɥɸɞɟɧɢɹ ɨɬ ɫɨɫɟɞɧɢɯ ɢɡɥɭɱɚɬɟɥɟɣ, ɪɚɜɧɚ 2Sm, ɝɞɟ m — ɰɟɥɨɟ ɱɢɫɥɨ.

ȼ ɪɚɡɧɵɯ ɪɚɛɨɬɚɯ ɢɯ ɧɚɡɵɜɚɸɬ ɩɨ-ɪɚɡɧɨɦɭ. Ɇɵ ɧɚɡɨɜɟɦ ɢɯ ɢɧɬɟɪɮɟɪɟɧɰɢɨɧɧɵɦɢ ɝɥɚɜɧɵɦɢ ɦɚɤɫɢɦɭ- ɦɚɦɢ (ɂȽɆ).

ȼ ɥɢɬɟɪɚɬɭɪɟ ɩɨɞɪɨɛɧɨ ɢɫɫɥɟɞɨɜɚɧɵ ɡɚɤɨɧɨɦɟɪɧɨɫɬɢ ɩɨɹɜɥɟɧɢɹ ɢɧɬɟɪɮɟɪɟɧɰɢɨɧɧɵɯ ɦɚɤɫɢɦɭɦɨɜ ɜ ɫɥɭɱɚɟ [d1. ȼ ɧɚɫɬɨɹɳɟɣ ɪɚɛɨɬɟ ɨɫɧɨɜɧɨɟ ɜɧɢɦɚɧɢɟ ɭɞɟɥɹɟɬɫɹ ɫɢɫɬɟɦɟ ɳɟɥɟɣ ɜ ɜɨɥɧɨɜɨɞɟ ɫ ɡɚɦɟɞɥɟɧɧɨɣ ɜɨɥɧɨɣ ɫ ɤɨɷɮɮɢɰɢɟɧɬɨɦ ɡɚɦɟɞɥɟɧɢɹ [!1. ɉɪɨɰɟɫɫ ɩɨɹɜɥɟɧɢɹ ȽɆ ɢ ɡɚɤɨɧɨɦɟɪɧɨɫɬɢ ɫɤɚɧɢɪɨɜɚɧɢɹ ɢɡɭ- ɱɢɦ ɧɚ ɩɪɨɫɬɟɣɲɟɦ ɩɪɢɦɟɪɟ ɨɞɧɨɪɨɞɧɨɣ ɞɢɫɤɪɟɬɧɨɣ ɫɢɫɬɟɦɵ N ɢɡɥɭɱɚɬɟɥɟɣ, ɪɚɡɧɟɫɟɧɧɵɯ ɧɚ ɪɚɫɫɬɨɹɧɢɟ

dz ɞɪɭɝ ɨɬ ɞɪɭɝɚ. Ʉɨɦɩɥɟɤɫɧɵɣ ɦɧɨɠɢɬɟɥɶ ɬɚɤɨɣ ɫɢɫɬɟɦɵ ɢɦɟɟɬ ɜɢɞ [1]:

(5)

9 Ɂɚɤɨɧɨɦɟɪɧɨɫɬɢ ɱɚɫɬɨɬɧɨɝɨ ɫɤɚɧɢɪɨɜɚɧɢɹ ɜ ɜɨɥɧɨɜɨɞɧɨ-ɳɟɥɟɜɨɣ ɚɧɬɟɧɧɟ, ɜɨɡɛɭɠɞɚɟɦɨɣ ɡɚɦɟɞɥɟɧɧɨɣ ɜɨɥɧɨɣ

sin 2 sin

z c

z

f Nkd

N kd

[ -

- [ -

ª º

¬ ¼

ª º

¬ ¼. (2)

ɗɬɨ ɜɵɪɚɠɟɧɢɟ ɫɜɨɞɢɬɫɹ ɤ ɮɨɪɦɭɥɟ ɞɥɹ ɤɨɦɩɥɟɤɫɧɨɝɨ ɦɧɨɠɢɬɟɥɹ ɧɟɩɪɟɪɵɜɧɨɣ ɫɢɫɬɟɦɵ ɢɡɥɭɱɚɬɟɥɟɣ, ɟɫɥɢ ɨɞɧɨɜɪɟɦɟɧɧɨ ɭɜɟɥɢɱɢɜɚɬɶ N ɢ ɭɦɟɧɶɲɚɬɶ d_z ɬɚɤ, ɱɬɨɛɵ Nd_z ɨɫɬɚɜɚɥɨɫɶ ɩɨɫɬɨɹɧɧɨɣ ɜɟɥɢɱɢɧɨɣ, ɪɚɜɧɨɣ ɞɥɢɧɟ ɚɧɬɟɧɧɵ. ȼ ɫɥɭɱɚɟ [ 1 ɩɪɢ - S 2 ɚɪɝɭɦɟɧɬ ɫɢɧɭɫɚ ɜ ɡɧɚɦɟɧɚɬɟɥɟ ɨɛɪɚɳɚɟɬɫɹ ɜ ɧɨɥɶ, ɱɬɨ ɫɨɨɬɜɟɬɫɬɜɭɟɬ ɩɨɹɜɥɟɧɢɸ ɜ ɷɬɨɦ ɧɚɩɪɚɜɥɟɧɢɢ, ɧɟɡɚɜɢɫɢɦɨ ɨɬ ɜɟɥɢɱɢɧɵ d_z, ɨɫɧɨɜɧɨɝɨ ȽɆ. ȿɫɥɢ [!1, ɭɪɨɜɟɧɶ f_c ɜ ɷɬɨɦ ɧɚɩɪɚɜɥɟɧɢɢ ɛɭɞɟɬ ɦɟɧɶɲɟ ɟɞɢɧɢɰɵ, ɚɧɚɥɨɝɢɱɧɨ ɫɥɭɱɚɸ ɧɟɩɪɟɪɵɜɧɨɣ ɫɢɫɬɟɦɵ ɢɡɥɭɱɚ- ɬɟɥɟɣ. ɋ ɭɦɟɧɶɲɟɧɢɟɦ ɭɝɥɚ - ɡɧɚɱɟɧɢɟ ɚɪɝɭɦɟɧɬɚ ɫɢɧɭɫɚ ɜ ɡɧɚɦɟɧɚɬɟɥɟ ɪɚɫɬɟɬ ɢ ɩɪɢɧɢɦɚɟɬ ɦɚɤɫɢɦɚɥɶɧɨɟ ɡɧɚɱɟɧɢɟ ɩɪɢ - S 2, ɩɪɨɩɨɪɰɢɨɧɚɥɶɧɨɟ d_z. ɇɨ ɟɫɥɢ d_z ɦɚɥɨ (ɛɥɢɡɤɨ ɪɚɫɩɨɥɨɠɟɧɧɵɟ ɢɡɥɭɱɚɬɟɥɢ), ɢ ɚɪɝɭɦɟɧɬ ɫɢɧɭɫɚ ɜ ɡɧɚɦɟɧɚɬɟɥɟ ɞɚɥɟɤ ɨɬ ɡɧɚɱɟɧɢɹ, ɪɚɜɧɨɝɨ S, Ⱦɇ ɚɧɬɟɧɧɵ ɩɨ ɯɚɪɚɤɬɟɪɭ ɚɧɚɥɨɝɢɱɧɚ Ⱦɇ ɧɟɩɪɟɪɵɜɧɨɣ ɫɢɫɬɟɦɵ ɢɡɥɭɱɚɬɟɥɟɣ: ɨɧɚ ɧɟ ɢɦɟɟɬ ɦɚɤɫɢɦɭɦɨɜ, ɩɨ ɭɪɨɜɧɸ ɪɚɜɧɵɯ ɟɞɢɧɢɰɟ. ɉɪɢ ɭɜɟɥɢ- ɱɟɧɢɢ d_z ɚɪɝɭɦɟɧɬ ɫɢɧɭɫɚ ɜ ɡɧɚɦɟɧɚɬɟɥɟ ɦɨɠɟɬ ɞɨɫɬɢɝɧɭɬɶ ɜɟɥɢɱɢɧɵ, ɪɚɜɧɨɣ S. Ɍɨɝɞɚ ɜ ɜɵɪɚɠɟɧɢɢ (2) ɜɨɡɧɢɤɚɟɬ ɧɟɨɩɪɟɞɟɥɟɧɧɨɫɬɶ ɬɢɩɚ «ɧɨɥɶ ɧɚ ɧɨɥɶ», ɢ f_c ɫɬɚɧɨɜɢɬɫɹ ɪɚɜɧɵɦ ɟɞɢɧɢɰɟ. ɗɬɨ ɡɧɚɱɢɬ, ɱɬɨ ɜ ɧɚ- ɩɪɚɜɥɟɧɢɢ - S 2 ɩɨɹɜɢɥɫɹ ɩɟɪɜɵɣ ɢɧɬɟɪɮɟɪɟɧɰɢɨɧɧɵɣ ɦɚɤɫɢɦɭɦ. ɉɪɢ ɞɚɥɶɧɟɣɲɟɦ ɭɜɟɥɢɱɟɧɢɢ d_z ɥɭɱ ɨɬɤɥɨɧɹɟɬɫɹ ɨɬ ɨɬɪɢɰɚɬɟɥɶɧɨɝɨ ɧɚɩɪɚɜɥɟɧɢɹ ɨɫɢ z ɢ ɩɟɪɟɦɟɳɚɟɬɫɹ ɜ ɫɬɨɪɨɧɭ ɭɜɟɥɢɱɟɧɢɹ ɭɝɥɚ -. ɉɪɢ ɷɬɨɦ Ⱦɇ ɚɧɬɟɧɧɵ ɩɪɢɧɰɢɩɢɚɥɶɧɨ ɨɬɥɢɱɚɟɬɫɹ ɨɬ Ⱦɇ ɧɟɩɪɟɪɵɜɧɨɣ ɫɢɫɬɟɦɵ ɢɡɥɭɱɚɬɟɥɟɣ.

Ɏɢɡɢɱɟɫɤɚɹ ɢɧɬɟɪɩɪɟɬɚɰɢɹ ɮɨɪɦɢɪɨɜɚɧɢɹ ɩɟɪɜɨɝɨ ɢɧɬɟɪɮɟɪɟɧɰɢɨɧɧɨɝɨ ɝɥɚɜɧɨɝɨ ɦɚɤɫɢɦɭɦɚ (ɂȽɆ 1) ɬɚɤɨɜɚ. Ɉɧ ɩɨɹɜɥɹɟɬɫɹ ɜ ɧɚɩɪɚɜɥɟɧɢɢ - S 2 ɬɨɝɞɚ, ɤɨɝɞɚ ɜ ɬɨɱɤɟ ɧɚɛɥɸɞɟɧɢɹ ɩɨɥɟ, ɢɡɥɭɱɟɧɧɨɟ ɩɨɫɥɟɞɭɸɳɢɦ ɢɡɥɭɱɚɬɟɥɟɦ, ɡɚɩɚɡɞɵɜɚɟɬ ɩɨ ɮɚɡɟ ɧɚ ɭɝɨɥ 2S ɨɬ ɩɨɥɹ ɢɡɥɭɱɟɧɢɹ ɩɪɟɞɵɞɭɳɟɝɨ ɢɡɥɭɱɚɬɟɥɹ (ɧɨɦɟɪɚ ɳɟɥɟɣ ɜɨɡɪɚɫɬɚɸɬ ɜ ɧɚɩɪɚɜɥɟɧɢɢ ɪɚɫɩɪɨɫɬɪɚɧɟɧɢɹ ɛɟɝɭɳɟɣ ɜɨɥɧɵ). Ɂɞɟɫɶ ɢɦɟɟɬ ɦɟɫɬɨ ɧɚɤɚɩɥɢɜɚ- ɧɢɟ ɡɚɩɚɡɞɵɜɚɧɢɹ, ɚ ɧɟ ɟɝɨ ɤɨɦɩɟɧɫɚɰɢɹ, ɩɪɨɢɫɯɨɞɹɳɚɹ ɩɪɢ ɮɨɪɦɢɪɨɜɚɧɢɢ ɨɫɧɨɜɧɨɝɨ ȽɆ. Ʉ ɡɚɩɚɡɞɵɜɚɧɢɸ ɩɨ ɜɨɡɛɭɠɞɟɧɢɸ ɞɨɛɚɜɥɹɟɬɫɹ ɡɚɩɚɡɞɵɜɚɧɢɟ ɡɚ ɫɱɟɬ ɪɚɡɧɨɫɬɢ ɯɨɞɚ ɥɭɱɟɣ. ɑɬɨɛɵ ɫɭɦɦɚ ɡɚɩɚɡɞɵɜɚɧɢɣ ɦɨɝɥɚ ɫɬɚɬɶ ɪɚɜɧɨɣ 2S, ɪɚɫɫɬɨɹɧɢɟ ɦɟɠɞɭ ɫɨɫɟɞɧɢɦɢ ɢɡɥɭɱɚɬɟɥɹɦɢ ɞɨɥɠɧɨ ɛɵɬɶ ɧɟ ɦɟɧɶɲɟ ɧɟɤɨɬɨɪɨɣ ɜɟɥɢɱɢ- ɧɵ. ȿɫɥɢ ɦɢɧɢɦɚɥɶɧɨɟ ɪɚɫɫɬɨɹɧɢɟ, ɧɟɨɛɯɨɞɢɦɨɟ ɞɥɹ ɩɨɹɜɥɟɧɢɹ ɂȽɆ1, ɭɜɟɥɢɱɢɬɶ ɜɞɜɨɟ, ɜ ɧɚɩɪɚɜɥɟɧɢɢ - S 2 ɩɨɹɜɢɬɫɹ ɜɬɨɪɨɣ ɂȽɆ. ɇɚɡɨɜɟɦ ɟɝɨ ɂȽɆ2. ɉɪɢ ɷɬɨɦ ɂȽɆ1 ɩɨɞɧɢɦɟɬɫɹ ɧɚɞ ɨɬɪɢɰɚɬɟɥɶɧɨɣ ɱɚ- ɫɬɶɸ ɨɫɢ ɚɧɬɟɧɧɵ ɧɚ ɭɝɨɥ -_m₁. ɋɥɟɞɨɜɚɬɟɥɶɧɨ, ɱɬɨɛɵ ɩɪɢ ɫɤɚɧɢɪɨɜɚɧɢɢ ɜ ɫɟɤɬɨɪɟ ɞɟɣɫɬɜɢɬɟɥɶɧɵɯ ɭɝɥɨɜ ɛɵɥ ɟɞɢɧɫɬɜɟɧɧɵɣ ɂȽɆ, ɞɨɥɠɧɵ ɛɵɬɶ ɜɵɩɨɥɧɟɧɵ ɬɚɤɢɟ ɭɫɥɨɜɢɹ:

1. Ʉɨɷɮɮɢɰɢɟɧɬ ɡɚɦɟɞɥɟɧɢɹ [ ɞɨɥɠɟɧ ɩɪɟɜɵɲɚɬɶ ɟɞɢɧɢɰɭ, ɱɬɨɛɵ ɨɬɫɭɬɫɬɜɨɜɚɥ ɨɫɧɨɜɧɨɣ ȽɆ.

2. ȼɟɥɢɱɢɧɚ d_z ɞɨɥɠɧɚ ɧɚɯɨɞɢɬɶɫɹ ɜ ɩɪɟɞɟɥɚɯ ɦɟɠɞɭ ɦɢɧɢɦɚɥɶɧɵɦ ɢ ɦɚɤɫɢɦɚɥɶɧɵɦ ɡɧɚɱɟɧɢɹɦɢ. Ɇɢ- ɧɢɦɚɥɶɧɨɟ ɡɧɚɱɟɧɢɟ ɹɜɥɹɟɬɫɹ ɭɫɥɨɜɢɟɦ ɫɭɳɟɫɬɜɨɜɚɧɢɹ ɂȽɆ ɤɚɤ ɬɚɤɨɜɨɝɨ. Ɇɚɤɫɢɦɚɥɶɧɨɟ ɡɚɩɪɟɳɚɟɬ ɩɨ- ɹɜɥɟɧɢɟ ɜɬɨɪɨɝɨ ɂȽɆ.

ɇɚɣɞɟɦ ɤɨɥɢɱɟɫɬɜɟɧɧɵɟ ɫɨɨɬɧɨɲɟɧɢɹ ɦɟɠɞɭ ɤɨɷɮɮɢɰɢɟɧɬɨɦ ɡɚɦɟɞɥɟɧɢɹ [, ɪɚɫɫɬɨɹɧɢɟɦ d_z ɢ ɞɥɢ- ɧɨɣ ɜɨɥɧɵ, ɤɨɬɨɪɵɟ ɞɨɥɠɧɵ ɜɵɩɨɥɧɹɬɶɫɹ ɩɪɢ ɷɬɨɦ. Ʉɨɦɩɥɟɤɫɧɵɣ ɦɧɨɠɢɬɟɥɶ (2) ɢɦɟɟɬ ɦɚɤɫɢɦɭɦɵ, ɪɚɜ- ɧɵɟ ɟɞɢɧɢɰɟ, ɫɬɨɥɶɤɨ ɪɚɡ, ɫɤɨɥɶɤɨ ɪɚɡ ɜɵɪɚɠɟɧɢɟ

kdz ²

[^sin-

ɩɪɢ ɢɡɦɟɧɟɧɢɢ ɭɝɥɚ - ɜ ɩɪɟɞɟɥɚɯ

2 2

S - S

d d ɩɪɢɧɢɦɚɟɬ ɡɧɚɱɟɧɢɟ, ɪɚɜɧɨɟ ɰɟɥɨɦɭ ɱɢɫɥɭ S. ɍɫɥɨɜɢɸ

kdz ²

[ ¹

mS (m 1, 2,...) ɫɨɨɬɜɟɬɫɬɜɭɟɬ ɩɨɹɜɥɟɧɢɟ ɜ ɧɚɩɪɚɜɥɟɧɢɢ - S 2 ɂȽɆ (ɭɠɟ ɧɟ ɨɫɧɨɜɧɵɯ) Ⱦɇ ɩɟɪɜɨɝɨ, ɜɬɨɪɨɝɨ ɢ ɬ.ɞ. ɩɨ- ɪɹɞɤɚ. ɇɚɡɨɜɟɦ ɢɯ ȽɆ ɜɵɫɲɟɝɨ ɩɨɪɹɞɤɚ. Ⱦɥɹ ɬɨɝɨ ɱɬɨɛɵ ɩɨɹɜɢɥɫɹ ɯɨɬɹ ɛɵ ɨɞɢɧ ɢɡ ɧɢɯ, ɚɪɝɭɦɟɧɬ ɫɢɧɭɫɚ ɜ ɡɧɚɦɟɧɚɬɟɥɟ ɞɨɥɠɟɧ ɫɬɚɬɶ ɪɚɜɧɵɦ S. ɑɬɨɛɵ ɧɟ ɩɨɹɜɢɥɫɹ ɜɬɨɪɨɣ ɦɚɤɫɢɦɭɦ ɜɵɫɲɟɝɨ ɩɨɪɹɞɤɚ, ɚɪɝɭɦɟɧɬ ɫɢɧɭɫɚ ɧɟ ɞɨɥɠɟɧ ɞɨɫɬɢɝɚɬɶ ɡɧɚɱɟɧɢɹ 2S. Ⱦɥɹ ɬɨɝɨ ɱɬɨɛɵ ɜ ɫɟɤɬɨɪɟ ɞɟɣɫɬɜɢɬɟɥɶɧɵɯ ɭɝɥɨɜ ɫɭɳɟɫɬɜɨɜɚɥ ɂȽɆ ɢ ɨɧ ɛɵɥ ɟɞɢɧɫɬɜɟɧɧɵɦ, ɞɨɥɠɧɨ ɜɵɩɨɥɧɹɬɶɫɹ ɭɫɥɨɜɢɟ

kdz ²

¹

²

Sd [ S. (3)

Ɉɬɫɸɞɚ ɧɟɩɨɫɪɟɞɫɬɜɟɧɧɨ ɫɥɟɞɭɟɬ

2 1O d^z 1 O

[^d [

. (4)

ȿɫɥɢ

z 1

d O

! [

, ɬɨ ɭɝɨɥ -, ɩɪɢ ɤɨɬɨɪɨɦ ɜɵɩɨɥɧɹɟɬɫɹ ɭɫɥɨɜɢɟ

kdz ²

[^sin-

S, (5)

ɭɜɟɥɢɱɢɜɚɟɬɫɹ. ɇɚɡɨɜɟɦ ɷɬɨɬ ɭɝɨɥ -_1m. ɂɡ (5) ɫɥɟɞɭɟɬ

1

1 1

sin _m

g dz dz

- O [ O

O

§ · § ·

¨ ¸ ¨© ¸¹

(6)

10

ɗɬɨ ɜɵɪɚɠɟɧɢɟ ɨɤɚɡɵɜɚɟɬɫɹ ɫɩɪɚɜɟɞɥɢɜɵɦ ɢ ɩɪɢ [d1 [1, 4]. Ⱥɧɚɥɨɝɢɱɧɵɣ ɜɢɞ ɢɦɟɟɬ ɢɡɜɟɫɬɧɚɹ ɮɨɪɦɭ- ɥɚ ɞɥɹ ɨɩɪɟɞɟɥɟɧɢɹ ɧɚɩɪɚɜɥɟɧɢɹ ɂȽɆ1 ɜ ɫɥɭɱɚɟ ɩɟɪɟɦɟɧɧɨ-ɮɚɡɧɨɣ ɚɧɬɟɧɧɵ, ɤɨɝɞɚ ɫɨɫɟɞɧɢɟ ɩɪɨɞɨɥɶɧɵɟ ɳɟɥɢ ɪɚɫɩɨɥɨɠɟɧɵ ɩɨ ɪɚɡɧɵɟ ɫɬɨɪɨɧɵ ɨɬ ɨɫɢ ɲɢɪɨɤɨɣ ɫɬɟɧɤɢ ɜɨɥɧɨɜɨɞɚ

1 1

sin- O _g 2d_z O

§ ·

¨ ¸

Ʉɨɝɞɚ ɪɚɫɫɬɨɹɧɢɟ ɦɟɠɞɭ ɳɟɥɹɦɢ d_z ɜ ɩɪɹɦɨɮɚɡɧɨɣ ɚɧɬɟɧɧɟ ɪɚɜɧɨ ɞɥɢɧɟ ɜɨɥɧɵ ɜ ɜɨɥɧɨɜɨɞɟ (d_z O_g), ɚ ɜ ɩɟɪɟɦɟɧɧɨ-ɮɚɡɧɨɣ — ɟɟ ɩɨɥɨɜɢɧɟ (d_z O_g 2), ɢɡ (6) ɢ (7) ɫɥɟɞɭɟɬ, ɱɬɨ ɥɭɱ ɫɜɟɬɢɬ ɩɨ ɧɨɪɦɚɥɢ ɤ ɚɧɬɟɧɧɟ ɢ ɨɬɤɥɨɧɹɟɬɫɹ ɨɬ ɧɟɟ, ɟɫɥɢ ɭɤɚɡɚɧɧɵɟ ɭɫɥɨɜɢɹ ɧɟ ɜɵɩɨɥɧɹɸɬɫɹ. ȼ ɷɬɨɦ, ɤɚɤ ɞɚɜɧɨ ɢɡɜɟɫɬɧɨ, ɫɨɫɬɨɢɬ ɫɭɬɶ ɱɚɫɬɨɬɧɨɝɨ ɫɤɚɧɢɪɨɜɚɧɢɹ. Ȼɨɥɶɲɨɣ ɢɧɬɟɪɟɫ ɩɪɟɞɫɬɚɜɥɹɸɬ ɫɨɛɨɣ ɡɚɤɨɧɨɦɟɪɧɨɫɬɢ ɷɬɨɝɨ ɩɪɨɰɟɫɫɚ, ɤɨɝɞɚ ɤɨɷɮɮɢɰɢɟɧɬ ɡɚɦɟɞɥɟɧɢɹ ɛɨɥɶɲɟ ɟɞɢɧɢɰɵ.

Ʉɨɥɢɱɟɫɬɜɟɧɧɵɟ ɫɨɨɬɧɨɲɟɧɢɹ, ɨɩɪɟɞɟɥɹɸɳɢɟ ɩɨɥɨɠɟɧɢɟ ɂȽɆ1 ɧɚ ɮɢɤɫɢɪɨɜɚɧɧɨɣ ɱɚɫɬɨɬɟ

Ɋɚɫɫɦɨɬɪɢɦ ɛɨɥɟɟ ɩɨɞɪɨɛɧɨ, ɤɚɤ ɢɡɦɟɧɹɟɬɫɹ ɩɨɥɨɠɟɧɢɟ ɂȽɆ1 ɫ ɢɡɦɟɧɟɧɢɟɦ d_z ɢ ɤɨɷɮɮɢɰɢɟɧɬɚ ɡɚ- ɦɟɞɥɟɧɢɹ [. ɋɧɚɱɚɥɚ ɩɪɨɚɧɚɥɢɡɢɪɭɟɦ ɷɬɨɬ ɜɨɩɪɨɫ ɜ ɛɨɥɟɟ ɩɪɨɫɬɨɦ ɫɥɭɱɚɟ ɩɪɹɦɨɮɚɡɧɨɣ ɚɧɬɟɧɧɵ. ȿɫɥɢ d_z ɨɩɪɟɞɟɥɹɟɬɫɹ ɝɪɚɧɢɱɧɵɦ ɡɧɚɱɟɧɢɟɦ ɥɟɜɨɣ ɱɚɫɬɢ ɧɟɪɚɜɟɧɫɬɜɚ (4), ɬ.ɟ.

z 1

d O

[

, ɢɡ (6) ɫɥɟɞɭɟɬ, ɱɬɨ ɩɟɪɜɵɣ ɝɥɚɜɧɵɣ ɦɚɤɫɢɦɭɦ (ȽɆ1) ɧɚɩɪɚɜɥɟɧ ɜ ɫɬɨɪɨɧɭ

2

- S. ɉɭɫɬɶ d_z ɭɜɟɥɢɱɢɬɫɹ ɧɚ ɧɟɤɨɬɨɪɭɸ ɞɨɥɸ ɷɬɨɝɨ ɪɚɫɫɬɨɹɧɢɹ ɢ ɩɪɢ ɥɸɛɨɦ [ ɛɭɞɟɬ ɪɚɜɧɵɦ

¹

z 1

d O D

[

, (8)

ɝɞɟ 0d dD 1.

ɉɪɚɜɨɣ ɝɪɚɧɢɰɟ ɧɟɪɚɜɟɧɫɬɜɚ (4) ɫɨɨɬɜɟɬɫɬɜɭɟɬD 1. ɉɨɞɫɬɚɜɥɹɹ ɜɵɪɚɠɟɧɢɟ (8) ɜ ɮɨɪɦɭɥɭ (6), ɩɨɥɭɱɚɟɦ

1

sin 1

m 1 - D[

D

. (9)

Ⱦɢɮɮɟɪɟɧɰɢɪɭɹ (9) ɩɨɨɱɟɪɟɞɧɨ ɩɨ [ ɢ D, ɭɛɟɠɞɚɟɦɫɹ ɜ ɬɨɦ, ɱɬɨ ɜ ɨɛɨɢɯ ɫɥɭɱɚɹɯ ɩɪɨɢɡɜɨɞɧɵɟ d ^1m d

- ɢ d ^1m [

d -

D ɩɨɥɨɠɢɬɟɥɶɧɵɟ, ɱɬɨ ɫɨɨɬɜɟɬɫɬɜɭɟɬ ɭɜɟɥɢɱɟɧɢɸ sin-₁_m ɫ ɪɨɫɬɨɦ [ ɢ D. ɉɪɢ D 0 ɩɨɥɭɱɚɟɦ

1^m 2

- S. ȿɫɥɢ ɪɚɫɫɬɨɹɧɢɟ ɭɞɜɨɢɥɨɫɶ (D 1), ȽɆ1 ɜ ɫɨɨɬɜɟɬɫɬɜɢɢ ɫ (9) ɫɜɟɬɢɬ ɜ ɧɚɩɪɚɜɥɟɧɢɢ, ɨɩɪɟɞɟɥɹ- ɟɦɨɦ ɜɵɪɚɠɟɧɢɟɦ

1

sin 1

m 2

- [ . (10)

ɉɪɢ ɷɬɨɦ ɜ ɧɚɩɪɚɜɥɟɧɢɢ, ɨɩɪɟɞɟɥɹɟɦɨɦ ₂

m 2

- S , ɩɨɹɜɥɹɟɬɫɹ ɜɬɨɪɨɣ ɝɥɚɜɧɵɣ ɦɚɤɫɢɦɭɦ (ɤɚɤ ɩɪɚɜɢɥɨ, ɷɬɨ ɜɟɫɶɦɚ ɧɟɠɟɥɚɬɟɥɶɧɨ). ȽɆ1 ɧɚɩɪɚɜɥɟɧ ɩɨ ɧɨɪɦɚɥɢ ɤ ɪɟɲɟɬɤɟ, ɤɨɝɞɚ D 1[. ɂɡ (8) ɫɥɟɞɭɟɬ, ɱɬɨ ɩɪɢ ɷɬɨɦ d_z O_g (ɪɚɫɫɬɨɹɧɢɟ ɦɟɠɞɭ ɳɟɥɹɦɢ ɪɚɜɧɨ ɞɥɢɧɟ ɜɨɥɧɵ ɜ ɜɨɥɧɨɜɨɞɟ O_g — ɢɡɜɟɫɬɧɵɣ ɮɚɤɬ).

Ɇɢɧɢɦɚɥɶɧɵɣ ɭɪɨɜɟɧɶ ɛɨɤɨɜɵɯ ɥɟɩɟɫɬɤɨɜ ɧɚɛɥɸɞɚɟɬɫɹ ɜ ɧɚɩɪɚɜɥɟɧɢɢ ɭɝɥɚ -_1min, ɨɩɪɟɞɟɥɹɟɦɨɝɨ ɜɵ- ɪɚɠɟɧɢɟɦ

sin 1min

2d_z

- [ O , (11)

ɩɨɥɭɱɟɧɧɵɦ ɢɡ ɭɫɥɨɜɢɹ

kdz ²

[^sin-

S ². (12)

ɉɪɢ ɩɨɞɫɬɚɧɨɜɤɟ ɜ (11) ɪɚɫɫɬɨɹɧɢɹ d_z ɢɡ (8) ɧɚɯɨɞɢɦ

1min

sin 1 2

2 1

[ [D

- D . (13)

ɂɡ (13) ɫɥɟɞɭɟɬ, ɱɬɨ ɫ ɭɜɟɥɢɱɟɧɢɟɦ ɤɨɷɮɮɢɰɢɟɧɬɚ ɡɚɦɟɞɥɟɧɢɹ [ ɧɚɩɪɚɜɥɟɧɢɟ ɦɢɧɢɦɭɦɚ ɛɨɤɨɜɵɯ ɥɟ- ɩɟɫɬɤɨɜ ɫɦɟɳɚɟɬɫɹ ɜ ɫɬɨɪɨɧɭ ɩɨɥɨɠɢɬɟɥɶɧɵɯ ɭɝɥɨɜ -. Ɇɨɠɧɨ ɧɚɣɬɢ ɡɧɚɱɟɧɢɟ [, ɩɪɢ ɤɨɬɨɪɨɦ -_1min S 2 ɢɥɢ ɦɢɧɢɦɚɥɶɧɵɣ ɭɪɨɜɟɧɶ ɜɨɜɫɟ ɨɬɫɭɬɫɬɜɭɟɬ (ɭɯɨɞɢɬ ɢɡ ɨɛɥɚɫɬɢ ɞɟɣɫɬɜɢɬɟɥɶɧɵɯ ɭɝɥɨɜ). Ȼɨɥɟɟ ɬɨɝɨ, ɦɨɠɟɬ ɢɫɱɟɡɧɭɬɶ ɢ ȽɆ1. ɉɨɤɚɠɟɦ ɷɬɨ.

(7)

ȼ ɫɥɭɱɚɟ D 0 ɢɦɟɟɦ _1min 1

sin 2

- [ , ɚ ȽɆ1 ɧɚɩɪɚɜɥɟɧ ɜ ɫɬɨɪɨɧɭ -₁_m S 2. ɉɪɢ ɭɞɜɨɟɧɧɨɦ ɪɚɫ-

ɫɬɨɹɧɢɢ (D 1) ɩɨɥɭɱɚɟɦ _1min 3 1

sin 4

- [ , ɬɨɝɞɚ ɤɚɤ ₁ 1

sin ^m 2

- [ (10). ȼ ɝɢɩɨɬɟɬɢɱɟɫɤɨɦ ɫɥɭɱɚɟ [ 3 ɭɝɨɥ -₁_m S 2, ɚ -₂_m S 2. ɉɪɢ ɷɬɨɦ ɨɤɚɡɵɜɚɟɬɫɹ, ɱɬɨ sin-_1min!1 (ɛɨɤɨɜɵɟ ɥɟɩɟɫɬɤɢ ɫɩɪɚɜɚ ɨɬ ɩɟɪɜɨ- ɝɨ ɝɥɚɜɧɨɝɨ ɦɚɤɫɢɦɭɦɚ ɭɲɥɢ ɢɡ ɨɛɥɚɫɬɢ ɞɟɣɫɬɜɢɬɟɥɶɧɵɯ ɭɝɥɨɜ). ȼ ɞɚɥɶɧɟɣɲɟɦ ɛɭɞɟɬ ɩɨɤɚɡɚɧɨ, ɱɬɨ ɬɚɤɚɹ ɤɚɪɬɢɧɚ ɧɟɪɟɚɥɶɧɚ.

Ɂɚɤɨɧɨɦɟɪɧɨɫɬɢ ɱɚɫɬɨɬɧɨɝɨ ɫɤɚɧɢɪɨɜɚɧɢɹ ɩɪɢ ɮɢɤɫɢɪɨɜɚɧɧɨɦ d_z

ȼ ɪɚɫɫɦɨɬɪɟɧɧɵɯ ɪɚɧɟɟ ɫɥɭɱɚɹɯ ɢɡɭɱɟɧɵ ɡɚɤɨɧɨɦɟɪɧɨɫɬɢ ɢɡɦɟɧɟɧɢɹ ɢɧɬɟɪɮɟɪɟɧɰɢɨɧɧɨɣ ɤɚɪɬɢɧɵ ɩɪɢ ɭɫɥɨɜɢɢ, ɱɬɨ ɱɚɫɬɨɬɚ ɢ ɫɨɨɬɜɟɬɫɬɜɭɸɳɢɣ ɤɨɷɮɮɢɰɢɟɧɬ ɡɚɦɟɞɥɟɧɢɹ [ ɡɚɮɢɤɫɢɪɨɜɚɧɵ, ɚ ɢɡɦɟɧɹɟɬɫɹ ɬɨɥɶɤɨ ɪɚɫɫɬɨɹɧɢɟ d_z ɦɟɠɞɭ ɫɨɫɟɞɧɢɦɢ ɳɟɥɹɦɢ. ɇɚ ɩɪɚɤɬɢɤɟ ɜɚɠɧɨ ɜɵɹɫɧɢɬɶ, ɱɬɨ ɩɪɨɢɫɯɨɞɢɬ ɫ Ⱦɇ ȼɓȺ ɩɪɢ ɢɡɦɟɧɟɧɢɢ ɱɚɫɬɨɬɵ ɩɪɢ ɮɢɤɫɢɪɨɜɚɧɧɨɦ ɡɧɚɱɟɧɢɢ d_z ɜ ɢɡɝɨɬɨɜɥɟɧɧɨɣ ɚɧɬɟɧɧɟ ɢ ɤɚɤɢɦ ɧɚɞɨ ɜɵɛɪɚɬɶ ɷɬɨ

dz ɞɥɹ ɷɮɮɟɤɬɢɜɧɨɝɨ ɫɤɚɧɢɪɨɜɚɧɢɹ ɥɭɱɨɦ ɩɪɢ ɦɢɧɢɦɚɥɶɧɨɦ ɢɡɦɟɧɟɧɢɢ ɱɚɫɬɨɬɵ.

Ɋɚɫɫɦɨɬɪɢɦ ɮɢɡɢɱɟɫɤɢɟ ɩɪɨɰɟɫɫɵ ɜ ɷɬɨɦ ɫɥɭɱɚɟ. Ɂɚɮɢɤɫɢɪɭɟɦ ɪɚɫɫɬɨɹɧɢɟ d_z (ɟɝɨ ɟɳɟ ɧɚɞɨ ɜɵɛɪɚɬɶ), ɩɨɥɚɝɚɟɦ, ɱɬɨ ɞɥɹ ɜɵɛɪɚɧɧɨɣ ɡɚɦɟɞɥɹɸɳɟɣ ɫɢɫɬɟɦɵ ɡɚɜɢɫɢɦɨɫɬɶ [ O ɢɡɜɟɫɬɧɚ. ɉɭɫɬɶ ɧɚ ɮɢɤɫɢɪɨɜɚɧɧɨɣ ɱɚɫɬɨɬɟ f₁ (ɫɨɨɬɜɟɬɫɬɜɟɧɧɨ O₁) ɧɚɞɨ, ɱɬɨɛɵ ȽɆ1 ɛɵɥ ɧɚɩɪɚɜɥɟɧ ɩɨɞ ɭɝɥɨɦ -_1m. ɉɭɫɬɶ ɤɨɷɮɮɢɰɢɟɧɬ ɡɚ- ɦɟɞɥɟɧɢɹ ɧɚ ɷɬɨɣ ɱɚɫɬɨɬɟ ɪɚɜɟɧ [₁. ɇɚɞɨ ɧɚɣɬɢ ɪɚɫɫɬɨɹɧɢɟ d_z (8), ɩɪɢ ɤɨɬɨɪɨɦ ɨɛɟɫɩɟɱɢɜɚɟɬɫɹ ɡɚɞɚɧɧɨɟ ɩɨɥɨɠɟɧɢɟ ɥɭɱɚ. ɂɡ ɜɵɪɚɠɟɧɢɹ (9) ɩɨɥɭɱɚɟɦ

1 1

1 sin sin

m m

D -

[ -

. (14)

ɉɪɢ ɷɬɨɦ ɪɚɫɫɬɨɹɧɢɟ d_z ɩɪɢɧɢɦɚɟɬ ɡɧɚɱɟɧɢɟ

1

1 1

dz O D

[

, (15)

ɝɞɟ D₁ ɨɩɪɟɞɟɥɹɟɬɫɹ ɮɨɪɦɭɥɨɣ (14).

ɉɭɫɬɶ ɞɚɥɟɟ ɱɚɫɬɨɬɚ f (ɫɨɨɬɜɟɬɫɬɜɟɧɧɨ ɢ ɪɚɛɨɱɚɹ ɞɥɢɧɚ ɜɨɥɧɵ O) ɢɡɦɟɧɹɟɬɫɹ, ɚ ɫ ɧɟɣ ɢ ɤɨɷɮɮɢɰɢɟɧɬ ɡɚɦɟɞɥɟɧɢɹ [. ɉɨ ɨɬɧɨɲɟɧɢɸ ɤ ɧɨɜɵɦ ɡɧɚɱɟɧɢɹɦ [ ɢ O ɩɪɟɠɧɟɟ (ɡɚɮɢɤɫɢɪɨɜɚɧɧɨɟ) ɡɧɚɱɟɧɢɟ d_z ɜɵɪɚ- ɡɢɬɫɹ ɫɥɟɞɭɸɳɢɦ ɨɛɪɚɡɨɦ

¹

z 1

d O D_O

[

. (16)

Ɂɞɟɫɶ ɢɧɞɟɤɫ D_O ɫɨɨɬɜɟɬɫɬɜɭɟɬ ɞɥɢɧɟ ɜɨɥɧɵ O. ɉɪɢɪɚɜɧɢɜɚɹ ɞɪɭɝ ɞɪɭɝɭ d_z ɢɡ (15) ɢ (16), ɧɚɯɨɞɢɦ

1

1 1

1 1 1

O 1

[

D O D

O [

. (17)

ɗɬɨ ɡɧɚɱɟɧɢɟ D_O ɢɫɩɨɥɶɡɭɟɦ ɞɥɹ ɨɩɪɟɞɟɥɟɧɢɹ ɩɨ ɮɨɪɦɭɥɟ (9) ɧɨɜɨɝɨ ɧɚɩɪɚɜɥɟɧɢɹ ȽɆ1 ɧɚ ɱɚɫɬɨɬɟ f (ɩɪɢ ɞɥɢɧɟ ɜɨɥɧɵ O):

1

sin 1

m 1

O O

- D [ D

. (18)

ɇɚɱɚɥɶɧɭɸ ɱɚɫɬɨɬɭ ɧɚɞɨ ɜɵɛɢɪɚɬɶ ɧɟ ɨɛɹɡɚɬɟɥɶɧɨ ɬɚɦ, ɝɞɟ [ 1, ɚ ɪɚɫɫɬɨɹɧɢɟ d_z ɧɟ ɬɚɤɨɟ, ɱɬɨɛɵ ɥɭɱ ɛɵɥ ɧɚɩɪɚɜɥɟɧ ɜɞɨɥɶ ɨɬɪɢɰɚɬɟɥɶɧɨɣ ɨɫɢ z, ɚ ɫɜɟɬɢɥ ɜ ɧɭɠɧɨɦ ɧɚɩɪɚɜɥɟɧɢɢ. ɋɤɨɪɟɟ ɜɫɟɝɨ, ɧɚɞɨ ɧɚɱɢɧɚɬɶ ɜɫɟ-ɬɚɤɢ ɫ ɞɥɢɧɧɨɜɨɥɧɨɜɨɝɨ ɭɱɚɫɬɤɚ ɞɢɫɩɟɪɫɢɨɧɧɨɣ ɯɚɪɚɤɬɟɪɢɫɬɢɤɢ, ɧɚ ɤɨɬɨɪɨɦ ɤɪɭɬɢɡɧɚ ɭɠɟ ɞɨɫɬɚɬɨɱɧɨ ɛɨɥɶɲɚɹ. Ɋɚɫɫɬɨɹɧɢɟ d_z ɞɨɥɠɧɨ ɨɛɟɫɩɟɱɢɬɶ ɫɤɚɧɢɪɨɜɚɧɢɟ ɜ ɠɟɥɚɟɦɨɦ ɫɟɤɬɨɪɟ ɭɝɥɨɜ ɛɟɡ ɩɨɹɜɥɟɧɢɹ ɂȽɆ2.

Ʉɨɷɮɮɢɰɢɟɧɬ ɡɚɦɟɞɥɟɧɢɹ ɞɨɥɠɟɧ ɛɵɬɶ ɛɨɥɶɲɟ ɡɧɚɱɟɧɢɹ [ 1 ɜ ɬɚɤɨɣ ɫɬɟɩɟɧɢ, ɱɬɨɛɵ ɧɟ ɩɨɹɜɢɥɫɹ ɨɫɧɨɜɧɨɣ ȽɆ. ɇɟɨɛɯɨɞɢɦɨ ɧɚɣɬɢ ɞɨɩɭɫɬɢɦɵɟ ɡɧɚɱɟɧɢɹ [ ɢ d_z, ɤɨɬɨɪɵɟ ɫɥɭɠɚɬ ɷɬɨɣ ɰɟɥɢ.

ȼɬɨɪɨɣ ɝɥɚɜɧɵɣ ɦɚɤɫɢɦɭɦ Ⱦɇ ɚɧɬɟɧɧɵ, ɫɨɫɬɨɹɳɟɣ ɢɡ N ɳɟɥɟɣ, ɩɨɹɜɥɹɟɬɫɹ ɜ ɧɚɩɪɚɜɥɟɧɢɢ - S 2, ɤɨɝɞɚ dz ²O

¹[

. ɉɪɢ ɜɵɛɪɚɧɧɨɦ d_z ɤɨɷɮɮɢɰɢɟɧɬ ɡɚɦɟɞɥɟɧɢɹ [ ɞɨɥɠɟɧ ɛɵɬɶ ɬɚɤɢɦ, ɱɬɨɛɵ ɜɵɩɨɥɧɹ- ɥɚɫɶ ɩɪɚɜɚɹ ɱɚɫɬɶ ɧɟɪɚɜɟɧɫɬɜɚ (4). ɋɥɟɞɨɜɚɬɟɥɶɧɨ, ɧɚ ɤɨɷɮɮɢɰɢɟɧɬ [ ɧɚɤɥɚɞɵɜɚɟɬɫɹ ɨɝɪɚɧɢɱɟɧɢɟ ɫɜɟɪɯɭ. Ɉɧ ɞɨɥɠɟɧ ɛɵɬɶ ɬɚɤɢɦ, ɱɬɨɛɵ ɡɧɚɱɟɧɢɟ ²^O

¹^[

ɩɪɟɜɵɲɚɥɨ d_z. ɉɨɞɱɟɪɤɧɟɦ, ɱɬɨ ɩɪɢ ɫɤɚɧɢɪɨɜɚɧɢɢ ɷɬɨ ɨɛɟɫɩɟɱɢɜɚɟɬɫɹ ɧɟ ɩɭɬɟɦ ɭɦɟɧɶɲɟɧɢɹ, ɚ ɡɚ ɫɱɟɬ ɬɨɝɨ, ɱɬɨ ɩɪɚɜɚɹ ɱɚɫɬɶ ɨɫɬɚɟɬɫɹ ɛɨɥɶɲɟ

d

_z.

(8)

12

ɇɚɣɞɟɦ ɩɪɟɞɟɥɶɧɵɟ ɡɧɚɱɟɧɢɹ d_z ɢ

[

, ɩɪɢ ɤɨɬɨɪɵɯ ɜ ɨɬɪɢɰɚɬɟɥɶɧɨɦ ɧɚɩɪɚɜɥɟɧɢɢ ɨɫɢ z ɧɟ ɩɨɹɜ- ɥɹɟɬɫɹ ɂȽɆ2, ɚ ɜ ɩɨɥɨɠɢɬɟɥɶɧɨɦ ɧɟ ɩɨɹɜɥɹɟɬɫɹ ɨɫɧɨɜɧɨɣ ȽɆ. ɇɟɠɟɥɚɬɟɥɶɧɵɣ ɂȽɆ2 ɩɨɹɜɥɹɟɬɫɹ, ɤɨɝɞɚ

kdz ²

[^sin-

²S. Ɍɨɝɞɚ ɡɧɚɦɟɧɚɬɟɥɶ ɜ ɮɨɪɦɭɥɟ (2) ɜɬɨɪɨɣ ɪɚɡ ɨɛɪɚɳɚɟɬɫɹ ɜ ɧɨɥɶ. ɑɢɫɥɢɬɟɥɶ ɩɪɢ ɷɬɨɦ ɬɨɠɟ ɪɚɜɟɧ ɧɭɥɸ, ɩɨɫɤɨɥɶɤɭ ɚɪɝɭɦɟɧɬ ɫɢɧɭɫɚ ɜ ɧɟɦ ɪɚɜɟɧ N2S. ɇɟɨɩɪɟɞɟɥɟɧɧɨɫɬɶ ɜɢɞɚ « 0 0 » ɩɨɪɨɠɞɚɟɬ ɂȽɆ2. ȼ ɧɚɩɪɚɜɥɟɧɢɹɯ -_Q, ɜ ɤɨɬɨɪɵɯ

z ²

^sin

²

N kd [ -_Q N S QS , (19)

ɧɚɛɥɸɞɚɸɬɫɹ ɧɭɥɢ Ⱦɇ (ɦɟɠɞɭ ɧɢɦɢ ɛɨɤɨɜɵɟ ɥɟɩɟɫɬɤɢ). ɉɨɥɨɠɢɬɟɥɶɧɵɦ Q ɫɨɨɬɜɟɬɫɬɜɭɸɬ ɧɭɥɢ ɫɩɪɚɜɚ ɨɬ ɂȽɆ2, ɨɬɪɢɰɚɬɟɥɶɧɵɦ — ɫɥɟɜɚ ɨɬ ɧɟɝɨ (ɭɝɨɥ - ɪɚɫɬɟɬ ɫɥɟɜɚ ɧɚɩɪɚɜɨ). Ɇɨɠɧɨ ɩɨɬɪɟɛɨɜɚɬɶ, ɱɬɨɛɵ Q-ɣ ɧɨɥɶ ɫɩɪɚɜɚ ɨɬ ɜɬɨɪɨɝɨ ȽɆ ɩɪɢɯɨɞɢɥɫɹ ɧɚ -_Q S 2. Ɍɨɝɞɚ ɂȽɆ2 ɛɭɞɟɬ ɨɬɫɭɬɫɬɜɨɜɚɬɶ ɜ ɨɛɥɚɫɬɢ ɞɟɣɫɬɜɢ- ɬɟɥɶɧɵɯ ɭɝɥɨɜ. ɂɡ (19) ɦɨɠɧɨ ɧɚɣɬɢ d_z, ɩɪɢ ɤɨɬɨɪɨɦ ɷɬɨ ɩɪɨɢɫɯɨɞɢɬ

2 1

1 2

dz

N

O Q

[

§ ·

¨ ¸

© ¹

. (20)

ȼɫɟ ɥɨɝɢɱɧɨ: ɱɟɦ ɛɨɥɶɲɟ N (ɞɥɢɧɧɟɟ ɚɧɬɟɧɧɚ), ɬɟɦ ɭɠɟ ɥɟɩɟɫɬɤɢ, ɜɤɥɸɱɚɹ ɝɥɚɜɧɵɣ, ɬɟɦ ɫɥɚɛɟɟ ɞɨɥɠɧɨ ɭɦɟɧɶɲɚɬɶɫɹ ɪɚɫɫɬɨɹɧɢɟ ɦɟɠɞɭ ɢɡɥɭɱɚɬɟɥɹɦɢ ɩɨ ɫɪɚɜɧɟɧɢɸ ɫ dz ²O

¹[

, ɱɬɨɛɵ ɜ ɨɛɥɚɫɬɢ ɜɢɞɢɦɵɯ ɭɝɥɨɜ ɧɟ ɩɨɹɜɥɹɥɫɹ ɂȽɆ2. ɇɚɩɪɚɜɥɟɧɢɟ ɂȽɆ1 ɩɪɢ ɬɚɤɨɦ d_z (ɭɝɨɥ -_1m) ɨɩɪɟɞɟɥɹɟɬɫɹ ɮɨɪɦɭɥɨɣ

1

sin 1

2 1 2

m

dz N

O [

- [ [

Q

§ ·

¨ ¸

ȿɫɥɢ ɜɵɛɪɚɧɧɨɟ ɡɧɚɱɟɧɢɟ d_z ɨɩɪɟɞɟɥɟɧɨ ɮɨɪɦɭɥɨɣ (20), ɤɨɷɮɮɢɰɢɟɧɬ ɡɚɦɟɞɥɟɧɢɹ [, ɡɚɩɪɟɳɚɸɳɢɣ ɩɨɹɜɥɟɧɢɟ ɂȽɆ2, ɧɟ ɞɨɥɠɟɧ ɩɪɟɜɵɲɚɬɶ ɜɟɥɢɱɢɧɭ

²

1

z

N

d N

O Q

[ . (22)

ȼɵɹɫɧɢɦ ɬɟɩɟɪɶ, ɧɚɫɤɨɥɶɤɨ ɜɟɥɢɱɢɧɚ [ ɞɨɥɠɧɚ ɩɪɟɜɵɲɚɬɶ ɟɞɢɧɢɰɭ, ɞɥɹ ɬɨɝɨ ɱɬɨɛɵ ɧɟ ɫɮɨɪ- ɦɢɪɨɜɚɥɫɹ ɨɫɧɨɜɧɨɣ ȽɆ. ɉɨɬɪɟɛɭɟɦ, ɱɬɨɛɵ ɜ ɧɚɩɪɚɜɥɟɧɢɢ - S 2 ɧɚɯɨɞɢɥɫɹ ɧɟ ɨɫɧɨɜɧɨɣ ȽɆ, ɚ ɩɟɪ- ɜɵɣ ɧɨɥɶ Ⱦɇ ɜɛɥɢɡɢ ɧɟɝɨ. ɉɨɫɤɨɥɶɤɭ ɩɪɢ [!1 ɚɪɝɭɦɟɧɬ ɫɢɧɭɫɚ ɜ ɱɢɫɥɢɬɟɥɟ ɧɟ ɦɨɠɟɬ ɨɛɪɚɬɢɬɶɫɹ ɜ ɧɨɥɶ, ɧɚɣɞɟɦ ɡɧɚɱɟɧɢɟ [, ɩɪɢ ɤɨɬɨɪɨɦ ɨɧ ɫɬɚɥ ɛɵ ɪɚɜɧɵɦ QS (Q 1, 2,3...). ɂɡ ɭɫɥɨɜɢɹ N kd

z ²

[^sin-

S ɩɨɥɭɱɚɟɦ ɫ ɭɱɟɬɨɦ ɬɨɝɨ, ɱɬɨ sin- sin S 2 1 ɫɥɟɞɭɸɳɟɟ ɜɵɪɚɠɟɧɢɟ ɞɥɹ ɢɫɤɨɦɨɝɨ [:

1

Ndz

[_¨Q^§ O ^·_¸

ɗɬɨ ɡɧɚɱɟɧɢɟ [ ɩɪɢ ɭɫɥɨɜɢɢ L Nd_z (L — ɞɥɢɧɚ ɚɧɬɟɧɧɵ) ɫɨɜɩɚɞɚɟɬ ɫ ɪɟɡɭɥɶɬɚɬɨɦ, ɩɨɥɭɱɟɧɧɵɦ ɜ [1], ɞɥɹ ɧɟɩɪɟɪɵɜɧɨɣ ɥɢɧɟɣɧɨɣ ɫɢɫɬɟɦɵ ɢɡɥɭɱɚɬɟɥɟɣ ɞɥɢɧɨɣ L. Ʉɚɤ ɢ ɜ ɩɪɟɞɵɞɭɳɟɦ ɫɥɭɱɚɟ, ɫ ɭɜɟɥɢɱɟɧɢɟɦ ɞɥɢɧɵ ɚɧɬɟɧɧɵ ɨɫɥɚɛɥɹɟɬɫɹ ɬɪɟɛɨɜɚɧɢɟ ɩɨɜɵɲɟɧɢɹ ɭɪɨɜɧɹ [ ɩɨ ɫɪɚɜɧɟɧɢɸ ɫ ɟɞɢɧɢɰɟɣ ɞɥɹ ɭɩɪɚɡɞɧɟɧɢɹ ɜ ɨɛɥɚɫɬɢ ɜɢɞɢɦɵɯ ɭɝɥɨɜ ɨɫɧɨɜɧɨɝɨ ɝɥɚɜɧɨɝɨ ɦɚɤɫɢɦɭɦɚ.

Ɏɢɡɢɱɟɫɤɢɟ ɩɪɨɰɟɫɫɵ ɩɪɢ ɱɚɫɬɨɬɧɨɦ ɫɤɚɧɢɪɨɜɚɧɢɢ

ɉɭɫɬɶ ɜɵɛɪɚɧɚ ɡɚɦɟɞɥɹɸɳɚɹ ɫɢɫɬɟɦɚ (ɢɡɜɟɫɬɧɚ ɱɚɫɬɨɬɧɚɹ ɡɚɜɢɫɢɦɨɫɬɶ ɤɨɷɮɮɢɰɢɟɧɬɚ ɡɚɦɟɞɥɟɧɢɹ [ f ) ɢ ɡɚɞɚɧ ɫɟɤɬɨɪ ɫɤɚɧɢɪɨɜɚɧɢɹ. ȼɵɛɟɪɟɦ ɝɪɚɧɢɰɭ ɷɬɨɝɨ ɫɟɤɬɨɪɚ ɭɝɥɨɜ -₁, ɦɚɤɫɢɦɚɥɶɧɨ ɨɬɤɥɨɧɟɧɧɭɸ ɨɬ ɧɨɪ- ɦɚɥɢ ɤ ɥɢɧɟɣɧɨɣ ɫɢɫɬɟɦɟ ɢɡɥɭɱɚɬɟɥɟɣ ɜ ɨɛɥɚɫɬɢ ɨɬɪɢɰɚɬɟɥɶɧɵɯ ɭɝɥɨɜ -. ɉɭɫɬɶ ɬɪɟɛɭɟɬɫɹ, ɱɬɨɛɵ ɥɭɱ ɜ ɷɬɨɦ ɧɚɩɪɚɜɥɟɧɢɢ ɧɚɛɥɸɞɚɥɫɹ ɧɚ ɱɚɫɬɨɬɟ f_min (O_max), ɝɞɟ ɤɨɷɮɮɢɰɢɟɧɬ ɡɚɦɟɞɥɟɧɢɹ ɩɪɢɧɢɦɚɟɬ ɦɢɧɢɦɚɥɶ- ɧɨɟ ɡɧɚɱɟɧɢɟ [_min. ȼɟɥɢɱɢɧɚ [_min ɧɚ ɱɚɫɬɨɬɟ f_min ɨɩɪɟɞɟɥɹɟɬɫɹ ɯɚɪɚɤɬɟɪɨɦ ɡɚɦɟɞɥɹɸɳɟɣ ɫɢɫɬɟɦɵ. ɂɡ (6) ɫɥɟɞɭɟɬ, ɱɬɨ ɭɤɚɡɚɧɧɨɟ ɜɵɲɟ ɩɨɥɨɠɟɧɢɟ ɥɭɱɚ ɨɛɟɫɩɟɱɢɜɚɟɬɫɹ, ɟɫɥɢ ɪɚɫɫɬɨɹɧɢɟ d_z ɦɟɠɞɭ ɫɨɫɟɞɧɢɦɢ ɢɡ- ɥɭɱɚɬɟɥɹɦɢ ɜ ɫɢɫɬɟɦɟ ɫɜɹɡɚɧɨ ɫ [_min ɢ O_max ɫɨɨɬɧɨɲɟɧɢɟɦ

max min sin 1

dz O

[ - . (24)

Ɍɚɤɢɦ ɨɛɪɚɡɨɦ, ɜɵɛɨɪɨɦ d_z ɧɚ ɱɚɫɬɨɬɟ f_min ɡɚɮɢɤɫɢɪɨɜɚɥɢ ɧɚɱɚɥɶɧɨɟ ɩɨɥɨɠɟɧɢɟ ɥɭɱɚ (ɭɝɨɥ -₁). ɉɪɢ ɭɜɟɥɢɱɟɧɢɢ ɱɚɫɬɨɬɵ ɥɟɜɚɹ ɱɚɫɬɶ ɧɟɪɚɜɟɧɫɬɜɚ (4) ɧɟ ɧɚɪɭɲɚɟɬɫɹ, ɚ ɩɪɚɜɚɹ ɝɪɚɧɢɰɚ ɧɟɪɚɜɟɧɫɬɜɚ ɭɦɟɧɶɲɚɟɬ- ɫɹ. ɍɦɟɧɶɲɚɹɫɶ, ɨɧɚ ɧɟ ɞɨɥɠɧɚ ɫɬɚɬɶ ɪɚɜɧɨɣ (ɢ ɦɟɧɶɲɟɣ) ɪɚɧɟɟ ɜɵɛɪɚɧɧɨɝɨ ɡɧɚɱɟɧɢɹ d_z. ȼ ɷɬɨɦ ɫɦɵɫɥɟ ɫɭɳɟɫɬɜɭɟɬ ɨɝɪɚɧɢɱɟɧɢɟ ɞɥɹ ɪɨɫɬɚ ɱɚɫɬɨɬɵ, ɩɪɢ ɤɨɬɨɪɨɣ ɨɛɹɡɚɧɨ ɜɵɩɨɥɧɹɬɶɫɹ ɧɟɪɚɜɟɧɫɬɜɨ

2

z 1

d O

[

. (25)

(9)

Ɍɚɤɢɦ ɨɛɪɚɡɨɦ, ɱɬɨɛɵ ɧɟ ɩɨɹɜɢɥɫɹ ɂȽɆ2, ɩɪɢ ɜɵɛɪɚɧɧɨɦ d_z ɩɚɪɚɦɟɬɪɵ O ɢ [ ɞɨɥɠɧɵ ɛɵɬɶ ɬɚɤɢɦɢ, ɱɬɨɛɵ ɜɵɩɨɥɧɹɥɨɫɶ ɪɚɜɟɧɫɬɜɨ (20). ȼ ɪɟɚɥɶɧɨɣ ɫɢɫɬɟɦɟ, ɤɪɨɦɟ ɨɝɪɚɧɢɱɟɧɢɹ (25), ɩɪɢ ɜɵɛɨɪɟ ɦɚɤɫɢɦɚɥɶ- ɧɨɣ ɱɚɫɬɨɬɵ ɧɚɞɨ ɫɥɟɞɢɬɶ ɡɚ ɬɟɦ, ɱɬɨɛɵ ɜ ɜɨɥɧɨɜɟɞɭɳɟɣ ɫɢɫɬɟɦɟ ɧɟ ɩɨɹɜɢɥɚɫɶ ɩɚɪɚɡɢɬɧɚɹ ɦɨɞɚ. ɗɬɨ ɨɩɪɟ- ɞɟɥɹɟɬɫɹ ɞɢɫɩɟɪɫɢɨɧɧɵɦɢ ɯɚɪɚɤɬɟɪɢɫɬɢɤɚɦɢ ɡɚɦɟɞɥɹɸɳɟɣ ɫɢɫɬɟɦɵ.

ɑɬɨɛɵ ɢɡɛɟɠɚɬɶ ɩɨɹɜɥɟɧɢɹ ɨɫɧɨɜɧɨɝɨ ɝɥɚɜɧɨɝɨ ɦɚɤɫɢɦɭɦɚ, ɤɨɷɮɮɢɰɢɟɧɬ ɡɚɦɟɞɥɟɧɢɹ [ ɞɨɥɠɟɧ ɩɪɟɜɵɲɚɬɶ ɟɞɢɧɢɰɭ ɧɚ ɜɟɥɢɱɢɧɭ, ɨɩɪɟɞɟɥɟɧɧɭɸ ɜ (23), ɝɞɟ Nd_z ɫɥɟɞɭɟɬ ɜɨɫɩɪɢɧɢɦɚɬɶ ɤɚɤ ɞɥɢɧɭ ɚɧɬɟɧɧɵ.

ȼɢɞɢɦ, ɩɨɩɪɚɜɤɚ ɤ ɟɞɢɧɢɰɟ ɬɟɦ ɦɟɧɶɲɟ, ɱɟɦ ɛɨɥɶɲɟ ɷɥɟɤɬɪɢɱɟɫɤɚɹ ɞɥɢɧɚ ɚɧɬɟɧɧɵ.

Ȼɨɥɶɲɨɣ ɢɧɬɟɪɟɫ ɩɪɟɞɫɬɚɜɥɹɟɬ ɜɨɩɪɨɫ, ɤɚɤɢɦɢ ɫɜɨɣɫɬɜɚɦɢ ɞɨɥɠɧɚ ɯɚɪɚɤɬɟɪɢɡɨɜɚɬɶɫɹ ɡɚɦɟɞɥɹɸɳɚɹ ɫɢɫɬɟɦɚ, ɱɬɨɛɵ ɨɧɚ ɨɛɟɫɩɟɱɢɜɚɥɚ ɧɚɢɛɨɥɟɟ ɷɮɮɟɤɬɢɜɧɨɟ ɫɤɚɧɢɪɨɜɚɧɢɟ ɥɭɱɨɦ ɂȽɆ1. ɇɚ ɩɟɪɜɵɣ ɜɡɝɥɹɞ ɤɚɠɟɬɫɹ, ɱɬɨ ɧɚɞɨ ɪɚɛɨɬɚɬɶ ɧɚ ɧɚɢɛɨɥɟɟ ɤɪɭɬɨɦ ɭɱɚɫɬɤɟ ɞɢɫɩɟɪɫɢɨɧɧɨɣ ɯɚɪɚɤɬɟɪɢɫɬɢɤɢ. Ɋɚɫɫɦɨɬɪɢɦ ɝɢ- ɩɨɬɟɬɢɱɟɫɤɢɣ ɫɥɭɱɚɣ, ɤɨɝɞɚ ɜ ɤɚɱɟɫɬɜɟ ɦɚɤɫɢɦɚɥɶɧɨɣ ɞɥɢɧɵ ɜɨɥɧɵ O_max ɩɪɢ ɫɤɚɧɢɪɨɜɚɧɢɢ ɜɵɛɪɚɧɚ ɞɥɢɧɚ ɜɨɥɧɵ, ɩɪɢ ɤɨɬɨɪɨɣ [ 1 G (G1 ɜɵɛɢɪɚɟɬɫɹ ɜ ɫɨɨɬɜɟɬɫɬɜɢɢ ɫ (23)). ɂɡ ɢɧɬɟɪɟɫɭɸɳɟɝɨ ɧɚɫ ɞɢɚɩɚɡɨɧɚ ɤɨɷɮɮɢɰɢɟɧɬɨɜ ɡɚɦɟɞɥɟɧɢɹ ɷɬɨ ɦɢɧɢɦɚɥɶɧɨɟ ɡɧɚɱɟɧɢɟ (ɧɚɡɨɜɟɦ ɟɝɨ [_min). ɂɡ (24) ɫɥɟɞɭɟɬ, ɱɬɨ ɟɫɥɢ ɩɪɢ ɷɬɨɦ ɥɭɱ ɫɜɟɬɢɬ ɜ ɧɚɩɪɚɜɥɟɧɢɢ - S 2, ɪɚɫɫɬɨɹɧɢɟ _min ^max

z 2

d O

| . Ɂɚɮɢɤɫɢɪɭɟɦ ɷɬɨ ɪɚɫɫɬɨɹɧɢɟ. ɉɪɢ ɷɬɨɦ ɧɚ ɫɚɦɨɣ ɜɵɫɨɤɨɣ ɱɚɫɬɨɬɟ ɫɤɚɧɢɪɨɜɚɧɢɹ ɞɨɥɠɧɚ ɜɵɩɨɥɧɹɬɶɫɹ ɜɬɨɪɚɹ ɱɚɫɬɶ ɧɟɪɚɜɟɧɫɬɜɚ (4)

max min

min

max

2

2 1

dz O O

| [

, (26)

ɨɬɤɭɞɚ ɫɥɟɞɭɟɬ

max

min max

4 1 O

O [ . (27)

ɋɨɨɬɧɨɲɟɧɢɟ (27) ɧɚɤɥɚɞɵɜɚɟɬ ɨɝɪɚɧɢɱɟɧɢɟ ɧɚ ɜɟɥɢɱɢɧɭ ɤɨɷɮɮɢɰɢɟɧɬɚ ɡɚɦɟɞɥɟɧɢɹ [_max. ɇɚɩɪɢɦɟɪ, ɩɪɢ [_max t3 ɧɟɪɚɜɟɧɫɬɜɨ (27) ɬɟɪɹɟɬ ɫɦɵɫɥ. ɗɬɨ ɡɧɚɱɢɬ, ɱɬɨ ɩɪɢ ɡɧɚɱɟɧɢɢ [, ɛɥɢɡɤɨɦ ɤ [ 3, ɤɨɷɮɮɢɰɢɟɧɬ ɡɚɦɟɞɥɟɧɢɹ ɩɪɢ ɭɜɟɥɢɱɟɧɢɢ ɱɚɫɬɨɬɵ ɞɨɥɠɟɧ ɩɪɚɤɬɢɱɟɫɤɢ ɦɝɧɨɜɟɧɧɨ ɢɡɦɟɧɢɬɶɫɹ ɨɬ ɟɞɢɧɢɰɵ ɞɨ [ 3. ɉɨ- ɫɬɪɨɟɧɢɟ ɬɚɤɨɣ ɡɚɦɟɞɥɹɸɳɟɣ ɫɢɫɬɟɦɵ ɧɟɪɟɚɥɶɧɨ, ɩɨɷɬɨɦɭ ɫɥɟɞɭɟɬ ɧɚɯɨɞɢɬɶ ɨɩɬɢɦɚɥɶɧɵɣ ɤɨɷɮɮɢɰɢɟɧɬ ɡɚɦɟɞɥɟɧɢɹ.

Ɋɟɡɭɥɶɬɚɬɵ ɪɚɫɱɟɬɚ

Ⱦɥɹ ɢɥɥɸɫɬɪɚɰɢɢ ɩɨɥɭɱɟɧɧɵɯ ɡɚɤɨɧɨɦɟɪɧɨɫɬɟɣ ɧɚ ɪɚɡɧɵɯ ɱɚɫɬɨɬɚɯ ɩɪɨɜɟɞɟɧ ɪɚɫɱɟɬ Ⱦɇ ɜ ɝɢɩɨɬɟɬɢ- ɱɟɫɤɨɦ ɫɥɭɱɚɟ ɨɞɧɨɪɨɞɧɨɣ ɷɤɜɢɞɢɫɬɚɧɬɧɨɣ ɫɢɫɬɟɦɵ N ɢɡɥɭɱɚɬɟɥɟɣ, ɜɨɡɛɭɠɞɚɟɦɨɣ ɜɨɥɧɨɣ ɫ ɤɨɷɮɮɢɰɢ- ɟɧɬɨɦ ɡɚɦɟɞɥɟɧɢɹ, ɛɨɥɶɲɢɦ ɟɞɢɧɢɰɵ ([!1). ɉɚɪɚɥɥɟɥɶɧɨ ɪɚɫɫɱɢɬɚɧɵ Ⱦɇ ɢ ɤɨɷɮɮɢɰɢɟɧɬ ɢɡɥɭɱɟɧɢɹ ɫɢ- ɫɬɟɦɵ ɩɨɩɟɪɟɱɧɵɯ ɳɟɥɟɣ ɜ ɲɢɪɨɤɨɣ ɫɬɟɧɤɟ ɩɪɹɦɨɭɝɨɥɶɧɨɝɨ ɜɨɥɧɨɜɨɞɚ ɫɟɱɟɧɢɟɦ 23×10 ɦɦ ɫ ɱɚɫɬɢɱɧɵɦ ɞɢɷɥɟɤɬɪɢɱɟɫɤɢɦ ɡɚɩɨɥɧɟɧɢɟɦ. ɋɥɨɣ ɞɢɷɥɟɤɬɪɢɤɚ ɬɨɥɳɢɧɨɣ 3 ɦɦ ɫ ɞɢɷɥɟɤɬɪɢɱɟɫɤɨɣ ɩɪɨɧɢɰɚɟɦɨɫɬɶɸ H 7 ɩɚɪɚɥɥɟɥɟɧ ɲɢɪɨɤɨɣ ɫɬɟɧɤɟ; ɜɵɛɪɚɧɵ ɩɚɪɚɦɟɬɪɵ ɫɢɫɬɟɦɵ, ɨɛɟɫɩɟɱɢɜɚɸɳɢɟ [!1. Ʉɨɷɮɮɢɰɢɟɧɬɵ ɡɚɦɟɞɥɟɧɢɹ ɧɚ ɪɚɡɧɵɯ ɱɚɫɬɨɬɚɯ ɨɩɪɟɞɟɥɟɧɵ ɢɡ ɞɢɫɩɟɪɫɢɨɧɧɨɝɨ ɭɪɚɜɧɟɧɢɹ ɞɥɹ ɨɫɧɨɜɧɨɣ LM -ɜɨɥɧɵ. Ɋɚɫ- ɱɟɬ ɚɦɩɥɢɬɭɞɧɨ-ɮɚɡɨɜɨɝɨ ɪɚɫɩɪɟɞɟɥɟɧɢɹ (ȺɎɊ) ɩɪɨɜɟɞɟɧ ɦɨɞɢɮɢɰɢɪɨɜɚɧɧɵɦ ɦɟɬɨɞɨɦ ɩɨɫɥɟɞɨɜɚɬɟɥɶɧɵɯ ɩɪɢɛɥɢɠɟɧɢɣ [7] ɫ ɭɱɟɬɨɦ ɬɨɥɳɢɧɵ ɫɬɟɧɤɢ ɜɨɥɧɨɜɨɞɚ. Ⱦɥɹ ɩɨɥɭɱɟɧɢɹ ɟɞɢɧɫɬɜɟɧɧɨɝɨ ɢɧɬɟɪɮɟɪɟɧɰɢɨɧɧɨɝɨ ɝɥɚɜɧɨɝɨ ɦɚɤɫɢɦɭɦɚ ɜ ɫɟɤɬɨɪɟ ɞɟɣɫɬɜɢɬɟɥɶɧɵɯ ɭɝɥɨɜ ɩɨ ɮɨɪɦɭɥɟ (24) ɜɵɛɪɚɧɨ ɪɚɫɫɬɨɹɧɢɟ d_z ɦɟɠɞɭ ɫɨ- ɫɟɞɧɢɦɢ ɳɟɥɹɦɢ ɜ ɫɨɨɬɜɟɬɫɬɜɢɢ ɫ ɦɟɬɨɞɢɤɨɣ, ɩɪɟɞɥɨɠɟɧɧɨɣ ɜ ɩɪɟɞɵɞɭɳɟɦ ɩɭɧɤɬɟ. ɇɚɱɚɥɶɧɨɟ ɩɨɥɨɠɟɧɢɟ ɥɭɱɚ ɩɪɢ ɞɥɢɧɟ ɜɨɥɧɵ ɜ ɫɜɨɛɨɞɧɨɦ ɩɪɨɫɬɪɚɧɫɬɜɟ O_max 37,5^{ɦɦ (}[ 1, 07) ɜɵɛɪɚɧɨ ɩɪɢ ɭɝɥɟ -₁ 45⁰. ɉɪɢ ɷɬɨɦ ɩɨɥɭɱɢɥɨɫɶ, ɱɬɨ d_z 21ɦɦ. ɉɪɢ ɭɜɟɥɢɱɟɧɢɢ ɱɚɫɬɨɬɵ (ɭɦɟɧɶɲɟɧɢɢ O ɞɨ 32 ɦɦ) ɭɝɨɥ - ɭɜɟɥɢ- ɱɢɜɚɟɬɫɹ ɞɨ -₂ 11⁰. ɗɬɨ ɢɦɟɟɬ ɦɟɫɬɨ ɜ ɝɢɩɨɬɟɬɢɱɟɫɤɨɦ ɜɚɪɢɚɧɬɟ ɩɨɫɬɨɹɧɧɨɝɨ ȺɎɊ. ȼ ɪɟɚɥɶɧɨɣ ɫɢɫɬɟɦɟ ɫɟɤɬɨɪ ɫɤɚɧɢɪɨɜɚɧɢɹ ɧɟɫɤɨɥɶɤɨ ɭɦɟɧɶɲɚɟɬɫɹ — ɨɬ 42⁰ ɞɨ 12⁰. ɍɪɨɜɟɧɶ ɛɨɤɨɜɨɝɨ ɥɟɩɟɫɬɤɚ ɩɪɢ - S 2 ɞɨɫɬɢɝɚɟɬ ɡɧɚɱɟɧɢɹ 0,26 (ɧɟɫɤɨɥɶɤɨ ɛɨɥɶɲɟ, ɱɟɦ ɭɪɨɜɟɧɶ ɩɟɪɜɨɝɨ ɥɟɩɟɫɬɤɚ ɜ ɚɧɬɟɧɧɟ ɫ ɩɨɫɬɨɹɧɧɵɦ ɚɦ- ɩɥɢɬɭɞɧɵɦ ɪɚɫɩɪɟɞɟɥɟɧɢɟɦ). ɗɬɢ ɢɡɦɟɧɟɧɢɹ ɜ ɪɟɚɥɶɧɨɣ ɚɧɬɟɧɧɟ, ɫɤɨɪɟ ɜɫɟɝɨ, ɩɪɨɢɫɯɨɞɹɬ ɢɡ-ɡɚ ɜɡɚɢɦɧɨɝɨ ɜɥɢɹɧɢɹ ɢɡɥɭɱɚɬɟɥɟɣ ɢ ɫɩɚɞɚɸɳɟɝɨ ɤ ɤɨɧɰɭ ɚɦɩɥɢɬɭɞɧɨɝɨ ɪɚɫɩɪɟɞɟɥɟɧɢɹ. ȼ ɢɧɬɟɪɜɚɥɟ ɭɤɚɡɚɧɧɨɝɨ ɜɵɲɟ ɫɟɤɬɨɪɚ ɫɤɚɧɢɪɨɜɚɧɢɹ ɤɨɷɮɮɢɰɢɟɧɬ ɢɡɥɭɱɟɧɢɹ ɧɟ ɨɩɭɫɤɚɟɬɫɹ ɧɢɠɟ ɭɪɨɜɧɹ 0,8. Ɋɚɫɱɟɬɧɵɟ ɞɚɧɧɵɟ ɩɪɟɞ- ɫɬɚɜɥɟɧɵ ɧɚ ɪɢɫ. 2, ɝɞɟ ɤɪɢɜɵɟ 1 ɢ 2 ɫɨɨɬɜɟɬɫɬɜɭɸɬ ɝɢɩɨɬɟɬɢɱɟɫɤɨɦɭ ɜɚɪɢɚɧɬɭ, ɚ ɤɪɢɜɵɟ 3 ɢ 4 — ɪɚɫɱɟɬɭ ɦɟɬɨɞɨɦ ɩɨɫɥɟɞɨɜɚɬɟɥɶɧɵɯ ɩɪɢɛɥɢɠɟɧɢɣ (1, 3 ɩɪɢ O = 37,5 ɦɦ, 2, 4 ɩɪɢ O = 32 ɦɦ). Ɋɚɫɱɟɬ ɩɪɨɜɨɞɢɥɢ ɞɥɹ ɫɢɫɬɟɦɵ ɢɡ 20 ɳɟɥɟɣ ɨɞɢɧɚɤɨɜɨɣ ɞɥɢɧɵ 16 ɦɦ.

(10)

14

Ɋɢɫ. 2. Ⱦɢɚɝɪɚɦɦɵ ɧɚɩɪɚɜɥɟɧɧɨɫɬɢ ɧɚ ɞɜɭɯ ɱɚɫɬɨɬɚɯ

ɋɥɟɞɭɟɬ ɨɬɦɟɬɢɬɶ, ɱɬɨ, ɧɟɫɦɨɬɪɹ ɧɚ ɛɨɥɶɲɨɟ ɪɚɡɥɢɱɢɟ ɜ ɚɦɩɥɢɬɭɞɧɵɯ ɪɚɫɩɪɟɞɟɥɟɧɢɹɯ ɜ ɝɢɩɨɬɟɬɢɱɟ- ɫɤɨɣ ɚɧɬɟɧɧɟ (ɚɦɩɥɢɬɭɞɧɨɟ ɪɚɫɩɪɟɞɟɥɟɧɢɟ ɩɨɫɬɨɹɧɧɨɟ) ɢ ɪɟɚɥɶɧɨɣ, ɝɞɟ ɨɧɨ ɫɩɚɞɚɸɳɟɟ, ɚ ɬɚɤɠɟ ɛɨɥɶɲɨɟ ɜɡɚɢɦɨɞɟɣɫɬɜɢɟ ɩɨɩɟɪɟɱɧɵɯ ɳɟɥɟɣ, ɯɚɪɚɤɬɟɪ Ⱦɇ ɜ ɷɬɢɯ ɞɜɭɯ ɫɥɭɱɚɹɯ ɦɟɧɹɟɬɫɹ ɧɟ ɨɱɟɧɶ ɫɢɥɶɧɨ. ɇɚ ɪɢɫ.

3 ɤɪɢɜɵɦɢ 1, 2, 3 ɩɪɟɞɫɬɚɜɥɟɧɵ ɡɚɜɢɫɢɦɨɫɬɢ ɨɬ O ɤɨɷɮɮɢɰɢɟɧɬɚ ɢɡɥɭɱɟɧɢɹ ɞɥɹ ɬɪɟɯ ɫɢɫɬɟɦ ɨɞɢɧɚɤɨɜɵɯ ɳɟɥɟɣ, ɜ ɤɚɠɞɨɣ ɢɡ ɤɨɬɨɪɵɯ ɞɥɢɧɵ L ɳɟɥɟɣ ɪɚɜɧɵ 15, 16 ɢ 17 ɦɦ. ɇɚɢɛɨɥɟɟ ɪɚɜɧɨɦɟɪɧɨ ɩɨ ɞɢɚɩɚɡɨɧɭ ɜɟ- ɞɟɬ ɫɟɛɹ ɤɨɷɮɮɢɰɢɟɧɬ ɢɡɥɭɱɟɧɢɹ ɫɢɫɬɟɦɵ ɳɟɥɟɣ ɞɥɢɧɨɣ 17 ɦɦ ɩɪɢ ɬɨɣ ɠɟ ɲɢɪɢɧɟ ɫɟɤɬɨɪɚ ɫɤɚɧɢɪɨɜɚɧɢɹ, ɱɬɨ ɢ ɞɥɹ ɳɟɥɟɣ ɞɥɢɧɨɣ 16 ɦɦ.

Ɋɢɫ. 3. Ɂɚɜɢɫɢɦɨɫɬɶ ɤɨɷɮɮɢɰɢɟɧɬɚ ɢɡɥɭɱɟɧɢɹ ɨɬ ɞɥɢɧɵ ɜɨɥɧɵ

ȼɵɜɨɞɵ

1) Ⱦɥɹ ɬɨɝɨ ɱɬɨɛɵ ɜ Ⱦɇ ɚɧɬɟɧɧɵ ɡɚɦɟɞɥɟɧɧɨɣ ɛɟɝɭɳɟɣ ɜɨɥɧɵ ɫ ɞɢɫɤɪɟɬɧɨɣ ɫɢɫɬɟɦɨɣ ɢɡɥɭɱɚɬɟɥɟɣ ɩɨ- ɹɜɢɥɫɹ ɢɧɬɟɪɮɟɪɟɧɰɢɨɧɧɵɣ ɦɚɤɫɢɦɭɦ, ɧɟɨɛɯɨɞɢɦɨ, ɱɬɨɛɵ ɪɚɫɫɬɨɹɧɢɟ d_z ɦɟɠɞɭ ɫɨɫɟɞɧɢɦɢ ɢɡɥɭɱɚɬɟɥɹɦɢ ɛɵɥɨ ɧɟ ɦɟɧɶɲɟ ɧɟɤɨɬɨɪɨɣ ɜɟɥɢɱɢɧɵ: d_z Ot1 1

[

.

2) Ⱦɥɹ ɬɨɝɨ ɱɬɨɛɵ ɷɬɨɬ ɦɚɤɫɢɦɭɦ ɛɵɥ ɟɞɢɧɫɬɜɟɧɧɵɦ (ɨɬɫɭɬɫɬɜɨɜɚɥɢ ɨɫɧɨɜɧɨɣ ɝɥɚɜɧɵɣ ɦɚɤɫɢɦɭɦ ɢ ɜɬɨɪɨɣ ɢɧɬɟɪɮɟɪɟɧɰɢɨɧɧɵɣ), ɧɟɨɛɯɨɞɢɦɨ ɱɬɨɛɵ

ɚ) ɤɨɷɮɮɢɰɢɟɧɬ ɡɚɦɟɞɥɟɧɢɹ [ ɩɪɟɜɵɲɚɥ ɟɞɢɧɢɱɧɵɣ ɭɪɨɜɟɧɶ ɧɚ ɜɟɥɢɱɢɧɭ, ɭɤɚɡɚɧɧɭɸ ɜ (23);

ɛ) ɪɚɫɫɬɨɹɧɢɟ ɦɟɠɞɭ ɢɡɥɭɱɚɬɟɥɹɦɢ ɛɵɥɨ ɦɟɧɶɲɟ dz ²O

¹[

ɧɚ ɜɟɥɢɱɢɧɭ, ɭɤɚɡɚɧɧɭɸ ɜ (20).

(11)

ɋɩɢɫɨɤ ɥɢɬɟɪɚɬɭɪɵ

1. Ɏɟɥɶɞ ə.ɇ., Ȼɟɧɟɧɫɨɧ Ʌ.ɋ. Ⱥɧɬɟɧɧɵ ɫɚɧɬɢɦɟ- ɬɪɨɜɵɯ ɢ ɞɟɰɢɦɟɬɪɨɜɵɯ ɜɨɥɧ. Ɇ.: ɂɡɞ-ɜɨ ȼȼɂȺ ɢɦ. ɇ.ȿ. ɀɭɤɨɜɫɤɨɝɨ, 1955. ɑ. 1. 208 ɫ.

2. Ɏɟɥɶɞ ə.ɇ., Ȼɟɧɟɧɫɨɧ Ʌ.ɋ. Ⱥɧɬɟɧɧɨ-ɮɢɞɟɪɧɵɟ ɭɫɬɪɨɣɫɬɜɚ. Ɇ.: ɂɡɞ-ɜɨ ȼȼɂȺ ɢɦ. ɇ.ȿ. ɀɭɤɨɜ- ɫɤɨɝɨ, 1959. ɑ. 2. 551 ɫ.

3. ɒɭɛɚɪɢɧ ɘ.ȼ. Ⱥɧɬɟɧɧɵ ɫɜɟɪɯɜɵɫɨɤɢɯ ɱɚɫɬɨɬ.

ɏɚɪɶɤɨɜ: ɂɡɞ. ɏȽɍ, 1960. 284 ɫ.

4. Ⱥɧɬɟɧɧɵ ɢ ɭɫɬɪɨɣɫɬɜɚ ɋȼɑ. ɉɪɨɟɤɬɢɪɨɜɚɧɢɟ ɮɚ- ɡɢɪɨɜɚɧɧɵɯ ɚɧɬɟɧɧɵɯ ɪɟɲɟɬɨɤ / ɉɨɞ ɪɟɞ. Ⱦ.ɂ. ȼɨɫ- ɤɪɟɫɟɧɫɤɨɝɨ. Ɇ.:Ɋɚɞɢɨ ɢ ɫɜɹɡɶ, 1994. 592 ɫ.

5. Ȼɚɯɪɚɯ Ʌ.Ⱦ., Ɇɚɥɨɜ Ⱥ.ȼ. ɇɟɤɨɬɨɪɵɟ ɜɨɩɪɨɫɵ ɱɚ- ɫɬɨɬɧɨɝɨ ɫɤɚɧɢɪɨɜɚɧɢɹ // Ⱥɧɬɟɧɧɵ. 2001. ȼɵɩ. 2 (48). ɋ. 14–20.

6. ȼɟɧɞɢɤ Ɉ.Ƚ., ɉɚɪɧɟɫ Ɇ.Ⱦ. Ⱥɧɬɟɧɧɵ ɫ ɷɥɟɤɬɪɢɱɟ- ɫɤɢɦ ɫɤɚɧɢɪɨɜɚɧɢɟɦ. ȼɜɟɞɟɧɢɟ ɜ ɬɟɨɪɢɸ. Ɇ.: ɇɚ- ɭɤɚ, 2001. 250 ɫ.

7. əɰɭɤ Ʌ.ɉ., Ȼɥɢɧɨɜɚ ɇ.Ʉ., ɀɢɪɨɧɤɢɧɚ Ⱥ.ȼ.

Ɇɚɬɟɦɚɬɢɱɟɫɤɚɹ ɦɨɞɟɥɶ ɥɢɧɟɣɧɨɣ ɫɢɫɬɟɦɵ ɳɟɥɟɣ ɜ ɜɨɥɧɨɜɨɞɟ ɫ ɩɪɨɢɡɜɨɥɶɧɨɣ ɨɬɪɚɠɚɸ- ɳɟɣ ɧɚɝɪɭɡɤɨɣ // Ɋɚɞɢɨɬɟɯɧɢɤɚ. 1992. ʋ 7–8.

ɋ. 73–78.

REGULARITIES OF FREQUENCY SCANNING IN SLOTTED-

WAVEGUIDE ANTENNA EXCITED WITH SLOWED-DOWN WAVE

Yatsuk L.P., Blinova N.K., Lyakhovsky A.A., Lyakhovsky A.F.

The scanning discrete system of radiators is under consideration. It is shown that the discrete system of radiators principally differs from continuous one. In the case when the slowing down coef¿ cient [ is greater than unit the continuous system radiates the electromagnetic

¿ eld only along its axis in the direction where excitation wave is spreading. The level of this radiation diminishes when [ is growing. In the opposite direction the radiation is absent. But in the case of discrete system the radiation appears here when the distance d_z between neighboring radiators is enough to provide in the point of observation the phase difference 2S between their

¿ elds, necessary for existing main maxima in the pattern.

But this difference must be less than 4S in order to prevent from appearance of the second main maxima.

So the certain correlation between d_z, [ O O_g and O

(O,O_g are wavelengths in free space and the waveguide correspondingly) was obtained in order to provide one- beam scanning inside a sector of real angles. The order of actions was elaborated in order to choose d_z for obtaining the radiation in given direction at the given lowest frequency. The limit frequency was de¿ ned while the further scanning.

The process of scanning was illustrated using a system of transversal narrow slots in the broad wall of a rectangular waveguide. For the slowdown system chosen (the dielectric layer parallel to broad walls inside the waveguide) the possibility was shown to obtain a beam scanning in the angle range from minus 42 to minus 12 degrees counted from the normal direction. The radiation coef¿ cient at that was not less then 0,8.