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Файл:Функции Грина и проводимость. Учебное пособие
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Ʉɨɧɬɪɨɥɶɧɵɟ ɜɨɩɪɨɫɵ
1. Ʉɚɤɢɦ ɨɛɪɚɡɨɦ ɫɥɭɱɚɣɧɵɟ ɩɪɢɦɟɫɢ ɜɥɢɹɸɬ ɧɚ ɬɟɦɩɟɪɚɬɭɪɭ ɩɚɣɟɪɥɫɨɜ-
ɫɤɨɝɨ ɩɟɪɟɯɨɞɚ?
2. ɉɪɢ ɤɚɤɨɦ ɜɪɟɦɟɧɢ ߬ ɩɪɢɦɟɫɢ ɩɨɥɧɨɫɬɶɸ ɩɨɞɚɜɥɹɸɬ ɩɚɣɟɪɥɫɨɜɫɤɢɣ ɩɟ-
ɪɟɯɨɞ?
3. ȼɵɩɢɫɚɬɶ ɜɵɪɚɠɟɧɢɟ, ɫɜɹɡɵɜɚɸɳɟɟ ɧɟɩɪɟɪɵɜɧɭɸ ɞɪɨɛɶ (12.45) ɫ ɨɬɧɨɲɟ-
ɧɢɟɦ ɮɭɧɤɰɢɣ Ȼɟɫɫɟɥɹ. Ⱦɨɤɚɡɚɬɶ ɷɬɨ ɜɵɪɚɠɟɧɢɟ.
4. Ʉɚɤ ɜɥɢɹɸɬ ɩɪɢɦɟɫɢ ɧɚ ɩɚɣɟɪɥɫɨɜɫɤɭɸ ɷɧɟɪɝɟɬɢɱɟɫɤɭɸ ɳɟɥɶ ɜ ɫɩɟɤɬɪɟ
ɷɥɟɤɬɪɨɧɨɜ?
141
Ƚɥɚɜɚ 13
ɉɊɈȼɈȾɂɆɈɋɌɖ
ɉȺɃȿɊɅɋɈȼɋɄɈȽɈ ȾɂɗɅȿɄɌɊɂɄȺ
ȼ ɷɬɨ ɝɥɚɜɟ ɦɵ ɪɚɫɫɦɨɬɪɢɦ ɫɬɚɬɢɱɟɫɤɭɸ ɩɪɨɜɨɞɢɦɨɫɬɶ ɨɞɧɨɦɟɪɧɨɝɨ ɦɟɬɚɥɥɚ
ɩɪɢ ɬɟɦɩɟɪɚɬɭɪɚɯ, ɦɟɧɶɲɢɯ ɬɟɦɩɟɪɚɬɭɪɵ ɩɚɣɟɪɥɫɨɜɫɤɨɝɨ ɩɟɪɟɯɨɞɚ. ȼɵɱɢɫɥɢɜ
ɷɬɭ ɩɪɨɜɨɞɢɦɨɫɬɶ, ɦɵ ɫɦɨɠɟɦ ɩɨɥɭɱɢɬɶ ɨɬɜɟɬ ɧɚ ɜɨɩɪɨɫ ɨ ɬɨɦ, ɤɚɤ ɜɥɢɹɟɬ ɩɚɣɟɪɥɫɨɜɫɤɢɣ ɩɟɪɟɯɨɞ ɧɚ ɹɜɥɟɧɢɟ ɥɨɤɚɥɢɡɚɰɢɢ ɷɥɟɤɬɪɨɧɧɵɯ ɫɨɫɬɨɹɧɢɣ, ɤɨɬɨɪɨɟ
ɩɪɢɫɭɳɟ ɨɞɧɨɦɟɪɧɨɦɭ ɷɥɟɤɬɪɨɧɧɨɦɭ ɝɚɡɭ ɜ ɩɨɥɟ ɩɪɢɦɟɫɟɣ. Ⱦɥɹ ɪɟɲɟɧɢɹ ɷɬɨɣ
ɦɵ ɢɫɩɨɥɶɡɭɟɦ ɩɨɞɯɨɞ,
ɪɚɡɜɢɬɵɣ ɜ ɝɥɚɜɟ 9, ɜ ɤɨɬɨɪɨɦ ɦɵ ɫɜɟɥɢ ɡɚɞɚɱɭ ɜɵɱɢɫɥɟɧɢɹ ɫɬɚɬɢɱɟɫɤɨɣ ɩɪɨɜɨɞɢɦɨɫɬɢ ɨɞɧɨɦɟɪɧɨɝɨ ɦɟɬɚɥɥɚ ɤ ɪɟɲɟɧɢɸ ɭɪɚɜɧɟɧɢɹ ɞɢɮɮɭɡɢɢ ɧɚ ɩɥɨɫɤɨɫɬɢ Ʌɨɛɚɱɟɜɫɤɨɝɨ. Ʉɚɤ ɛɭɞɟɬ ɩɨɤɚɡɚɧɨ ɜ ɷɬɨɣ ɝɥɚɜɟ, ɜ ɫɥɭɱɚɟ ɨɬ-
ɥɢɱɧɨɝɨ ɨɬ ɧɭɥɹ ɩɚɪɚɦɟɬɪɚ ɩɨɪɹɞɤɚ ɩɚɣɟɪɥɫɨɜɫɤɨɝɨ ɩɟɪɟɯɨɞɚ ȟ, ɢɡɨɬɪɨɩɧɵɣ ɩɪɨ-
ɰɟɫɫ ɞɢɮɮɭɡɢɢ ɡɚɦɟɧɹɟɬɫɹ ɧɚ ɞɢɮɮɭɡɢɸ ɫɨ ɫɧɨɫɨɦ, ɱɬɨ ɩɪɢɜɨɞɢɬ ɤ
ɭɜɟɥɢɱɟɧɢɸ
ɫɤɨɪɨɫɬɢ ɫɜɚɥɢɜɚɧɢɹ ɧɚ ɚɛɫɨɥɸɬ ɱɚɫɬɢɰɵ ɜ ɞɢɮɮɭɡɧɨɦ ɩɪɨɰɟɫɫɟ, ɚ ɷɬɨ, ɜ ɫɜɨɸ
ɨɱɟɪɟɞɶ, ɫɨɨɬɜɟɬɫɬɜɭɟɬ ɭɦɟɧɶɲɟɧɢɸ ɞɥɢɧɵ ɥɨɤɚɥɢɡɚɰɢɢ ɞɥɹ ɷɥɟɤɬɪɨɧɨɜ ɜ ɫɥɭɱɚɣɧɨɦ ɩɨɬɟɧɰɢɚɥɟ.
ɇɚɱɧɟɦ ɫ ɮɨɪɦɭɥɵ ɞɥɹ ɭɫɪɟɞɧɟɧɧɨɣ ɩɨ ɫɥɭɱɚɣɧɵɦ ɩɪɢɦɟɫɧɵɦ ɩɨɥɹɦ ɩɪɨɜɨ-
ɞɢɦɨɫɬɢ ɨɞɧɨɦɟɪɧɨɝɨ ɦɟɬɚɥɥɚ (8.45), ɜ ɤɨɬɨɪɨɣ ɱɚɫɬɨɬɭ ȳ ɜɧɟɲɧɟɝɨ ɷɥɟɤɬɪɢɱɟ-
ɫɤɨɝɨ ɩɨɥɹ ɩɨɥɨɠɢɦ ɪɚɜɧɨɣ ɧɭɥɸ
ߪൌെ
ଶ
݁
ݒ
Ԝන݀ݐଵԜන ݀
ʹߨ
ିஶ
ஶ
߱Ԝቆ
݀݊
ி
݀߱
ሺ߱ሻ
ቇԜۃሾ݃ො
ோ
ఠ
ሺ
ݐǡݐ
ሻ
ሺ
ሻ
݃ො
ଵ
ሿۄ
ݐ
ǡݐ
ఠ
ଵ
Ǥሺͳ͵Ǥͳ
ȼ ɝɥɚɜɟ 9 ɩɨɤɚɡɚɧɨ, ɱɬɨ ɩɨɞɫɬɚɧɨɜɤɚ ɮɭɧɤɰɢɣ Ƚɪɢɧɚ ɜ ɷɬɭ ɮɨɪɦɭɥɭ, ɜɵɪɚɠɟɧ-
ɧɵɯ ɱɟɪɟɡ ɷɥɟɦɟɧɬɵ ܵ-ɦɚɬɪɢɰɵ, ɩɪɢɜɨɞɢɬ ɤ ɫɥɟɞɭɸɳɟɦɭ ɜɵɪɚɠɟɧɢɸ
ஶ
ɝɞɟ ɮɭɧɤɰɢɹ ܴ
ఠ
ሺ
ݐ
ߪൌെ
ሻ
ɟɫɬɶ
ଶ
݁
ܮ
න݀
ିஶ
ൌൣܵ
መ
ఠԜଶଶ
߱Ԝቆ
ʹߨ
ሺ
ሻ
ݐ
ܴ
ఠ
ሺ߱ሻ
݀݊
ி
݀߱
መ
ሺ
ݐ
ǡͲሻܵ
ఠԜଵଵ
ቇۃܴ
ሺ
ఠ
ݐ
ǡͲሻ൧ିଵǤ
ሺ
ሻ
ۄ
ݐ
ǡ
ሺ
ͳ͵Ǥʹ
ሺ
ͳ͵Ǥ͵
ɉɪɢ ɜɵɜɨɞɟ (13.2) ɦɵ ɩɟɪɟɲɥɢ ɤ ɨɛɪɚɡɰɭ ɤɨɧɟɱɧɨɣ ɞɥɢɧɵ ܮ, ɩɪɢ ɷɬɨɦ ɜɜɟɥɢ
ɨɛɨɡɧɚɱɟɧɢɟ ݐ
ൌܮȀݒ
.
13.1. ɍɪɚɜɧɟɧɢɟ ɞɢɮɮɭɡɢɢ ɫɨ ɫɧɨɫɨɦ
ɧɚ ɩɥɨɫɤɨɫɬɢ Ʌɨɛɚɱɟɜɫɤɨɝɨ
ሻ
ሻ
ሻ
ȼ ɝɥɚɜɟ 9 ɛɵɥɨ ɩɨɤɚɡɚɧɨ, ɱɬɨ ɭɫɪɟɞɧɟɧɢɟ ɩɨ ɫɥɭɱɚɣɧɵɦ ɩɨɥɹɦ ɜ ɮɨɪɦɭɥɟ (13.2)
ɦɨɠɧɨ ɨɫɭɳɟɫɬɜɢɬɶ ɫ ɩɨɦɨɳɶɸ ɮɭɧɤɰɢɢ ɪɚɫɩɪɟɞɟɥɟɧɢɹ ɩɨ ɮɨɪɦɭɥɟ (9.30):
142
ሺ
ሻ
ۄ
ݐ
ൌͶߨන݀ߠ
ۃܴ
ఠ
Ɏɭɧɤɰɢɹ ɪɚɫɩɪɟɞɟɥɟɧɢɹ
ఠ
ஶ
ሺ
ݐǡߠǡ߮
ሺߠ
ሺߠሻͳ
ሻ
ɭɞɨɜɥɟɬɜɨɪɹɟɬ ɭɪɚɜɧɟɧɢɸ ɞɢɮɮɭɡɢɢ ɧɚ
ଶగ
ሻ
න݀߮
ሺ
ఠ
ݐ
ǡߠǡ߮
ሻ
Ǥ
ሺ
ͳ͵ǤͶ
ɩɥɨɫɤɨɫɬɢ Ʌɨɛɚɱɟɜɫɤɨɝɨ, ɢɦɟɸɳɟɝɨ ɜɢɞ
μ
ͳ
μݐ
ൌ
ᶭ
߬
ଶ
Ԝ
ͳ
μଶ
ଶ
߬
μ߮
െʹ߱
߲
߲߮
݅οܮ
െ݅οכܮ
ି
Ǥ
ା
ሺ
ͳ͵Ǥͷ
ȼɵɜɨɞ ɷɬɨɝɨ ɭɪɚɜɧɟɧɢɹ ɫɨɜɟɪɲɟɧɧɨ ɚɧɚɥɨɝɢɱɟɧ ɜɵɜɨɞɭ ɭɪɚɜɧɟɧɢɹ (9.45) ɢ
ɩɪɢɜɨɞɢɬɶɫɹ ɡɞɟɫɶ ɧɟ ɛɭɞɟɬ. ȼ ɭɪɚɜɧɟɧɢɢ (13.5) ɤɚɤ ɢ ɜ (9.45) ᶭ
– ɨɩɟɪɚɬɨɪ
Ʌɚɩɥɚɫɚ – Ȼɟɥɶɬɪɚɦɢ ɧɚ ɩɥɨɫɤɨɫɬɢ Ʌɨɛɚɱɟɜɫɤɨɝɨ
ɚ ɨɩɟɪɚɬɨɪɵ ܮ
ଶ
μ
ൌ
ᶭ
ଶ
μߠ
ɞɚɸɬɫɹ ɜɵɪɚɠɟɧɢɹɦɢ
േ
ൌ݁
ܮ
േ
ሺɅ
േఝ
൬
μߠ
μ
ሻ
μߠ
μ
േ݅ሺɅ
ଶ
ͳ
ሺߠሻ
ሻ
߲߮
ଶ
μ
ǡ
ଶ
μ߮
߲
൰
ሺ
ͳ͵Ǥ
ሺ
ͳ͵Ǥ
ɢ ɹɜɥɹɸɬɫɹ ɩɨɜɵɲɚɸɳɢɦ ɢ ɩɨɧɢɠɚɸɳɢɦ ɨɩɟɪɚɬɨɪɚɦɢ ɜ ɫɦɵɫɥɟ ɚɧɚɥɨɝɢɱɧɨɦ
ɤɨɦɩɨɧɟɧɬɚɦ ɤɜɚɧɬɨɜɨ-ɦɟɯɚɧɢɱɟɫɤɨɝɨ ɨɩɟɪɚɬɨɪɚ ɤɢɧɟɬɢɱɟɫɤɨɝɨ ɦɨɦɟɧɬɚ, ɚ
ɜɪɟɦɹ ߬ ɞɚɟɬɫɹ ɜɵɪɚɠɟɧɢɟɦ ߬
ିଵ
ൌʹ߬
ିଵ
ିଵ
߬
ଵ
.
ଶ
Ⱦɥɹ ɩɨɫɬɪɨɟɧɢɹ ɪɟɲɟɧɢɹ ɭɪɚɜɧɟɧɢɹ (13.5) ɪɚɫɫɦɨɬɪɢɦ ɫɨɛɫɬɜɟɧɧɵɟ ɮɭɧɤɰɢɢ
ɢ ɫɨɛɫɬɜɟɧɧɵɟ ɡɧɚɱɟɧɢɹ ɨɩɟɪɚɬɨɪɚ ɜ ɩɪɚɜɨɣ ɱɚɫɬɢ ɭɪɚɜɧɟɧɢɹ (ɛɟɡ ɫɥɚɝɚɟɦɵɯ ɫ
ɨɩɟɪɚɬɨɪɚɦɢ ܮ
)
േ
Ԝ߯
ͳ
μଶ߯
ଶ
߬
μ߮
ͳ
ᶭ
߬
ଶ
െʹ߱
߲߯
߲߮
ൌ߳߯Ǥ
ሺ
ͳ͵Ǥͺ
Ɋɟɲɟɧɢɟ ɷɬɨɝɨ ɭɪɚɜɧɟɧɢɹ ɦɨɠɧɨ ɡɚɩɢɫɚɬɶ ɜ ɜɢɞɟ
߯
ሺ
ߠǡ߮ሻൌݓ
ఔ
ఔ
ሺߠሻ
ɝɞɟ ݉ൌͲǡേͳǡǥ , ɚ ɮɭɧɤɰɢɢ ݓ
ଶ
ఝ
݁
ǡ߳ఔൌെ
ሺߠሻ
ɹɜɥɹɸɬɫɹ ɫɨɛɫɬɜɟɧɧɵɦɢ ɮɭɧɤɰɢɹɦɢ ɨɩɟ-
ఔ
݉
െʹ݅߱݉
߬
ߥሺߥͳ
ሻ
ሺ
ǡ
߬
ଶ
ͳ͵Ǥͻ
ɪɚɬɨɪɚ Ʌɚɩɥɚɫɚ – Ȼɟɥɶɬɪɚɦɢ
ଶ
μ
μߠ
ݓ
ఔ
ଶ
ሺɅ
μݓ
ఔ
ሻ
μߠ
ଶ
݉
െ
ݓఔൌߥሺߥͳሻݓఔǡ
ଶ
ሺߠሻ
ሺ
ͳ͵ǤͳͲ
ሻ
ሻ
ሻ
ሻ
ሻ
ሻ
ሻ
ɩɟɪɟɣɞɹ ɤ ɧɟɡɚɜɢɫɢɦɨɣ ɩɟɪɟɦɟɧɧɨɣ ݑൌߠ, ɩɨɥɭɱɢɦ ɭɪɚɜɧɟɧɢɟ
ሺݑ
ଶ
െͳ
ଶ
μ
ݓ
ఔ
ሻ
ଶ
μݑ
ʹݑ
μݓ
ఔ
െቆߥሺߥͳሻ
μݑ
Ʉɚɤ ɢɡɜɟɫɬɧɨ ɪɟɲɟɧɢɟɦ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɨɝɨ ɭɪɚɜɧɟɧɢɹ (13.11) ɹɜɥɹɸɬɫɹ ɬɚɤ
ɧɚɡɵɜɚɟɦɵɟ ɩɪɢɫɨɟɞɢɧɟɧɧɵɟ ɮɭɧɤɰɢɢ Ʌɟɠɚɧɞɪɚ ܲ
ଶ
݉
ቇݓఔൌͲǤሺͳ͵Ǥͳͳ
ݑଶെͳ
ሺݑሻ. ȼ ɧɚɲɟɦ ɫɥɭɱɚɟ ɷɬɢ
ఔ
143
ሻ
ɮɭɧɤɰɢɢ ɧɭɠɧɨ ɜɡɹɬɶ ɩɪɢ ɜɟɳɟɫɬɜɟɧɧɨɦ ɡɧɚɱɟɧɢɢ ɚɪɝɭɦɟɧɬɚ ɧɚ ɩɨɥɭɢɧɬɟɪɜɚɥɟ
ሾͳǡλሻ, ɬɚɤ ɤɚɤ ݑൌߠ, ɤɪɨɦɟ ɬɨɝɨ ɜ ɷɬɨɦ ɫɥɭɱɚɟ ߥሺߥͳሻ൏Ͳ, ɱɬɨ ɚɜɬɨɦɚɬɢɱɟ-
ɫɤɢ ɜɵɩɨɥɧɹɟɬɫɹ, ɟɫɥɢ ɩɨɥɨɠɢɬɶ ߥൌെͳȀʹ݅ߣ, ɬɚɤ ɤɚɤ:
െߥሺߥͳሻൌͳȀͶߣ
ଶ
Ǥ
ሺ
ͳ͵Ǥͳʹ
Ɍɚɤɢɦ ɨɛɪɚɡɨɦ, ɞɥɹ ɭɪɚɜɧɟɧɢɹ (13.8) ɩɨɥɭɱɚɟɦ ɩɨɥɧɭɸ ɫɢɫɬɟɦɭ ɪɟɲɟɧɢɣ
ݓ
ఒ
ሺݑሻ
ൌܲ
ିଵȀଶାఒ
ሺݑሻ
ǡ߯
ఒ
ሺ
ߠǡ߮ሻൌܲ
ିଵȀଶାఒ
ሺ
ߠሻ݁
ఝ
ሺ
ǡ
ͳ͵Ǥͳ͵
ɞɥɹ ɜɫɟɯ ɰɟɥɵɯ ݉ ɢ ɞɥɹ ɥɸɛɵɯ ߣאሾͲǡλሻ.
ɢ ܮ
Ⱦɥɹ ɞɚɥɶɧɟɣɲɟɝɨ ɨɱɟɧɶ ɜɚɠɧɨ ɡɧɚɬɶ ɞɟɣɫɬɜɢɟ ɨɩɟɪɚɬɨɪɨɜ ܮ
ା
ି
ɧɚ ɮɭɧɤɰɢɢ
(13.13). Ɉɧɨ ɬɚɤɨɜɨ:
ܮ
ି߯ఒ
ܮ
ା߯ఒ
ሺ
ߠǡ߮ሻൌെ
ሺ
ߠǡ߮ሻൌ߯
ሺ
ሾ
݉െͳȀʹ
ఒାଵ
ଶߣଶ
ሻ
ሺ
ߠǡ߮ሻǡ
ሿ
߯
ఒିଵ
ሺ
ߠǡ߮ሻǤ
ሺ
ͳ͵ǤͳͶ
ሺ
ͳ͵Ǥͳͷ
ɗɬɢ ɪɚɜɟɧɫɬɜɚ ɨɫɧɨɜɚɧɵ ɧɚ ɫɥɟɞɭɸɳɢɯ ɪɟɤɭɪɪɟɧɬɧɵɯ ɫɨɨɬɧɨɲɟɧɢɹɯ ɞɥɹ
ɮɭɧɤɰɢɣ Ʌɟɠɚɧɞɪɚ ([14], ɫɬɪ. 156)
ሺݑሻ
݀ܲ
ሺݑ
ଶ
െͳ
ఔ
ሻ
െ݉ݑܲ
݀ݑ
ଶ
ሺ
ሻ
ݑ
െͳ
ൌሺߥ݉ሻሺߥെ݉ͳ
݀ܲ
ఔ
݀ݑ
ఔ
ሺݑሻ
ሺݑሻ
ൌඥݑ
݉ݑܲ
ଶ
ඥ
ሻ
ݑ
ଶ
ሺݑሻ
ఔ
െͳܲ
െͳܲ
ൌ
ିଵ
ఔ
ାଵ
ఔ
ሺݑሻ
ሺݑሻ
ǡ
Ǥ
ሺ
ͳ͵Ǥͳ
ሺ
ͳ͵Ǥͳ
Ⱦɥɹ ɩɨɫɬɪɨɟɧɢɹ ɪɟɲɟɧɢɹ ɭɪɚɜɧɟɧɢɹ ɞɢɮɮɭɡɢɢ ɛɭɞɟɬ ɬɚɤɠɟ ɩɨɥɟɡɧɨ ɫɥɟɞɭɸ-
ɳɟɟ ɫɨɨɬɧɨɲɟɧɢɟ
ሺݑሻ
ൌ
Ȟሺߥͳെ݉
Ȟሺߥͳ݉
ି
ܲ
ఔ
ሻ
ሺݑሻ
Ǥ
ܲ
ఔ
ሻ
ሺ
ͳ͵Ǥͳͺ
13.2. ɉɨɫɬɪɨɟɧɢɟ ɪɟɲɟɧɢɹ ɭɪɚɜɧɟɧɢɟ ɞɢɮɮɭɡɢɢ
Ȼɭɞɟɦ ɢɫɤɚɬɶ ɪɟɲɟɧɢɟ ɭɪɚɜɧɟɧɢɹ (13.5) ɜ ɫɥɟɞɭɸɳɟɦ ɜɢɞɟ
ஶ
ஶ
ሺ
ሺݐǡߠǡ߮ሻൌන݀ߣߩ
ୀିஶ
ఒܣ
ߣǡݐሻ݁
ɝɞɟ
ߣሺߨߣ
ൌ
ߩ
ఒ
ɚ ܣ
ሺߣǡݐሻ – ɢɫɤɨɦɵɟ ɮɭɧɤɰɢɢ. ȼɟɫɨɜɚɹ ɮɭɧɤɰɢɹ ߩ
ɭɫɥɨɜɢɹ ɞɥɹ ɮɭɧɤɰɢɢ ɪɚɫɩɪɟɞɟɥɟɧɢɹ
ሺͲǡߠǡ߮ሻൌ
ͳ
ߨ
ଵ
ଵ
మ
ି
ାఒ
ቀ
ସ
ఛ
మ
ሻ
ǡ
ߨ
ߜሺߠെͳሻǡ
௧
ቁ
ఒ
݁
ఝ
ܲ
ିଵȀଶାఒ
ሺ
ߠሻǡሺͳ͵Ǥͳͻ
ሺ
͵ǤʹͲ
ɜɵɛɢɪɚɟɬɫɹ ɢɡ ɧɚɱɚɥɶɧɨɝɨ
ሺ
ͳ͵Ǥʹͳ
ሻ
ሻ
ሻ
ሻ
ሻ
ሻ
ሻ
ሻ
ሻ
ሻ
144
ɤɪɨɦɟ ɬɨɝɨ, ɞɥɹ ɜɵɩɨɥɧɟɧɢɹ ɷɬɨɝɨ ɭɫɥɨɜɢɹ ɧɟɨɛɯɨɞɢɦɨ
ሺ
ߣǡͲሻൌߜ
ܣ
ɉɨɞɫɬɚɧɨɜɤɚ (13.16) ɜ ɮɨɪɦɭɥɭ (13.4) ɞɥɹ ɭɫɪɟɞɧɟɧɢɹ ܴ
ஶ
ሺ
ሻ
ۃ
ۄ
ݐ
ܴ
ఠ
ൌʹߨන݀ߣߣ
ൌʹන݀ߠ
ஶ
ሺߨߣ
ሺߠሻͳ
ሻ
ଶ
ሺߨߣሻ
ሺߠ
ܣ
Ǥ
ଶగ
ሻ
නሺݐǡߠǡ߮ሻൌ
ߣǡݐ
ି
ሻ
݁
ሺ
ሺ
ݐ
ఠ
ଵ
ଵ
మ
௧
ାఒ
ቀ
ቁ
ಽ
ସ
ఛ
మ
ǡ
ɬɚɤɢɦ ɨɛɪɚɡɨɦ, ɩɪɨɜɨɞɢɦɨɫɬɶ ɨɩɪɟɞɟɥɹɟɬɫɹ ɬɨɥɶɤɨ ɮɭɧɤɰɢɟɣ ܣ
ሺ
ሻ
ߣǡݐ
ɋɢɫɬɟɦɚ ɭɪɚɜɧɟɧɢɣ ɞɥɹ ܣ
ɩɨɥɭɱɚɟɬɫɹ ɩɨɞɫɬɚɧɨɜɤɨɣ (13.16) ɜ (13.5).
ሻ
ɞɚɟɬ
ሺ
ߣǡݐ
ሺ
ͳ͵Ǥʹʹ
ሺ
ͳ͵Ǥʹ͵
ሻ
.
ȼɜɟɞɹ ɛɟɡɪɚɡɦɟɪɧɨɟ ɜɪɟɦɹ ݏൌݐȀ߬ ɢ ɛɟɡɪɚɡɦɟɪɧɵɟ ɩɚɪɚɦɟɬɪɵ ߚൌʹ߱߬, ߛൌ߬ο,
(ɤɚɤ ɜ ɝɥɚɜɟ 12), ɩɨɥɭɱɢɦ ɫɢɫɬɟɦɭ ɭɪɚɜɧɟɧɢɣ
߲ܣ
ൌെሺ݉ଶ݅ߚ݉ሻܣെ݅ߛ
߲ݏ
ሺ
ሾ
݉ͳȀʹ
ଶߣଶ
ሻ
ሿ
ܣ
ାଵ
െ݅ɀכܣ
ିଵ
ǡሺͳ͵ǤʹͶ
ɪɟɲɟɧɢɟ ɤɨɬɨɪɨɣ ɢɳɟɦ ɜ ɜɢɞɟ
ஃሺሻ௦
ଶߣଶ
ሻ
ሺߣሻ
ǡ
ܽ
ሿ
ܽ
݅ɀכܽ
ାଵ
ିଵ
ሺ
ͳ͵Ǥʹͷ
ൌͲǤሺͳ͵Ǥʹ
ɬɨɝɞɚ ɞɥɹ ɤɨɷɮɮɢɰɢɟɧɬɨɜ ܽ
ଶ
ሺ݉
݅ߚ݉Ȧሺɉሻሻܽ݅ߛ
ሺ
ܣ
ߣǡݏሻൌ݁
ሺߣሻ
ɩɨɥɭɱɚɟɦ ɫɢɫɬɟɦɭ ɭɪɚɜɧɟɧɢɣ
ሺ
ሾ
݉ͳȀʹ
ȼ ɨɛɳɟɦ ɜɢɞɟ ɧɚɣɬɢ ɪɟɲɟɧɢɟ ɫɢɫɬɟɦɵ (13.26) ɧɟ ɭɞɚɟɬɫɹ. ɉɨɩɵɬɚɟɦɫɹ ɧɚɣɬɢ
ɩɪɢɛɥɢɠɟɧɧɵɟ ɪɟɲɟɧɢɹ. Ⱦɥɹ ɷɬɨɝɨ ɫɧɚɱɚɥɚ ɪɚɫɫɦɨɬɪɢɦ ɩɪɨɜɨɞɢɦɨɫɬɶ ɩɪɢ ɧɭɥɟ
ɬɟɦɩɟɪɚɬɭɪ.
13.3. ɋɬɚɬɢɱɟɫɤɚɹ ɩɪɨɜɨɞɢɦɨɫɬɶ. ɋɥɭɱɚɣ ࢀൌ
ሺ߱ሻ
ɉɪɢ ܶൌͲ ɜ ɮɨɪɦɭɥɟ (13.2) ɮɭɧɤɰɢɹ ݀݊
ሺ߱ሻ
݀݊
ி
ൌെߜሺ߱ሻǡ
݀߱
ɬɨɝɞɚ ɜɵɪɚɠɟɧɢɟ ɞɥɹ ɩɪɨɜɨɞɢɦɨɫɬɢ ɩɪɢɦɟɬ ɜɢɞ
ଶ
ܮ
݁
ۃ
ʹߨ
ܴ
ߪൌ
Ɍɚɤɢɦ ɨɛɪɚɡɨɦ, ɜ ɷɬɨɦ ɫɥɭɱɚɟ ɞɨɫɬɚɬɨɱɧɨ ɧɚɣɬɢ ɪɟɲɟɧɢɟ ɫɢɫɬɟɦɵ (13.26) ɩɪɢ
ߚൌͲ. Ⱦɥɹ ɩɪɨɫɬɨɬɵ ɦɵ ɪɚɫɫɦɨɬɪɢɦ ɫɥɭɱɚɣ ȁߛȁاͳ, ɱɬɨ ɩɨɡɜɨɥɹɟɬ ɨɛɪɟɡɚɬɶ ɫɢɫɬɟɦɭ
ɭɫɥɨɜɢɟɦ ܽ
ൌͲɞɥɹ ɜɫɟɯ ȁ݉ȁʹ. ȼ ɷɬɨɦ ɫɥɭɱɚɟ, ɫɢɫɬɟɦɚ (13.26) ɩɪɢɦɟɬ ɜɢɞ
ሺ
ͳȦሻܽ
݅ߛߢܽଵ݅ɀכܽିଵൌͲǡ
Ȧܽ
ቐ
ሺ
݅ɀכܽൌͲǡ
ଵ
ሻ
݅ߛߢܽൌͲǡ
ିଵ
Ȁ݀߱ ɩɪɟɜɪɚɳɚɟɬɫɹ ɜ ߜ-ɮɭɧɤɰɢɸ:
ி
ሺ
ሻ
ۄ
ݐ
Ǥ
ሺ
ͳ͵Ǥʹ
ሺ
ͳ͵Ǥʹͺ
ሺ
ͳ͵Ǥʹͻ
ሻ
ሻ
ሻ
ሻ
ሻ
ሻ
ሻ
ሻ
145
ɝɞɟ ɞɥɹ ɭɩɪɨɳɟɧɢɹ ɮɨɪɦɭɥ ɜɜɟɞɟɧɨ ɨɛɨɡɧɚɱɟɧɢɟ ߢൌͳȀͶߣ
ɲɟɦ Ȧ ɜɦɟɫɬɨ Ȧሺɉሻ. Ɇɚɬɪɢɰɚ ɫɢɫɬɟɦɵ ɢɦɟɟɬ ɜɢɞ
כ
ࣞൌ
ͳ݅ɀ
݅ߛߢ Ͳ ݅ɀ
Ͳ
כ
൩ǡ
Ͳ݅ߛߢͳ
ɚ ɯɚɪɚɤɬɟɪɢɫɬɢɱɟɫɤɨɟ ɭɪɚɜɧɟɧɢɟ
ଶ
, ɢ ɜɪɟɦɟɧɧɨ ɩɢ-
ሺ
ͳ͵Ǥ͵Ͳ
ሻ
൫ࣞȦܫመ൯ൌȦሺͳȦ
Ʉɨɪɧɢ (13.31) ɫ ɬɨɱɧɨɫɬɶɸ ɞɨ
Ȧ
ൌെͳǡȦାൎെʹȁߛ
ଶ
ሻ
ଶ
ȁߛȁ
ɬɚɤɨɜɵ
ଶ
ȁ
ߢǡȦିൌെͳʹȁߛ
ʹȁߛ
ଶ
ȁ
ߢሺͳȦሻൌͲǤ
ȁ
ሺ
ͳ͵Ǥ͵ͳ
ଶ
ߢǡ
ሺ
ͳ͵Ǥ͵ʹ
ɚ ɫɨɨɬɜɟɬɫɬɜɭɸɳɢɟ ɫɨɛɫɬɜɟɧɧɵɟ ɜɟɤɬɨɪɚ ɟɫɬɶ
ݒො
ൌ൭
כ
ߛ
Ͳ
െߛߢ
൱ǡݒො
ା
ൌቌ
െ݅ߛ
ͳെʹȁߛ
െ݅ߛߢ
כ
ି
ଶ
ȁ
ߢ
ቍǡݒො
ൌቌ
െ݅ߛ
ʹȁߛ
െ݅ߛߢ
כ
ଶ
ቍǤ
ȁ
ߢ
ሺ
ͳʹǤ͵͵
ɗɬɨɝɨ ɞɨɫɬɚɬɨɱɧɨ, ɱɬɨɛɵ ɩɨɫɬɪɨɢɬɶ ɜ ɷɬɨɦ ɩɪɢɛɥɢɠɟɧɢɢ ɮɭɧɤɰɢɸ ɪɚɫɩɪɟɞɟ-
ɥɟɧɢɹ ሺݐǡߠǡ߮
ሻ
, ɹɜɥɹɸɳɭɸɫɹ ɜɟɳɟɫɬɜɟɧɧɨɣ ɢ ɭɞɨɜɥɟɬɜɨɪɹɸɳɟɣ ɧɚɱɚɥɶɧɨɦɭ
ɭɫɥɨɜɢɸ (13.22). ɉɪɢ ɷɬɨɦ ɧɟɨɛɯɨɞɢɦɨ ɢɫɩɨɥɶɡɨɜɚɬɶ ɫɨɨɬɧɨɲɟɧɢɟ (13.18), ɤɨɬɨɪɨɟ ɜ ɞɚɧɧɨɦ ɤɨɧɤɪɟɬɧɨɦ ɫɥɭɱɚɟ ɩɪɢɦɟɬ ɜɢɞ
ܲ
ିଵ
ିଵȀଶାఒ
ሺ
ߠሻൌെ
ͳ
ͳȀͶߣ
ܲ
ଶ
ଵ
ିଵȀଶାఒ
ሺ
ߠሻൌെ
ͳ
ܲ
ߢ
ଵ
ିଵȀଶାఒ
ሺ
ߠሻǤሺͳ͵Ǥ͵Ͷ
Ɏɭɧɤɰɢɹ ɪɚɫɩɪɟɞɟɥɟɧɢɹ ɜ ɢɬɨɝɟ ɢɦɟɟɬ ɜɢɞ
ஶ
ሺݐǡߠǡ߮ሻൌන ݀ߣߩ
ሺ
ߣǡݐȀ߬ሻʹ൫െ݅ߛ
ܤ
ଵ
ଵ
ଵ
మ
ି
ఛ
݁
ఒ
௧
ାఒ
ቀ
ቁ
ସ
మ
כ݁ఝ
ൣܣ
൯ܲ
ሺ
ߣǡݐȀ߬ሻܲ
ଵ
ିଵȀଶାఒ
ିଵȀଶାఒ
ሺ
ߠሻ൧ǡ
ሺ
ߠሻ
ሺ
ͳ͵Ǥ͵ͷ
ɝɞɟ
ܣ
ሺ
ߣǡݏሻൌ
ͳെʹȁߛ
ͳെͶȁߛ
ଶ
ȁ
ߢ
ଶ
ȁ
ߢ
݁
ିଶȁఊ
మ
ȁ
௦
െ
ȁ
ʹȁߛ
ͳെͶȁߛ
ଶ
ߢ
݁
ଶ
ȁ
ߢ
ି௦ାଶȁఊ
మ
ȁ
௦
ሺ
ǡ
ͳ͵Ǥ͵
ሻ
ሻ
ሻ
ሻ
ሻ
ሻ
ܤ
ሺ
ߣǡݏሻൌ
ଵ
ͳ
ͳെͶȁߛ
ଶ
ȁ
ߢ
൫݁
మ
ȁ
ିଶȁఊ
௦െ݁ି௦ାଶȁఊ
మ
ȁ
௦
൯Ǥ
ሺ
ͳ͵Ǥ͵
ȼɵɪɚɠɟɧɢɟ (13.35) ɞɥɹ ɮɭɧɤɰɢɢ ɪɚɫɩɪɟɞɟɥɟɧɢɹ ɞɟɦɨɧɫɬɪɢɪɭɟɬ, ɱɬɨ ɩɪɢ ȟ്Ͳ
ɞɢɮɮɭɡɢɹ ɩɟɪɟɫɬɚɟɬ ɛɵɬɶ ɢɡɨɬɪɨɩɧɨɣ, ɞɟɣɫɬɜɢɬɟɥɶɧɨ, ɜɬɨɪɨɟ ɫɥɚɝɚɟɦɨɟ, ɤɚɤ ɧɟ-
ɬɪɭɞɧɨ ɜɢɞɟɬɶ, ɩɪɨɩɨɪɰɢɨɧɚɥɶɧɨ ο
߮െοଶ߮, ɝɞɟ ȟൌοଵ݅ο
ଵ
.
ଶ
146
ሻ
Ɍɟɩɟɪɶ ɦɵ ɦɨɠɟɦ ɩɨɞɫɬɚɜɢɬɶ (13.36) ɜ (13.23), ɩɪɢ ɷɬɨɦ ɨɩɭɫɬɢɦ ɫɥɚɝɚɟɦɵɟ,
ɩɪɨɩɨɪɰɢɨɧɚɥɶɧɵɟ ʹȁߛ
ଶ
ȁ
ߢ, ɤɨɬɨɪɨɟ ɩɪɢ ɭɫɥɨɜɢɢ ȁߛȁاͳ ɞɚɸɬ ɧɟɫɭɳɟɫɬɜɟɧɧɵɣ
ɜɤɥɚɞ, ɷɬɨ ɞɚɟɬ
ஶ
ሺ
ሻ
ۃ
ۄ
ܴ
ݐ
ఠ
ൌʹߨන݀ߣߣ
ሺߨߣ
ଶ
ሺߨߣሻ
ሻ
ሼെݐ
ሺ
ͳȀͶߣ
ଶ
ሻሺ
ͳȀ߬
ʹ߬ȁο
ଶ
ȁ
ଶ
ሻሽሺ
ͳ͵Ǥ͵ͺ
ɜɵɱɢɫɥɹɹ ɢɧɬɟɝɪɚɥ, ɭɱɢɬɵɜɚɹ ɩɪɢ ɷɬɨɦ, ɱɬɨ ɝɥɚɜɧɵɣ ɜɤɥɚɞ ɜɧɨɫɢɬ ɨɛɥɚɫɬɶ
ߣͳȀξݐ
ɝɞɟ ߣ
اͳ ɢ ɩɨɞɫɬɚɜɥɹɹ ɷɬɨ ɜ (13.28), ɩɨɥɭɱɢɦ
ଶ
݁
ܮ
ሺ
ۃ
ߪൌ
ɷɮɮɟɤɬɢɜɧɚɹ ɞɥɢɧɚ ɥɨɤɚɥɢɡɚɰɢɢ, ɞɚɜɚɟɦɚɹ ɜɵɪɚɠɟɧɢɟɦ
ʹߨ
ܴ
ሻ
ۄ
ܮȀݒ
ͳ
ߣ
ൌ
ଶ
݁
ൌ
ͳ
ݒ
߬ଶ
ሺ
ߨߣ
Ͷξܮ
ʹ߬ȁο
ݒ
ଷȀଶ
ሻ
൜െ
Ͷߣ
ଶ
ȁ
ǡ
ܮ
ൠǡ
ሺ
ͳ͵Ǥ͵ͻ
ሺ
ͳ͵ǤͶͲ
ɢɡ ɤɨɬɨɪɨɝɨ ɫɥɟɞɭɟɬ, ɱɬɨ ɨɛɪɚɡɨɜɚɧɢɟ ɜɨɥɧɵ ɡɚɪɹɞɨɜɨɣ ɩɥɨɬɧɨɫɬɢ ɭɫɢɥɢɜɚɟɬ ɥɨɤɚɥɢɡɚɰɢɸ.
13.4. ɋɬɚɬɢɱɟɫɤɚɹ ɩɪɨɜɨɞɢɦɨɫɬɶ.
ɋɥɭɱɚɣ ɤɨɧɟɱɧɨɣ ɬɟɦɩɟɪɚɬɭɪɵ
ȼ ɷɬɨɦ ɫɥɭɱɚɟ ɦɵ ɞɨɥɠɧɵ ɜɟɪɧɭɬɶɫɹ ɤ ɮɨɪɦɭɥɟ (13.2), ɤɨɬɨɪɨɣ ɦɨɠɧɨ ɩɪɢ-
ɞɚɬɶ ɫɥɟɞɭɸɳɢɣ ɜɢɞ
ஶ
ଶ
݁
ߪሺܶሻൌ
ͺߨܶ
ܮ
න݀
ିஶ
߱Ԝ
ିଶ
ሺ
߱Ȁʹܶሻۃܴ
ሺ
ሻ
ۄ
ܮȀݒ
ఠ
ǡ
ሺ
ͳ͵ǤͶͳ
ሻ
ሻ
ሻ
ሻ
ሺ
ሻ
ɢ ɜɵɱɢɫɥɢɬɶ ɜɟɥɢɱɢɧɭ
ۃ
ܴ
ఠ
ܮȀݒ
ۄ
ɩɪɢ ɤɨɧɟɱɧɨɣ ɱɚɫɬɨɬɟ. Ʉɚɤ ɛɭɞɟɬ ɜɢɞɧɨ
ɧɢɠɟ, ɫɭɳɟɫɬɜɟɧɧɚɹ ɡɚɜɢɫɢɦɨɫɬɶ ɨɬ ɱɚɫɬɨɬɵ ɛɭɞɟɬ ɩɪɢ ߱߬بͳ. ȼɨɡɜɪɚɳɚɹɫɶ ɤ
ɫɢɫɬɟɦɟ (13.26), ɩɨɥɭɱɢɦ, ɱɬɨ ɜ ɷɬɨɦ ɫɥɭɱɚɟ ɦɚɬɪɢɰɚ ɫɢcɬɟɦɵ ɨɝɪɚɧɢɱɟɧɧɨɣ
ɭɫɥɨɜɢɟɦ ܽ
ൌͲɞɥɹ ɜɫɟɯ ȁ݉ȁʹ ɟɫɬɶ
ͳ݅ߚ ݅ɀ
ࣞൌ
݅ߛߢ Ͳ ݅ɀ
כ
Ͳ
כ
൩ǡ
ሺ
ͳ͵ǤͶʹ
Ͳ݅ߛߢͳെ݅ߚ
ɚ ɟɟ ɯɚɪɚɤɬɟɪɢɫɬɢɱɟɫɤɨɟ ɭɪɚɜɧɟɧɢɟ ɢɦɟɟɬ ɜɢɞ
ଶ
൫ࣞȦܫመ൯ൌȦሾߚ
ሺͳȦ
Ʉɨɪɧɢ ɷɬɨɝɨ ɭɪɚɜɧɟɧɢɹ ɫ ɬɨɱɧɨɫɬɶɸ ɞɨ ʹȁߛ
ൌെ
Ȧ
ሻ
ȁ
ʹȁߛ
ͳߚ
ଶ
ሿʹȁߛȁ
ଶ
ȁ
ଶ
ߢ
ǡ
ଶ
ଶ
ߢሺͳȦሻൌͲǤ
ሺ
ͳ͵ǤͶ͵
ሺ
ͳ͵ǤͶͶ
147
ሻ
ሻ
ሻ
ଶ
ȁߛȁ
ߢ
ൌെͳേ݅ߚ
Ȧ
േ
ͳߚ
ሺ
ͳט݅ߚሻǤ
ଶ
ሺ
ͳ͵ǤͶͷ
ɉɨɫɬɪɨɟɧɢɟ ɮɭɧɤɰɢɢ ɪɚɫɩɪɟɞɟɥɟɧɢɹ ɩɪɨɜɨɞɢɬɫɹ ɚɧɚɥɨɝɢɱɧɨ ɬɨɦɭ, ɤɚɤ ɷɬɨ
ɛɵɥɨ ɫɞɟɥɚɧɨ ɞɥɹ ɫɥɭɱɚɹ ߚൌͲ, ɧɨ ɹɜɧɵɣ ɟɟ ɜɢɞ ɩɪɢɜɨɞɢɬɶ ɧɟ ɛɭɞɟɦ ɜ ɜɢɞɭ ɟɟ
ɝɪɨɦɨɡɞɤɨɫɬɢ. ɋɨɝɥɚɫɧɨ (13.23) ɧɚɦ ɩɨɬɪɟɛɭɟɬɫɹ ɬɨɥɶɤɨ ɮɭɧɤɰɢɹ ܣ
ሺߣǡݏሻ, ɤɨɬɨ-
ɪɚɹ ɨɤɚɡɵɜɚɟɬɫɹ ɪɚɜɧɨɣ
ݏ
ሻ
ିଷȀଶ
൰
ଶ
ȁ
ߢ
ȁ
൜െ൬
ି
Ԝ݁
ൠ
ଶ
ͳߚ
ଶ
. Ɍɨɝɞɚ ɜɦɟɫɬɨ (13.38) ɩɨɥɭɱɢɦ
ܮ
ͳ
൰൬
ߣଶ൰ൠൌ
Ͷ
ସఒ
ሺ߱ሻ
ߣ
ሺఠሻ
ǡ
ሺ
ͳ͵ǤͶ
ሺ
ͳ͵ǤͶ
ሺ
ߣǡݏሻൎ൜െʹȁߛ
ܣ
ɫ ɬɨɱɧɨɫɬɶ ɞɨ ɫɥɚɝɚɟɦɵɯ ɩɪɨɩɨɪɰɢɨɧɚɥɶɧɵɯ ȁߛ
ஶ
ۃܴ
ሺ
ሻ
ۄ
ܮȀݒ
ఠ
ൌʹߨන݀ߣߣ
ହȀଶ
ߨ
ൌ
ʹԜ
ሺߨߣ
ଶ
ሺߨߣሻ
ܮ
൬
ሺ߱ሻ
ߣ
ɝɞɟ
ଶ
ͳ
ሺ߱ሻ
ߣ
ͳ
ൌ
ݒ
߬ଶ
ʹ߬ȁο
ݒ
ȁ
ͳሺʹ߱߬
ͳ
Ǥ
ଶ
ሻ
ሺ
ͳ͵ǤͶͺ
ɉɨɞɫɬɚɜɥɹɹ (13.47) ɜ (13.41), ɩɨɥɭɱɢɦ
ߪሺܶሻൌ
ଶɎଷȀଶ
݁
ͺܶξܮ
ஶ
න݀
߱Ԝିଶቀ
߱
ʹܶ
ቁ൬
ߣ
ͳ
ሺ߱ሻ
൰
ିଷȀଶ
൜െ
Ͷߣ
ܮ
ൠǤሺͳ͵ǤͶͻ
ሺ߱ሻ
ሻ
ሻ
ሻ
ሻ
ሻ
ɂɧɬɟɝɪɚɥ ɩɨ ɱɚɫɬɨɬɟ ɦɨɠɧɨ ɜɵɱɢɫɥɢɬɶ ɦɟɬɨɞɨɦ ɧɚɢɫɤɨɪɟɣɲɟɝɨ ɫɩɭɫɤɚ. Ɉɛ-
ɥɚɫɬɶ, ɜɧɨɫɹɳɚɹ ɝɥɚɜɧɵɣ ɜɤɥɚɞ, ɷɬɨ ߱بܶ, ɩɨɷɬɨɦɭ ɦɨɠɧɨ ɡɚɩɢɫɚɬɶ
ஶ
ି
ସ
మ
න ݀
߱ቆ
ߪሺܶሻൌ
ଶɎଷȀଶ
݁
ξ
ɡɞɟɫɶ ɦɵ ɜɜɟɥɢ ɞɥɢɧɵ ɩɪɨɛɟɝɚ ݈ൌݒ
ɦɚɤɫɢɦɭɦ ɜ ɬɨɱɤɟ ߱
߱
ൌ൬ܶȁο
ͳ
݈
ଶ
߬ ɢ ݈ଶൌݒ߬
ȁ
ȁοȁ
ʹ݈߱
ଶ
Ͷ݈
ିଷȀଶ
ଶ
ቇ
ቊെ
ଶ
. Ɏɭɧɤɰɢɹ ɜ ɷɤɫɩɨɧɟɧɬɟ ɢɦɟɟɬ
ଶ
ଵȀଷ
ܮ
൰
ǡ
߱
ܮȁο
െ
ܶ
ͺ݈߱
ଶ
ȁ
ቋ
ଶ
ɜ ɢɬɨɝɟ, ɩɨɥɭɱɚɟɦ ɪɟɡɭɥɶɬɚɬ
ߪሺܶሻൌ
݁
ଶɎଶ
ξ
ଵȀଷ
ȁοȁ
݈
݈
ቆ
ቇ
ܶܮ
ଶȀଷ
݈
ଶ
൬
൰
൝െ
݈
Ͷ݈
ܮ
͵
െ
ʹ
ଶ
ଶȀଷ
ȁοȁ
ቆ
ቇ
ܶ
ܮ
൬
൰
݈
ଵȀଷ
ൡǤሺͳ͵Ǥͷʹ
148
ǡሺͳ͵ǤͷͲ
ሺ
ͳ͵Ǥͷͳ
ሻ
ሻ
ሻ
Ɏɨɪɦɭɥɚ ɩɨɥɭɱɟɧɚ ɜ ɩɪɟɞɩɨɥɨɠɟɧɢɹɯ ߱
بܶǡ߬
, ɤɨɬɨɪɵɟ ɦɨɝɭɬ ɛɵɬɶ ɜɵ-
ିଵ
ɪɚɠɟɧɵ ɜ ɬɟɪɦɢɧɚɯ ɨɝɪɚɧɢɱɟɧɢɣ ɧɚ ɬɟɦɩɟɪɚɬɭɪɭ
ܮିଵ
ȁοȁ
߬
اܶاȁο
ݒ
ିଶ
ሺ
ሻ
ଵȀଶ
ሺ
ሻ
ȁ
ܮȀ݈
ǡ
ሺ
ͳ͵Ǥͷ͵
ɱɬɨ ɜɫɟɝɞɚ ɜɵɩɨɥɧɹɟɬɫɹ ɩɪɢ ɞɨɫɬɚɬɨɱɧɨ ɛɨɥɶɲɢɯ ɪɚɡɦɟɪɚɯ ɨɛɪɚɡɰɚ ܮ.
13.5. ɉɪɨɜɨɞɢɦɨɫɬɶ ɜ ɩɟɪɟɦɟɧɧɨɦ ɩɨɥɟ
ɇɚ ɫɚɦɨɦ ɞɟɥɟ, ɩɨɥɭɱɟɧɧɵɟ ɮɨɪɦɭɥɵ (13.39), (13.52) ɞɥɹ ɫɢɫɬɟɦɵ ɤɨɧɟɱɧɨɣ
ɞɥɢɧɵ ܮ, ɦɨɠɧɨ ɫɪɚɜɧɢɬɶ ɬɨɥɶɤɨ ɫ ɪɟɡɭɥɶɬɚɬɚɦɢ ɤɨɦɩɶɸɬɟɪɧɨɝɨ ɦɨɞɟɥɢɪɨɜɚɧɢɹ.
Ɉɧɢ ɧɟ ɫɨɨɬɜɟɬɫɬɜɭɸɬ ɪɟɚɥɶɧɨɣ ɷɤɫɩɟɪɢɦɟɧɬɚɥɶɧɨɣ ɫɢɬɭɚɰɢɢ. ɇɨ ɨɧɢ ɩɪɟɞɫɬɚɜɥɹɸɬ ɩɨɥɟɡɧɵɟ ɤɚɱɟɫɬɜɟɧɧɵɟ ɞɨɤɚɡɚɬɟɥɶɫɬɜɚ ɬɨɝɨ, ɱɬɨ ɩɨɹɜɥɟɧɢɟ ɜɨɥɧɵ ɡɚɪɹɞɨɜɨɣ ɩɥɨɬɧɨɫɬɢ ɭɫɢɥɢɜɚɟɬ ɥɨɤɚɥɢɡɚɰɢɸ (ɬɨ ɟɫɬɶ ɭɦɟɧɶɲɚɟɬ ɞɥɢɧɭ ɥɨɤɚɥɢɡɚɰɢɢ)
ɩɪɢ ɥɸɛɨɣ ɬɟɦɩɟɪɚɬɭɪɟ.
ɑɬɨ ɤɚɫɚɟɬɫɹ ɷɤɫɩɟɪɢɦɟɧɬɚ
ɤɨɧɟɱɧɵɣ ɨɛɪɚɡɟɰ, ɧɨ ɤɨɧɟɱɧɭɸ ɱɚɫɬɨɬɭ ɷɥɟɤɬɪɢɱɟɫɤɨɝɨ ɩɨɥɹ. Ɇɨɠɧɨ ɩɨɤɚɡɚɬɶ,
ɱɬɨ ɞɥɹ ɱɚɫɬɨɬɵ ɩɨɥɹ ȳ, ɬɚɤɨɣ ɱɬɨ ߬
߬ȁοȁاͳ (߬
ିଵ
ൌʹ߬
ଵ
ିଵ
߬
ɧɢɟɦ ɮɨɪɦɭɥɵ (10.108)
ɫ ɩɨɦɨɳɶɸ ɡɚɦɟɧɵ
ɢ ɢɧɬɟɝɪɢɪɨɜɚɧɢɟɦ ɩɨ ɱɚɫɬɨɬɟ ɫ ɮɚɤɬɨɪɨɦ ሺʹܶሻ
ߪሺߗሻൌ
ʹ
݁ଶݒߗଶන݀
ߨܶ
ൈ
Τ
ȼ ɩɪɟɞɟɥɟ ͳ߬
ଶ
بȁο
ɞɨɫɬɚɬɨɱɧɨ ɱɢɫɬɵɣ ɢ ɬɟɦɩɟɪɚɬɭɪɚ ɛɥɢɡɤɚ ɤ ܶ
ଶȀܶଶ
ȁοȁ
߬ا߬
ൌ
ଶ
ɫɬɚɧɨɜɢɬɫɹ ɜɨɡɦɨɠɧɵɦ. ȼ ɷɬɨɦ ɫɥɭɱɚɟ
ଵଵ
ʹ
݁ଶݒ߬ሺߗ߬
ߨ
ͳ
ߨܶ
݁ଶݒߗ
ߪሺߗሻൌ
, ɬɨ ɞɥɹ ɫɪɚɜɧɟɧɢɹ ɫ ɧɢɦ ɫɥɟɞɭɟɬ ɪɚɫɫɦɨɬɪɟɬɶ ɛɟɫ-
ȳاͳ, ɚ ɬɚɤ ɠɟ ɩɪɢ ɭɫɥɨɜɢɹɯ ߬ا߬
Ͷ
݁ଶݒ߬
ߨ
ͳ
՜
߬
ଶ
ଶ
ஶ
߱Ԝିଶቀ
ͳ
ͳ
ቈ
ቆ
ߗ
߬
ଶ
ஶ
ଷ
න݀
ஶ
൰න݀
ଶ
ଷ
ߗଶଶ൬
ଶ
ʹ߬ȁο
ͳሺʹ߱߬
߱
ቁቆ
ʹܶ
ʹ߬ȁο
ͳሺʹ߱߬
߱߱Ԝିଶቀ
ݔݔԜ
ͳ
൰ǡ
ߗ߬
ଶ
ଶ
ȁ
ǡ
ଶ
ሻ
ିଵିଶ
ሺ
ሻ
߱Ȁʹܶ
, ɜ ɢɬɨɝɟ ɩɨɥɭɱɚɟɦ
ଶ
ͳ
߬
ଶ
ଶ
ȁ
ቇǤ
ଶ
ሻ
, ɬɨ ɫɥɭɱɚɣ ܶ߬بͳ, ߬ȁοȁاͳ,
˔
߱
ቁଶቈ
ʹܶ
ିଶ
ሺݔሻ
ȁ
ʹ߬ȁο
ͳሺʹ߱߬
ȁοȁ
ʹߗ߬߱
ȁοȁ
ଶ
ቈ
ͺߗ߬ܶ
ଶ
ሻ
ଶ
ଶ
ଶ
ݔିଶǡሺͳ͵Ǥͷ
ଶ
ቇିଷൈ
ൌ
ሺ
ͳ͵ǤͷͶ
ሺ
ͳ͵Ǥͷͷ
ሺ
ͳ͵Ǥͷ
ିଵ
) ɜɵɪɚɠɟɧɢɟ ɞɥɹ ɩɪɨɜɨɞɢɦɨɫɬɢ ɩɨɥɭɱɚɟɬɫɹ ɨɛɨɛɳɟ-
ଶ
ߪሺߗሻൌ
ͳ
߬
ଶ
ଶ
ȁ
Ȁሺ߬ܶଶሻ ɮɨɪɦɭɥɚ ɩɟɪɟɯɨɞɢɬ ɜ (13.54), ɧɨ ɟɫɥɢ ɨɛɪɚɡɟɰ
߬
ଶ
ȁοȁ
ܶ
ଶ
ሻ
൬
ȁοȁ
ଶ
ሻ
,
ሻ
ሻ
ሻ
ሻ
149
ɭɱɢɬɵɜɚɹ, ɱɬɨ ɥɨɝɚɪɢɮɦ – ɦɟɞɥɟɧɧɨ ɦɟɧɹɸɳɚɹɫɹ ɮɭɧɤɰɢɹ, ɦɧɨɠɢɬɟɥɶ ɫ ɥɨɝɚ-
ɪɢɮɦɨɦ ɜ ɤɜɚɞɪɚɬɟ ɦɨɠɧɨ ɜɵɧɟɫɬɢ ɢɡ-ɩɨɞ ɢɧɬɟɝɪɚɥɚ, ɩɨɥɨɠɢɦ ɜ ɧɟɦ ͺݔ
ଶ
ൌͳ,
ɬɨɝɞɚ ɢɧɬɟɝɪɚɥ ɫɜɟɞɟɬɫɹ ɤ ɬɚɛɥɢɱɧɨɦɭ
ஶ
න݀
ݔݔԜ
ିଶ
ሺݔሻ
ൌ
͵ͳߨ
͵ʹήͶʹ
ǡ
ሺ
ͳ͵Ǥͷͺ
ɜ ɢɬɨɝɟ
ߪሺߗሻൌ
ͻͻʹߨ
ʹͳ
ହ
݁ଶݒ߬ሺߗ߬
ܶ
ଶ
ሻ
൬
൰ଶቈ
ȁοȁ
ȁοȁ
ߗ߬ܶ
ଶ
Ǥ
ଶ
ሺ
ͳ͵Ǥͷͻ
ɗɬɚ ɮɨɪɦɭɥɚ ɩɪɟɞɫɤɚɡɵɜɚɟɬ ɫɢɥɶɧɭɸ ɬɟɦɩɟɪɚɬɭɪɧɭɸ ɡɚɜɢɫɢɦɨɫɬɶ ɩɪɨɜɨɞɢɦɨɫɬɢ ɜ ɨɤɪɟɫɬɧɨɫɬɢ ɩɚɣɟɪɥɫɨɜɫɤɨɝɨ ɩɟɪɟɯɨɞɚ. Ⱦɟɣɫɬɜɢɬɟɥɶɧɨ, ɢɦɟɹ ɜɜɢɞɭ, ɱɬɨ
ɜ ɨɤɪɟɫɬɧɨɫɬɢ ɤɪɢɬɢɱɟɫɤɨɣ ɬɟɦɩɟɪɚɬɭɪɵ
ȁοȁ
ሺܶ
െܶሻ
ଵȀଶ
(ɮɨɪɦɭɥɚ (11.45)), ɩɨ-
ɥɭɱɢɦ
ߪሺߗሻߗ
ଶ
ሺ
ܶ
െܶ
ିଷ
ଶ
ሺ
ሾ
ܶ
െܶሻȀߗሿǡ
ሺ
ͳ͵ǤͲ
ሻ
ሻ
ሻ
ሻ
ɷɬɨ ɮɨɪɦɭɥɚ ɜɟɪɧɚ ɩɪɢ ܶ
െܶܶሺ߬Ȁ߬ଶሻ.
Ʌɢɬɟɪɚɬɭɪɚ
ɋɨɞɟɪɠɚɧɢɟ ɷɬɨɣ ɝɥɚɜɵ ɨɫɧɨɜɚɧɨ ɧɚ ɦɚɬɟɪɢɚɥɟ ɪɚɛɨɬɵ [8].
Ʉɨɧɬɪɨɥɶɧɵɟ ɜɨɩɪɨɫɵ
1. ȼɵɱɢɫɥɢɬɶ ɤɨɦɦɭɬɚɬɨɪ ɨɩɟɪɚɬɨɪɨɜ ܮ
2. ɉɪɨɞɟɥɚɬɶ ɜɵɜɨɞ ɭɪɚɜɧɟɧɢɹ (13.5).
3. Ⱦɚɬɶ ɤɚɱɟɫɬɜɟɧɧɨɟ ɨɛɴɹɫɧɟɧɢɟ ɫɢɥɶɧɨɣ ɡɚɜɢɫɢɦɨɫɬɢ ɩɪɨɜɨɞɢɦɨɫɬɢ ߪሺߗ
െܶ ɜɛɥɢɡɢ ɤɪɢɬɢɱɟɫɤɨɣ ɬɟɦɩɟɪɚɬɭɪɵ.
ɨɬ ܶ
4. Ʉɚɤ ɩɚɣɟɪɥɫɨɜɫɤɢɣ ɩɟɪɟɯɨɞ ɜɥɢɹɟɬ ɧɚ ɞɥɢɧɭ ɥɨɤɚɥɢɡɚɰɢɢ ɷɥɟɤɬɪɨɧɧɵɯ ɫɨ-
ɫɬɨɹɧɢɣ?
ା
ɢ ܮ
.
ି
ሻ
150
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