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Файл:Функции Грина и проводимость. Учебное пособие
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ɉɪɨɢɧɬɟɝɪɢɪɭɟɦ ɷɬɨ ɜɵɪɚɠɟɧɢɟ ɩɨ ߱:
ஶ
න
ିஶ
݀߱
ʹߨ
ߚ߱
th ൬
ൌʹԜ݅ԜImܩ
ோ
ሺ
߱ǡxǡxሻെܩ
൰ሾܩ
ʹ
˔
ሺ
ݐൌͲǡxǡxሻൌʹԜ݅ԜImԜ൫݅݊
ሺ
߱ǡxǡx
ሻ
ሿ
ൌʹԜ݅ԜImԜන
ሺxሻ
ஶ
ିஶ
൯ൌʹԜ݅Ԝ݊
݀߱
ʹߨ
ሺxሻ
˔
ሺ
߱ǡxǡxሻൌ
ܩ
Ǥ
ሺ
͵Ǥͺͳ
Ɍɚɤɢɦ ɨɛɪɚɡɨɦ, ɩɨɥɭɱɚɟɦ ɫɥɟɞɭɸɳɟɟ «ɩɪɚɜɢɥɨ ɫɭɦɦ»:
ஶ
݀߱
න
Ͷߨ݅
ିஶ
Ԝth ൬
ߚ߱
ʹ
൰ሾܩ
ோ
ሺ
߱ǡxǡxሻെܩ
ሺ
߱ǡxǡx
ሻ
ሿ
ൌ݊
ሺxሻ
Ǥ
ሺ
͵Ǥͺʹ
Ʌɢɬɟɪɚɬɭɪɚ
Ɉɩɪɟɞɟɥɟɧɢɟ ɜɪɟɦɟɧɧɵɯ (ɡɚɩɚɡɞɵɜɚɸɳɢɯ ɢ ɨɩɟɪɟɠɚɸɳɢɯ) ɮɭɧɤɰɢɣ Ƚɪɢɧɚ
ɩɪɢ ɤɨɧɟɱɧɵɯ ɬɟɦɩɟɪɚɬɭɪɚɯ ɦɨɠɧɨ ɧɚɣɬɢ ɜ [3], § 36 ɢ [5], § 17. ɉɪɚɜɢɥɚ ɦɚɰɭɛɚɪɨɜɫɤɨɣ ɞɢɚɝɪɚɦɦɧɨɣ ɬɟɯɧɢɤɢ ɢɡɥɨɠɟɧɵ ɜ [3], ɝɥ. IV ɢ ɜ [5], ɝɥ. III. ɋɜɹɡɶ ɦɟɠɞɭ
ɦɚɰɭɛɚɪɨɜɫɤɢɦɢ, ɡɚɩɚɡɞɵɜɚɸɳɢɦɢ ɢ ɨɩɟɪɟɠɚɸɳɢɦɢ ɮɭɧɤɰɢɹɦɢ Ƚɪɢɧɚ ɪɚɫɫɦɨɬɪɟɧɚ ɜ [3], § 37 ɢ [5], § 17.
Ʉɨɧɬɪɨɥɶɧɵɟ ɜɨɩɪɨɫɵ
1. ȼɵɩɢɫɚɬɶ ɚɧɬɢɤɨɦɦɭɬɚɰɢɨɧɧɵɟ ɫɨɨɬɧɨɲɟɧɢɹ ɞɥɹ ɮɟɪɦɢɟɜɫɤɢɯ ɨɩɟɪɚɬɨ-
ɪɨɜ ɪɨɠɞɟɧɢɹ ɢ ɭɧɢɱɬɨɠɟɧɢɹ.
2. Ʉɚɤ ɨɞɧɨɱɚɫɬɢɱɧɵɣ ɨɩɟɪɚɬɨɪ ɡɚɩɢɫɵɜɚɟɬɫɹ ɱɟɪɟɡ ɨɩɟɪɚɬɨɪɵ ɪɨɠɞɟɧɢɹ ɢ
ɭɧɢɱɬɨɠɟɧɢɹ?
3. Ʉɚɤɢɦ ɨɛɪɚɡɨɦ ɫɜɹɡɚɧɵ ɦɟɠɞɭ ɫɨɛɨɣ ɨɩɟɪɟɠɚɸɳɚɹ ɢ ɡɚɩɚɡɞɵɜɚɸɳɚɹ
ɮɭɧɤɰɢɢ Ƚɪɢɧɚ ɜ ɢɦɩɭɥɶɫɧɨ-ɱɚɫɬɨɬɧɨɦ ɩɪɟɞɫɬɚɜɥɟɧɢɢ?
4. Ʉɚɤɨɜɚ ɫɜɹɡɶ ɦɟɠɞɭ ɡɚɩɚɡɞɵɜɚɸɳɟɣ ɢ ɦɚɰɢɛɚɪɨɜɫɤɨɣ ɮɭɧɤɰɢɹɦɢ Ƚɪɢɧɚ?
5. Ʉɚɤɨɜɚ ɫɜɹɡɶ ɦɟɠɞɭ
6. ȼɵɩɢɫɚɬɶ ɫɨɨɬɧɨɲɟɧɢɹ, ɜɵɪɚɠɚɸɳɢɟ ɨɩɟɪɟɠɚɸɳɭɸ ɢ ɡɚɩɚɡɞɵɜɚɸɳɭɸ
ɮɭɧɤɰɢɢ Ƚɪɢɧɚ ɱɟɪɟɡ ɩɪɢɱɢɧɧɭɸ ɮɭɧɤɰɢɸ Ƚɪɢɧɚ.
ɨɩɟɪɟɠɚɸɳɟɣ ɢ ɦɚɰɢɛɚɪɨɜɫɤɨɣ ɮɭɧɤɰɢɹɦɢ Ƚɪɢɧɚ?
ሻ
ሻ
31
Ƚɥɚɜɚ 4
ە
ɈȻɓɂȿ ɋɈɈɌɇɈɒȿɇɂə
ȾɅə ɎɍɇɄɐɂɂ ɈɌɄɅɂɄȺ «ɌɈɄ-ɌɈɄ»
4.1. Ɏɨɪɦɭɥɚ Ʉɭɛɨ ɞɥɹ ɩɥɨɬɧɨɫɬɢ ɬɨɤɚ
ɉɭɫɬɶ ɬɟɩɟɪɶ ɧɚ ɫɢɫɬɟɦɭ ɷɥɟɤɬɪɨɧɨɜ ɧɚɥɨɠɟɧɨ ɷɥɟɤɬɪɨɦɚɝɧɢɬɧɨɟ ɩɨɥɟ ɫ ɜɟɤ-
ሺ
ɬɨɪɧɵɦ ɩɨɬɟɧɰɢɚɥɨɦ ܣ
ɜɢɞɟ
ɝɞɟ ɩɟɪɜɨɟ ɫɥɚɝɚɟɦɨɟ െ ɷɬɨ ɬɚɤ ɧɚɡɵɜɚɟɦɵɣ ɝɪɚɞɢɟɧɬɧɵɣ ɜɤɥɚɞ ɜ ɩɥɨɬɧɨɫɬɶ ɬɨɤɚ
ሺሻ
ଔƸ
ఈ
ȼɬɨɪɨɟ ɫɥɚɝɚɟɦɨɟ െ ɷɬɨ ɞɢɚɦɚɝɧɢɬɧɵɣ ɜɤɥɚɞ
Ʌɟɝɤɨ ɭɛɟɞɢɬɶɫɹ, ɱɬɨ ɜɵɪɚɠɟɧɢɟ ɞɥɹ ɩɥɨɬɧɨɫɬɢ ɬɨɤɚ (4.1) ɢɧɜɚɪɢɚɧɬɧɨ ɨɬɧɨɫɢɬɟɥɶɧɨ ɤɚɥɢɛɪɨɜɨɱɧɵɯ ɩɪɟɨɛɪɚɡɨɜɚɧɢɣ:
ɤɚɠɞɨɟ ɢɡ ɫɥɚɝɚɟɦɵɯ ɜ (4.1) ɩɪɢɨɛɪɟɬɚɟɬ ɩɪɢ ɷɬɢɯ ɩɪɟɨɛɪɚɡɨɜɚɧɢɹɯ ɪɚɜɧɵɟ, ɧɨ
ɩɪɨɬɢɜɨɩɨɥɨɠɧɵɟ ɩɨ ɡɧɚɤɭ ɞɨɛɚɜɤɢ. Ƚɚɦɢɥɶɬɨɧɢɚɧ ɷɥɟɤɬɪɨɧɨɜ ɜ ɷɥɟɤɬɪɨɦɚɝɧɢɬɧɨɟ ɩɨɥɟ ɩɨɥɭɱɚɟɬɫɹ ɢɡ (3.12) ɩɨ ɩɪɚɜɢɥɭ
ɬɚɤ ɱɬɨ ɩɨɥɭɱɚɟɦ
ܪ
ൌනܸ݀߰
ி
ɉɪɟɞɩɨɥɚɝɚɹ, ɱɬɨ ɜ ɞɚɥɶɧɟɣɲɟɦ ɛɭɞɟɬ ɪɚɫɫɦɚɬɪɢɜɚɬɶɫɹ ɫɢɫɬɟɦɚ ɷɥɟɤɬɪɨɧɨɜ ɜ
ɩɨɥɟ ɫɥɭɱɚɣɧɨ ɪɚɫɩɨɥɨɠɟɧɧɵɯ ɩɪɢɦɟɫɟɣ, ɦɵ ɜɤɥɸɱɢɥɢ ɜ ɝɚɦɢɥɶɬɨɧɢɚɧ (4.6) ɩɨ-
ɬɟɧɰɢɚɥɶɧɭɸ ɷɧɟɪɝɢɸ ܷ
ɡɚɰɢɹɦ ɩɨɬɟɧɰɢɚɥɚ ɧɚɲɚ ɫɢɫɬɟɦɚ ɧɟ ɹɜɥɹɟɬɫɹ ɩɪɨɫɬɪɚɧɫɬɜɟɧɧɨ-ɨɞɧɨɪɨɞɧɨɣ.
xǡݐሻǤ Ɉɩɟɪɚɬɨɪ ɩɥɨɬɧɨɫɬɢ ɬɨɤɚ ɷɥɟɤɬɪɨɧɨɜ ɡɚɩɢɲɟɦ ɜ
ఈ
ሺ
xǡݐሻൌ
ሺௗሻ
ଔƸ
ఈ
ሺ
xǡݐሻൌଔƸ
ଔƸ
ఈ
݅݁
ʹ݉
ሺ
xǡݐሻൌെ
ሺ
xǡݐሻ՜ܣ
ܣ
ఈ
߰
ା
߰
൫
ሺxሻ
ሺxሻ
ሺሻ
ఈ
ఈ
െ
x
݁
݉ܿ
՜݁
՜݁
ሺ
xǡݐሻଔƸ
ఈ
x
ଶ
߰
ఈ
ା
൯߰
భ
ା
ሺxሻ
߰
ሺ
xǡݐሻ
ሺxሻ
ఞ
ሺxሻ
ఞ
ି
ሺxሻ
ሺxሻ
Ԝ߰
Ԝ߰
ሺௗሻ
ఈ
߰
ሺxሻ
ା
ሺ
ሺ
ܣ
࢞
ሺxሻ
xǡݐሻǡ
x
ଵ
ሺ
ఈ
ሺxሻ
߯
ǡ
ǡ
ሻ
ǡԜԜԜx
xǡݐሻǤ
՜xǤ
ଵ
ǡ
ቀpොെ
݁
ԜAǡ
ܿ
݁
ԜAቁଶܷ
ܿ
ሺxሻ
െߤ൨߰
ሺxሻ
Ǥ
pො՜pොെ
ͳ
ା
ሺxሻ
ʹ݉
ሺxሻ
. ɗɬɨ ɩɪɢɜɨɞɢɬ ɤ ɬɨɦɭ, ɱɬɨ ɞɨ ɭɫɪɟɞɧɟɧɢɹ ɩɨ ɪɟɚɥɢ-
ሺ
ͶǤͳ
ሺ
ͶǤʹ
ሺ
ͶǤ͵
ሺ
ͶǤͶሻ
ሺͶǤͷሻ
ሺ
ͶǤ
ሻ
ሻ
ሻ
ሻ
32
Ɉɫɬɚɜɥɹɹ ɜ (4.6) ɬɨɥɶɤɨ ɥɢɧɟɣɧɨɟ ɫɥɚɝɚɟɦɨɟ, ɩɨ ܣ
ɝɚɦɢɥɶɬɨɧɢɚɧ ɦɨɠɧɨ ɩɟ-
ఈ
ɪɟɩɢɫɚɬɶ ɬɚɤ
ܪ
ൌܪെ
ி
݅݁
ʹ݉˔
ൌܪെ
නܸ݀߰
ͳ
නܸ݀ଔƸ
˔
ା
ሺ
ሻ
x
൫
ଵ
ሺሻ
ఈ
ఈ
x
భ
ሺ
xǡݐሻܣ
െ
ఈ
ఈ
൯߰
x
ሺ
xǡݐሻǤ
ሺxሻ
ܣ
ሺ
xǡݐሻൌ
ఈ
ሺ
ͶǤ
Ⱦɥɹ ɭɩɪɨɳɟɧɢɹ ɮɨɪɦɭɥ ɩɨ ɩɨɜɬɨɪɹɸɳɟɦɭɫɹ ɜɟɤɬɨɪɧɨɦɭ ɢɧɞɟɤɫɭ ɛɭɞɟɦ ɩɨɞɪɚɡɭɦɟɜɚɬɶ ɫɭɦɦɢɪɨɜɚɧɢɟ. ɂɬɚɤ, ɜ ɥɢɧɟɣɧɨɦ ɩɪɢɛɥɢɠɟɧɢɢ ɝɚɦɢɥɶɬɨɧɢɚɧ ɜɨɡɦɭ-
ɳɟɧɢɹ ɡɚɜɢɫɢɬ ɬɨɥɶɤɨ ɨɬ ଔƸ
ɦɭɥɚɦɢ (1.1), (1.2) ɢ ɮɨɪɦɭɥɨɣ Ʉɭɛɨ (1.25) ɩɪɢɦɟɧɢɦɚ ɬɨɥɶɤɨ ɤ ɫɪɟɞɧɟɦɭ ɨɬ ଔƸ
ሺሻ
, ɩɨ ɷɬɨɣ ɩɪɢɱɢɧɟ ɬɟɨɪɢɹ ɥɢɧɟɣɧɨɝɨ ɨɬɤɥɢɤɚ ɫ ɮɨɪ-
ఈ
ሺሻ
ఈ
Ɂɚɩɢɫɚɜ ɜɨɡɦɭɳɟɧɢɟ ɜ (4.7) ɜ ɜɢɞɟ
ሻ
.
ܸൌെනܸ݀ଔƸ
ሺሻ
ఈ
ሺ
xǡݐሻቆ
ͳ
ሺ
xǡݐሻቇǡ
ܣ
ఈ
ܿ
ሺ
ͶǤͺ
ɫɨɩɨɫɬɚɜɥɹɹ ɫ (1.1), ɩɨɥɭɱɢɦ
ሺሻ
ሺ
݆
xǡݐሻൌන݀
ఈ
ஶ
ݐଵන݀
ௗ
x
ଵߎఈఉ
ሺ
ݐ
ǡxǡx
ଵ
ͳ
ሻ
ଵ
ሺ
ቆ
x
ܣ
ఉ
ଵ
ܿ
ǡݐെݐ
ሻ
ቇ
ଵ
ɢɥɢ ɞɥɹ ɮɭɪɶɟ ɤɨɦɩɨɧɟɧɬ
݆
ሺሻ
ఈ
ሺ
xǡߗሻൌ
ͳ
ܿ
න݀
ௗ
x
ଵߎఈఉ
ሺ
ߗǡxǡx
ሻ
ሺ
ܣ
x
ଵ
ǡߗሻǡ
ఉ
ଵ
ሺ
ͶǤͳͲ
ɝɞɟ ɫɨɝɥɚɫɧɨ ɮɨɪɦɭɥɟ Ʉɭɛɨ
ஶ
ሺ
ߎ
ߗǡxǡx
ఈఉ
݅
ሻ
ൌ
න݀
ଵ
ݐ݁
ఆ௧
ർቂଔƸ
ሺሻ
ఈ
ሺ
xǡݐሻǡଔƸ
ఉ
ሺሻ
ሺ
x
ଵ
ǡͲሻቃ்ǡ
ሺ
ͶǤͳͳ
ɨɬɦɟɬɢɦ ɡɞɟɫɶ, ɱɬɨ ɢɡ-ɡɚ ɩɪɨɫɬɪɚɧɫɬɜɟɧɧɨɣ ɧɟɨɞɧɨɪɨɞɧɨɫɬɢ ɫɢɫɬɟɦɵ (ɷɥɟɤɬɪɨɧɵ
ɧɚɯɨɞɹɬɫɹ ɜɨ ɜɧɟɲɧɟɦ ɩɨɥɟ ܷሺxሻሻ ɮɭɧɤɰɢɹ ߎ
x െx
, ɚ ɨɬ ɤɚɠɞɨɝɨ ɚɪɝɭɦɟɧɬɚ ɜ ɨɬɞɟɥɶɧɨɫɬɢ.
ଵ
ఈఉ
ሺ
ߗǡxǡx
ሻ
ɡɚɜɢɫɢɬ ɧɟ ɨɬ ɪɚɡɧɨɫɬɢ
ଵ
ɑɬɨ ɤɚɫɚɟɬɫɹ ɜɤɥɚɞɚ ɜ ɫɪɟɞɧɢɣ ɬɨɤ ɞɢɚɦɚɝɧɢɬɧɨɝɨ ɫɥɚɝɚɟɦɨɝɨ (4.3), ɬɨ ɨɧ ɩɨɥɭɱɚɟɬɫɹ ɩɪɨɫɬɵɦ ɭɫɪɟɞɧɟɧɢɟɦ (4.3) ɩɨ ɧɟɜɨɡɦɭɳɟɧɧɨɦɭ ɫɨɫɬɨɹɧɢɸ
ሺௗሻ
݆
ఈ
ሺ
xǡݐሻൌർଔƸ
ሺௗሻ
ఈ
ሺ
xǡݐሻ
்
ൌെ
݁
݉ܿ
ଶ
ା
ሺxሻ
ൻ߰
ሺxሻ
߰
ൿ
்
ሺ
xǡݐሻൌെ
ܣ
ఈ
݁
݉ܿ
ଶ
ሺ
ሺxሻ
݊
xǡݐሻǤሺͶǤͳʹ
Ԝܣ
ఈ
ɉɪɟɞɫɬɚɜɢɜ ɷɬɨ ɜ ɜɢɞɟ
݆
ሺௗሻ
ሺ
xǡݐሻൌන݀
ఈ
ஶ
ݐଵන݀
ൈቆ
ௗ
x
ͳ
ܣ
ܿ
ଶ
݁
݉ܿ
ǡݐെݐ
ଵ
ሺxሻ
݊
ߜఈఉߜሺx െx
ሻ
ቇǡ
ଵ
൭െ
ଵ
ሺ
x
ఉ
ଵ
ሻߜሺ
ሻ
ݐ
൱ൈ
ଵ
ሺ
ͶǤͳ͵
ሻ
ሻ
ሻ
ሻ
ሻ
33
ɢɥɢ ɞɥɹ ɮɭɪɶɟ-ɤɨɦɩɨɧɟɧɬ
݆
ሺௗሻ
ሺ
xǡȳሻൌන݀x
ఈ
ଶ
݁
݉ܿ
ሺxሻߜሺ
݊
x െx
൭െ
ଵ
ͳ
ሻ
൱ቆ
ଵ
ሺ
x
ܣ
ǡȳሻቇǡ
ఉ
ଵ
ܿ
ɬɚɤɢɦ ɨɛɪɚɡɨɦ, ɞɥɹ ɩɨɥɧɨɝɨ ɬɨɤɚ ɢɦɟɟɦ
ሺ
ߗǡxǡx
ͳ
ௗ
݀
ܿ
݁
ሻ
െ
ଵ
݉
x
ଶ
ଵܳఈఉ
ሺxሻ
݊
ሺ
ȳǡxǡx
ሻ
ଵ
ߜఈఉߜሺx െx
ሺ
ܣ
x
ఉ
ଵ
ሻ
Ǥ
ଵ
ǡȳሻǡሺͶǤͳͷሻ
ɝɞɟ ܳ
݆
ఈఉ
ሺ
xǡߗሻൌ݆
ఈ
ሺ
߱ǡxǡx
ሺሻ
ሺ
xǡȳሻ݆
ఈ
ሻ
ɞɢɧɚɦɢɱɟɫɤɚɹ ɜɨɫɩɪɢɢɦɱɢɜɨɫɬɶ ɞɥɹ ɩɥɨɬɧɨɫɬɢ ɬɨɤɚ
ଵ
ሺ
ܳ
ߗǡxǡx
ఈఉ
ଵ
ሻ
ሺௗሻ
ఈ
ൌߎ
ሺ
xǡȳሻൌ
ఈఉ
Ɍɚɤɢɦ ɨɛɪɚɡɨɦ, ɡɚɞɚɱɚ ɢɫɫɥɟɞɨɜɚɧɢɹ ɩɪɨɜɨɞɢɦɨɫɬɢ ɫɢɫɬɟɦɵ ɷɥɟɤɬɪɨɧɨɜ ɫɜɨ-
ɞɢɬɫɹ ɤ ɜɵɱɢɫɥɟɧɢɸ ɮɭɧɤɰɢɢ ߎ
ఈఉ
ሺ
ߗǡxǡx
ሻ
.
ଵ
4.2. Ɇɚɰɭɛɚɪɨɜɫɤɚɹ ɜɨɫɩɪɢɢɦɱɢɜɨɫɬɶ ɞɥɹ ɬɨɤɚ
Ⱦɥɹ ɦɚɰɭɛɚɪɨɜɫɤɨɣ ɜɨɫɩɪɢɢɦɱɢɜɨɫɬɢ ɢɦɟɟɦ
ఉ
ெ
ሺ
ߗ
ߎ
ɝɞɟ ߗ
ൌʹߨ݇Ȁߚ. Ɇɚɰɭɛɚɪɨɜɫɤɚɹ ɩɥɨɬɧɨɫɬɶ ɬɨɤɚ ɩɨɥɭɱɚɟɬɫɹ ɢɡ (4.2):
ఈఉ
ଔƸ
ሺሻ
ఈ
ǡxǡx
ሺ
xǡ߬ሻൌ
ͳ
ሻ
ൌ
න݀
ଵ
݅݁
൫
ʹ݉
ఆೖఛ
߬݁
ఈ
ఈ
െ
൯߰
x
y
ർTఛଔƸ
ା
ሺ
ሺሻ
ఈ
xǡ߬ሻ߰
ሺሻ
ሺ
xǡ߬ሻǡଔƸ
ఉ
ሺ
yǡ߬ሻǡԜԜԜy ՜x
ሺ
x
ǡͲሻ்ǡ
ଵ
ሺ
ͶǤͳͶ
ሺ
ͶǤͳ
ሺ
ͶǤͳ
ሺ
ͶǤͳͺ
ሻ
ሻ
ሻ
ሻ
ɩɨɞɫɬɚɜɥɹɟɦ ɜ (4.17), ɩɨɥɭɱɚɟɦ
ெ
ሺ
ߗ
ఈఉ
ǡxǡx
ߎ
ɝɞɟ y ՜x ɢ y
࣬൫ɒǡxǡyǡ
ଶ
݁
ሻ
ൌെ
ଵ
՜xଵ. Ⱦɥɹ ࣬൫ɒǡxǡyǡ
ଵ
൫
ଶ
Ͷ݉
x
ǡyଵ൯ൌൻTఛ߰
ଵ
ఈ
x
െ
ఉ
ఈ
൯ቀ
y
x
x
ǡyଵ൯ ɢɦɟɟɦ
ଵ
ା
ሺ
െ
భ
xǡ߬ሻ߰
ఉ
ቁන݀
y
భ
ሺ
yǡ߬ሻ߰
ఉ
ఆೖఛ
߬݁
࣬൫ɒǡxǡyǡ
ା
ሺ
x
ǡͲሻ߰൫yଵǡͲ൯ൿ்Ǥ
ଵ
x
ǡyଵ൯ǡሺͶǤͳͻ
ଵ
ɗɬɨ ɫɪɟɞɧɟɟ ɞɥɹ ɱɟɬɵɪɟɯ ɩɫɢ-ɨɩɟɪɚɬɨɪɨɜ ɜɵɪɚɠɚɟɬɫɹ ɱɟɪɟɡ ɮɭɧɤɰɢɢ Ƚɪɢɧɚ:
࣬൫ɒǡxǡyǡ
ൌെܩ
x
ǡyଵ൯ൌെൻTఛ߰
ଵ
ெ
ሺ
߬ǡyǡx
ሺ
yǡ߬ሻ߰
y
ǡx൯ܩ
ଵ
ା
ሺ
x
ǡͲሻ߰൫yଵǡͲ൯߰
ଵ
ெ
ሺ
Ͳǡyǡxሻܩ
ெ
ሻ
ܩ
൫െ߬ǡ
ଵ
ெ
൫Ͳǡ
ା
ሺ
xǡ߬ሻൿ
y
ଵ
ǡxଵ൯Ǥ
ൌ
்
34
ሺ
ͶǤʹͲ
ሺ
ͶǤʹͳ
ሻ
ሻ
ሻ
ெ
ȼɬɨɪɨɟ ɫɥɚɝɚɟɦɨɟ ɧɟ ɞɚɟɬ ɜɤɥɚɞ ɜ ߎ
ெ
ߎ
ఉ
ൈන ݀
ఈఉ
ఆೖఛԜܩெ
߬݁
ሺ
ߗ
ǡxǡx
ሺ
ሻ
ଵ
߬ǡyǡx
ൌ
ଵ
, ɩɨɷɬɨɦɭ ɩɨɥɭɱɚɟɦ
ఈఉ
ଶ
݁
ఈ
൫
െ
x
ଶ
Ͷ݉
ெ
ሻ
ܩ
൫െ߬ǡyଵǡx൯ǡ ܡ՜ܠǡܡଵ՜ܠଵǡ
ఈ
൯ቀ
y
ఉ
ఉ
െ
x
y
భ
భ
ቁൈ
ሺ
ͶǤʹʹ
ɩɨɞɫɬɚɜɥɹɟɦ
ஶ
ሺ߬ሻ
ൌ
݁
ൌ
Ͷ݉
݅ߞ
݅ߗǡyǡx
௦
ଶ
ͳ
ߚ
ଶ
݁
௦ୀିஶ
ఈ
൫
x
െ
ሻ
ଵ
ିೞఛ
ܩ
ெ
ܩ
ఉ
ఈ
൯ቀ
y
x
భ
ெ
൫݅ߞ௦ǡyଵǡx൯ǡ
ሺ
ሻሺ
݅ߞ
௦
െ
ஶ
ͳ
ఉ
ቁ
Ԝ
y
భ
ߚ
௦ୀିஶ
ͶǤʹ͵
ሺ
ͶǤʹͶ
ɬɨɝɞɚ
ɡɞɟɫɶ ߞ
ெ
ܩ
ெ
ሺ
ߗ
ߎ
ఈఉ
ൌߨሺʹݏͳሻȀߚ. Ɂɚɞɚɱɚ ɬɟɩɟɪɶ ɫɨɫɬɨɢɬ ɜ ɩɨɫɬɪɨɟɧɢɢ ɜɵɪɚɠɟɧɢɹ ࣥሺߗ
௦
ǡxǡx
ሻ
ଵ
ெ
ሺ
ܩ
ɚɧɚɥɢɬɢɱɟɫɤɨɝɨ ɜ ɜɟɪɯɧɟɣ ɩɨɥɭɩɥɨɫɤɨɫɬɢ ɤɨɦɩɥɟɤɫɧɨɣ ɱɚɫɬɨɬɵ ߗ ɬɚɤɨɝɨ ɱɬɨ, ɜ
ɬɨɱɤɚɯ ɦɧɢɦɨɣ ɩɨɥɭɨɫɢ ݅ߗ
ࣥሺ݅ߗ
ɩɪɢ ߗൌʹߨ݇ȀߚǡԜ݇Ͳ ɢɦɟɥ ɦɟɫɬɨ ɪɚɜɟɧɫɬɜɨ
ஶ
ͳ
ሻ
ൌ
ߚ
௦ୀିஶ
Ԝ
ெ
ሺ
݅ߞ
ܩ
݅ߗǡyǡx
௦
ሻ
ܩ
ଵ
ெ
൫݅ߞ௦ǡyଵǡx൯Ǥ
ሺ
ͶǤʹͷ
4.3. ɉɨɫɬɪɨɟɧɢɟ ɮɭɧɤɰɢɢ ሺࢹ
ሻ
Ɋɚɫɫɦɨɬɪɢɦ ɢɧɬɟɝɪɚɥ
݀ݖ
th ൬
ߚݖ
ʹ
൰Ԝ݂ሺݖሻǡ
න
Ͷߨ݅
ሺ
ͶǤʹ
ɩɨ ɤɨɧɬɭɪɭ ܥ (ɪɢɫ. 4.1).
ሻ
ሻ
ሻ
ሻ
ሻ
ሻ
Ɋɢɫ. 4.1
35
Ɏɭɧɤɰɢɹ th
ሺ
ߚݖȀʹሻ ɢɦɟɟɬ ɩɨɥɸɫɚ ɜ ɬɨɱɤɚɯ ݖ
ResԜԜth ൬
ߚݖ
൰Ԝ݂ሺݖሻ൨
ʹ
గሺଶ௦ାଶ
௭ୀ
ఉ
ൌ݅ߨሺʹݏʹሻȀߚ, ɩɪɢ ɷɬɨɦ
௦
Ͷߨ݅
ߚ
݅ߨሺʹݏͳ
݂ቆ
ߚ
ൌ
ሻ
ሻ
ቇǡ
ɫɥɟɞɨɜɚɬɟɥɶɧɨ
ߚݖ
݀ݖ
Ԝth ൬
න
Ͷߨ݅
൰Ԝ݂ሺݖሻൌ
ʹ
ͳ
ߚ
௦ୀିஶ
ஶ
݂
݅ߨሺʹݏͳ
ቆ
ሻ
ቇǤ
ߚ
ɂɫɩɨɥɶɡɭɹ ɷɬɭ ɮɨɪɦɭɥɭ, ɧɚɩɢɲɟɦ (ɞɥɹ ɤɪɚɬɤɨɫɬɢ ɨɛɨɡɧɚɱɢɦ 1 ൌyǡx
ǡx)
2 ൌy
ଵ
ஶ
ࣥሺ݅ߗ
ൌන
ͳ
ሻ
ൌ
ߚ
௦ୀିஶ
݀ݖ
Ԝth ൬
Ͷߨ݅
Ԝ
ߚݖ
ʹ
ெ
ܩ
൰ܩ
ሺ
݅ߞ
ெ
ሺ
ݖ݅ߗ
௦
݅ߗǡ1ሻܩ
ǡ1ሻܩ
ெ
ெ
ሺ
݅ߞ
ሺ
ݖǡ2ሻǤ
ǡ2ሻൌ
௦
ሺ
ͶǤʹ
ሺ
ͶǤʹͺ
ሺ
ͶǤʹͻ
ሻ
ሻ
ɢ
ଵ
ሻ
Ɋɚɡɨɛɶɟɦ ɤɨɧɬɭɪ ܥ ɧɚ ɬɪɢ ɤɨɧɬɭɪɚ ܥ
ɩɪɢ ImԜݖ൏െߗ
ܥ
ଷ
(ɪɢɫ. 4.2).
ȼ ɨɛɥɚɫɬɢ ImԜݖͲ ɢɦɟɟɦ
ெ
ሺ
ܩ
ݖ݅ߗ
ǡ1ሻൌܩ
ோ
ሺ
ݖ݅ߗ
ɩɪɢ ImԜݖͲ, ܥଶ ɩɪɢ െߗ൏ImԜݖ൏Ͳ ɢ
ଵ
Ɋɢɫ. 4.2
ǡ1ሻǡԜԜԜܩ
ெ
ሺ
ݖǡ2ሻൌܩ
ோ
ሺ
ݖǡ2ሻǡ
ሺ
ͶǤ͵Ͳ
ሻ
36
ɜ ɨɛɥɚɫɬɢ െߗ
ܩ
ɜ ɨɛɥɚɫɬɢ ImԜݖ൏െߗ
ܩ
ɤɨɧɬɭɪ ܥ
൏ImԜݖ൏Ͳ
ெ
ሺ
ݖ݅ߗ
ெ
ሺ
ݖ݅ߗ
ɞɟɮɨɪɦɢɪɭɟɦ ɬɚɤ, ɱɬɨɛɵ ɨɧ ɩɪɢɠɢɦɚɥɫɹ ɤ ɞɟɣɫɬɜɢɬɟɥɶɧɨɣ ɨɫɢ ɫɜɟɪɯɭ
ଵ
ǡ1ሻൌܩ
ǡ1ሻൌܩ
ோ
ሺ
ሺ
ݖ݅ߗ
ݖ݅ߗ
ǡ1ሻǡԜԜܩ
ǡ1ሻǡԜԜԜܩ
ெ
ሺ
ݖǡ2ሻൌܩ
ெ
ሺ
ݖǡ2ሻൌܩ
ሺ
ݖǡ2ሻǡ
ሺ
ݖǡ2ሻǡ
ሺ
ͶǤ͵ͳ
ሺ
ͶǤ͵ʹ
(ɪɢɫ. 4.3), ɩɟɪɟɦɟɧɧɭɸ ɧɚ ɞɟɣɫɬɜɢɬɟɥɶɧɨɣ ɨɫɢ ɨɛɨɡɧɚɱɢɦ ߱, ɬɚɤ ɱɬɨ
ஶ
݀߱
ߚ߱
ܫ
ɤɨɧɬɭɪ ܥ
ோ
ሺ
ൌනǤ
భ
ǤǤൌන
ଵ
ɞɟɮɨɪɦɢɪɭɟɦ ɚɧɚɥɨɝɢɱɧɨ, ɱɬɨɛɵ ɨɧ ɩɪɢɠɢɦɚɥɫɹ ɤ ɨɫɢ ImԜݖൌെߗ
ଷ
ିஶ
Ͷߨ݅
th ൬
ʹ
൰ܩ
߱݅ߗ
ǡ1ሻܩ
ோ
ሺ
߱݅ߜǡ2ሻǡ
ሺ
ͶǤ͵͵
ɩɪɢ ɷɬɨɦ ɧɚɩɪɚɜɥɟɧɢɟ ɤɨɧɬɭɪɚ ɛɭɞɟɬ ɫɩɪɚɜɚ ɧɚɥɟɜɨ, ɩɨɷɬɨɦɭ ɩɨɥɭɱɢɦ ɞɨɩɨɥɧɢɬɟɥɶɧɵɣ ɡɧɚɤ ɦɢɧɭɫ;
ஶ
ܫ
ଷ
ɤɨɧɬɭɪ ܥ
ൌනǤ
ǤǤൌെන
య
ɞɟɮɨɪɦɢɪɭɟɬɫɹ ɜ ɞɜɚ ɤɨɧɬɭɪɚ ܥ
ଶ
ିஶ
݀߱
Ͷߨ݅
ߚሺ߱െ݅ߗ
th ቆ
ImԜݖൌͲ ɢ ɫɜɟɪɯɭ ɤ ɨɫɢ ImԜݖൌെߗ
ஶ
݀߱
Ͷߨ݅
ିஶ
݀߱
Ͷߨ݅
th ቆ
ߚሺ߱െ݅ߗ
ܫ
ܫ
ௗ
ൌනǤ
ଵ
ൌනǤ
ଵ
మ
ǤǤൌെන
ೠ
మ
ǤǤൌන
ஶ
ିஶ
th ൬
ʹ
ʹ
ߚ߱
ʹ
൰ܩ
ሻ
ሻ
ቇܩ
ோ
ቇܩ
௨
ଶ
ɢ ܥ
ሺ
߱݅ߗ
ሺ
߱െ݅ߜǡ1ሻܩ
ௗ
, ɫɨɨɬɜɟɬɫɬɜɟɧɧɨ ɫɧɢɡɭ ɤ ɨɫɢ
ଶ
ோ
ሺ
߱݅ߜǡ1ሻܩ
ሺ
߱െ݅ߗ
ሺ
߱െ݅ߜǡ2ሻǡሺͶǤ͵ͷ
ǡ1ሻܩ
ሺ
߱െ݅ߗ
ǡ2ሻǡሺͶǤ͵Ͷ
ǡ2ሻǤሺͶǤ͵
Ɍɟɩɟɪɶ ɭɱɬɟɦ
ሻ
ሻ
ሻ
,
ሻ
ሻ
ሻ
Ɋɢɫ. 4.3
37
ߚሺ߱െ݅ߗ
th ቆ
ɂ ɫɥɨɠɢɦ ɜɫɟ ɱɟɬɵɪɟ ɢɧɬɟɝɪɚɥɚ
ࣥሺ݅ߗ
ሼ
ܩ
െܩ
ோ
ሺ
߱݅ߗ
ோ
ሺ
߱݅ߗ
ǡ1ሻܩ
ǡ1ሻܩ
ோ
ሺ
߱݅ߜǡ2ሻെܩ
ሺ
߱െ݅ߜǡ2ሻܩ
ʹ
ሻ
ൌන
ሻ
ቇൌth ൬
ஶ
݀߱
Ͷߨ݅
ିஶ
ߚ߱
th ൬
ሺ
߱െ݅ߜǡ1ሻܩ
ோ
ሺ
߱݅ߜǡ1ሻܩ
ʹ
ߚ߱
ʹ
൰Ǥ
൰
ሺ
߱െ݅ߗ
ሺ
߱െ݅ߗ
ǡ2ሻെ
ǡ2
ሺ
ሻ
ሽ
ǤሺͶǤ͵ͺ
ͶǤ͵
ሻ
ሻ
Ɍɟɩɟɪɶ ɦɨɠɧɨ ɜ ɷɬɨɦ ɜɵɪɚɠɟɧɢɢ ݅ߗ
ɡɚɦɟɧɢɬɶ ɧɚ ɤɨɦɩɥɟɤɫɧɭɸ ɩɟɪɟɦɟɧɧɭɸ
ߗ c ImԜߗͲ
ஶ
݀߱
th ൬
Ͷߨ݅
ሺ
߱െ݅ߜǡ1ሻܩ
ோ
ሺ
߱݅ߜǡ1ሻܩ
ߚ߱
ʹ
൰
ሺ
߱െߗǡ2ሻെ
ሺ
߱െߗǡ2
ሻ
ሺ
ሽ
ǡ
ͶǤ͵ͻ
ሼ
െܩ
ோ
ሺ
ܩ
߱ߗǡ1ሻܩ
ோ
ሺ
߱ߗǡ1ሻܩ
ࣥሺߗሻൌන
ோ
ሺ
߱݅ߜǡ2ሻെܩ
ሺ
߱െ݅ߜǡ2ሻܩ
ିஶ
ɜɨɡɶɦɟɦ ɩɪɟɞɟɥ ɷɬɨɝɨ ɜɵɪɚɠɟɧɢɹ ɩɪɢ ImԜߗ՜Ͳ ɩɨɥɭɱɢɦ ɢɫɤɨɦɭɸ ɮɭɧɤɰɢɸ ɜɟɳɟɫɬɜɟɧɧɨɣ ɩɟɪɟɦɟɧɧɨɣ ߗ.
ߚሺ߱ߗ
ʹ
ஶ
ൌන
ିஶ
െܩ
ሻ
ɦɨɠɧɨ ɪɚɡɛɢɬɶ ɧɚ ɞɜɚ ɫɥɚɝɚɟɦɵɯ, ɢɦɟɸɳɢɯ ɪɚɡ-
ሺ
݀߱
Ͷߨ݅
ሺ
߱ǡ1ሻܩ
ሻ
ሺߗሻ
ሻ
ቇെth ቆ
ߚ߱
th ൬
ʹ
ሺ
ࣥ
ሻ
ߚሺ߱
ʹ
ோ
ሺ
൰ሺܩ
ሺ
߱െߗǡ2ሻሻǤ
ሻ
௦௧௧
ሺߗሻ
:
ோ
ሺ
߱ߗǡ1ሻܩ
ቇܩ
߱ߗǡ1ሻܩ
ሺ
߱ǡ2ሻǡሺͶǤͶͲ
ோ
ሺ
ሻ
߱ǡ2
ሺ
ͶǤͶͳ
ɉɨɥɭɱɟɧɧɭɸ ɮɭɧɤɰɢɸ ࣥሺߗ
ɧɵɣ ɮɢɡɢɱɟɫɤɢɣ ɫɦɵɫɥ ࣥሺߗሻൌࣥ
ஶ
ࣥ
ሺ
ሻ
ሺߗሻ
ൌන
ࣥ
ିஶ
ሺ
௦௧௧
݀߱
Ͷߨ݅
ሻ
ሺߗሻ
ቈth ቆ
ȼɵɪɚɡɢɦ ɝɢɩɟɪɛɨɥɢɱɟɫɤɢɟ ɬɚɧɝɟɧɫɵ ɱɟɪɟɡ ɮɟɪɦɢɟɜɫɤɭɸ ɮɭɧɤɰɢɸ ɪɚɫɩɪɟɞɟɥɟɧɢɹ, ɢɫɩɨɥɶɡɭɹ ɮɨɪɦɭɥɭ:
ሺ߱ሻ
݊
ி
ͳ
ൌ
ఉఠ
݁
ͳ
ͳ
ൌ
ʹ
൭ͳെth ൬
ߚ߱
ʹ
൰൱ǡ
ሺ
ͶǤͶʹ
ሻ
ሻ
ሻ
ሻ
ɬɨɝɞɚ (4.40) ɩɪɢ ߚߗاͳ ɦɨɠɧɨ ɡɚɩɢɫɚɬɶ ɜ ɜɢɞɟ
ஶ
ࣥ
ሺ
ሻ
ሺߗሻ
ൌെ
ൌන
ʹߨ݅
ିஶ
ߗ
݀߱
ሾ
݊
ʹߨ݅
ஶ
න݀
ିஶ
ሺ߱ሻ
ி
߱Ԝቆ
݀݊
െ݊
ሺ߱ሻ
ி
݀߱
ሺ
߱ߗ
ி
ቇԜܩ
ோ
ሺ
38
ோ
ሻ
ሺ
ሿ
ܩ
߱ߗǡ1ሻܩ
߱ߗǡ1ሻܩ
ሺ
߱ǡ2ሻǤ
ሺ
߱ǡ2ሻൌ
ሺ
ͶǤͶ͵
ሻ
ɉɨɞɫɬɚɜɥɹɹ (4.40) ɢ (4.41) ɜ ɮɨɪɦɭɥɭ (4.25), ɩɨɥɭɱɢɦ
ശ
ሬ
ஶ
ଶ
݁
ൈܩ
ൌ
ͳߨ݉
ͺߨ݉
ଶ
݁
න݀
ଶ
݅
ିஶ
ோ
ሺ
߱ߗǡyǡx
ஶ
න݀
ଶ
݅
ିஶ
߱Ԝቆ
߱th ൬
ሺ
ሻ
ሺ
ߎ
ఈఉ
ሺ
௦௧௧
ߎ
ఈఉ
ߗǡxǡx
ሻ
ሺ
ߗǡxǡx
ሻ
ଵ
ൌെߗ
ሻ
ଵ
݀݊
ଵ
݀߱
ሻ
ܩ
ሺ߱ሻ
ி
ቇ൫
൫߱ǡyଵǡx൯Ǥ
ߚ߱
൰൫
ʹ
ఈ
x
ఈ
x
െ
െ
ఈ
y
ఈ
൯ቀ
y
൯ቀ
ఉ
x
భ
ఉ
x
భ
െ
െ
ఉ
y
భ
ఉ
y
భ
ቁൈ
ቁൈ
ሺ
ͶǤͶͶ
ሻ
ோ
ሺ
ൈൣܩ
߱ߗǡyǡx
ɜ ɷɬɢɯ ɮɨɪɦɭɥɚɯ ɩɨɫɥɟ ɩɪɢɦɟɧɟɧɢɹ ɨɩɟɪɚɬɨɪɚ ɧɭɠɧɨ ɩɨɥɨɠɢɬɶ y ൌx, y
ோ
ሻ
ܩ
൫߱ǡ
ଵ
y
ଵ
ǡx൯െܩ
ሺ
߱ǡyǡx
ሻ
ଵ
ܩ
൫߱െߗǡ
y
ǡx൯൧ǡሺͶǤͶͷ
ଵ
ଵ
ൌx
ଵ
Ɍɚɤɢɦ ɨɛɪɚɡɨɦ, ɜɵɪɚɠɟɧɢɹ (4.44) ɢ (4.45) ɨɩɪɟɞɟɥɹɸɬ ɞɢɧɚɦɢɱɟɫɤɭɸ ɜɨɫɩɪɢɢɦɱɢɜɨɫɬɶ (4.16) ɞɥɹ ɩɥɨɬɧɨɫɬɢ ɬɨɤɚ ɧɟɜɡɚɢɦɨɞɟɣɫɬɜɭɸɳɢɯ ɷɥɟɤɬɪɨɧɨɜ.
4.4. «Ɍɨɠɞɟɫɬɜɚ ɍɨɪɞɚ»
ɂɫɩɨɥɶɡɭɹ ɤɚɥɢɛɪɨɜɨɱɧɭɸ ɢɧɜɚɪɢɚɧɬɧɨɫɬɶ ɬɟɨɪɢɢ, ɦɨɠɧɨ ɩɨɥɭɱɢɬɶ ɩɨɥɟɡɧɵɟ ɬɨɠɞɟɫɬɜɚ ɞɥɹ ɨɞɧɨɢɦɟɧɧɵɯ ɮɭɧɤɰɢɣ Ƚɪɢɧɚ, ɩɨɡɜɨɥɹɸɳɢɯ ɭɩɪɨɫɬɢɬɶ ɜɵɪɚɠɟɧɢɟ ɞɥɹ ɫɬɚɬɢɱɟɫɤɨɣ ɱɚɫɬɢ ɜɨɫɩɪɢɢɦɱɢɜɨɫɬɢ (4.45).
ɉɨɞ ɞɟɣɫɬɜɢɟɦ ɤɚɥɢɛɪɨɜɨɱɧɵɯ ɩɪɟɨɛɪɚɡɨɜɚɧɢɣ (4.4), ɫ ɨɞɧɨɣ ɫɬɨɪɨɧɵ, ɮɭɧɤ-
ɰɢɹ Ƚɪɢɧɚ (ɦɵ ɢɦɟɟɦ ɜ ɜɢɞɭ ɡɚɩɚɡɞɵɜɚɸɳɭɸ ɢɥɢ ɨɩɟɪɟɠɚɸɳɭɸ ɮɭɧɤɰɢɸ ɢ ɧɟ
ɛɭɞɟɦ ɷɬɨ ɨɬɦɟɱɚɬɶ) ɩɪɟɨɛɪɚɡɭɟɬɫɹ ɩɨ ɩɪɚɜɢɥɭ
ǡx
ሻ
՜݁
ଵ
ɝɞɟ ߯
ܩሺ߱ǡx
ሺxሻ
ɩɪɨɢɡɜɨɥɶɧɚɹ ɮɭɧɤɰɢɹ. ɉɪɟɞɩɨɥɚɝɚɹ, ɱɬɨ ߯
ɩɪɢɪɚɳɟɧɢɹ ɮɭɧɤɰɢɢ Ƚɪɢɧɚ ߜܩሺ߱ǡxǡx
ܩሺ߱ǡxǡx
ሻ
ɩɨɥɭɱɢɦ ɜ ɥɢɧɟɣɧɨɦ ɩɪɢɛɥɢɠɟɧɢɢ ɜɵ-
ଵ
ሻ
݁
ଵ
ሺxሻ
ఞ
ɪɚɠɟɧɢɟ
ߜܩሺ߱ǡxǡx
ሻ
ൌ
ଵ
ܿ
ሺxሻെ߯ሺ
൫߯
x
ሻ൯ܩሺ
ଵ
݅݁
ɋ ɞɪɭɝɨɣ ɫɬɨɪɨɧɵ, ɷɬɨ ɩɪɟɨɛɪɚɡɨɜɚɧɢɟ ɦɨɠɧɨ ɬɪɚɤɬɨɜɚɬɶ ɤɚɤ ɩɨɹɜɥɟɧɢɟ ɷɥɟɤ-
ɬɪɨɦɚɝɧɢɬɧɨɝɨ ɩɨɥɹ ɫ ɜɟɤɬɨɪɧɵɦ ɩɨɬɟɧɰɢɚɥɨɦ ܣ
ఈ
ɷɬɨ ɩɪɢɜɟɞɟɬ ɤ ɩɨɹɜɥɟɧɢɸ ɜ ɝɚɦɢɥɶɬɨɧɢɚɧɟ ɫɥɚɝɚɟɦɨɝɨ
ሻ
ሺ
ఞ
x
భ
ି
ǡ
ሺxሻ
ɫɤɨɥɶ ɭɝɨɞɧɨ ɦɚɥɨ, ɞɥɹ
ሻ
߱ǡxǡx
ሺ࢞ሻ
ൌ
Ǥ
ଵ
ఈ
ሺxሻ
߯
, ɫɨɝɥɚɫɧɨ (4.7)
x
ሺ
ͶǤͶ
ሺ
ͶǤͶ
ሻ
.
ሻ
ሻ
ɝɞɟ
ௗ
ା
ሺxሻ
x߰
න݀
݅݁
ఈ
ሺxሻ
ൌെ
ܸ
ʹ݉˔
ൣ
ఈ
൫
x
x
߯
ܸ
ሺxሻ
ሺxሻ
ሺxሻ
߰
൯െ൫
ሺ
ǡ
ఈ
x
ఈ
ሺxሻ
߯
൯
൧Ǥ
x
ͶǤͶͺ
ሺ
ͶǤͶͻ
ሻ
ሻ
39
ɉɨɩɪɚɜɤɚ ɤ ɮɭɧɤɰɢɢ Ƚɪɢɧɚ ɜ ɥɢɧɟɣɧɨɦ ɩɪɢɛɥɢɠɟɧɢɢ ɩɨ ܸ
ሺyሻ
ߜܩሺ߱ǡxǡx
ሻ
ଵ
ൌන݀
ௗ
y ܩሺ߱ǡxǡyሻቆ
ܸ
ቇܩሺ߱ǡyǡx
ɢɦɟɟɬ ɜɢɞ
ሻ
ൌ
ଵ
ൌ
݅݁
ʹ݉˔
න݀ௗy ቀ
ఈ
ሺyሻ
߯
ቁܩ൫߱ǡxǡyଵ൯ቀ
y
ఈ
y
െ
ఈ
ቁܩሺ߱ǡyǡx
y
భ
ሻ
ሺ
Ǥ
ଵ
ͶǤͷͲ
ɋɪɚɜɧɢɜɚɹ (4.47) ɢ (4.50), ɩɨɥɭɱɚɟɦ
ሻ൯ܩሺ
x
ଵ
ቁܩ൫߱ǡxǡyଵ൯ቀ
ൌ
ʹ݉
න݀ௗy ቀ
൫߯
ሺxሻെ߯ሺ
ఈ
ሺyሻ
߯
y
ɷɬɨ ɬɨɠɞɟɫɬɜɨ ɜɟɪɧɨ ɞɥɹ ɥɸɛɨɣ ɮɭɧɤɰɢɢ ߯
ሺxሻ
ሻ
߱ǡxǡx
ఈ
y
ଵ
െ
ൌ
, ɜɵɛɟɪɟɦ ߯
ఈ
ቁܩሺ߱ǡyǡx
y
భ
ሺxሻ
ൌݔ
ሻ
ଵ
ఉ
, ɝɞɟ ߚൌݔǡݕǡݖ,
ሺ
ǡ
ͶǤͷͳ
ɬɨɝɞɚ
ቀݔ
ఉ
െݔ
ఉ
ቁܩሺ߱ǡxǡx
ଵ
ሻ
ൌ
ଵ
න݀y ܩ൫߱ǡxǡyଵ൯ቀ
ʹ݉
ɩɪɢɦɟɧɢɜ ɤ ɩɪɚɜɨɦɭ ɢ ɥɟɜɨɦɭ ɜɵɪɚɠɟɧɢɸ ɨɩɟɪɚɬɨɪ
Ͷ݉
െ
ߜఈఉԜܩሺ߱ǡxǡx
ሻ
ൌන݀y ൫
ଵ
ఈ
x
భ
െ
ఈ
x
൯ቀ
ఉ
y
െ
ఉ
ఉ
െ
ఈ
x
െ
ቁܩሺ߱ǡyǡx
y
భ
ఈ
, ɩɨɥɭɱɢɦ
x
భ
y
ఉ
ቁܩ൫߱ǡxǡyଵ൯ܩሺ߱ǡyǡx
y
భ
ሻǡሺ
ଵ
ଵ
ሻǤሺ
ͶǤͷʹ
ͶǤͷ͵
ɗɬɨ ɢ ɟɫɬɶ ɢɫɤɨɦɨɟ ɬɨɠɞɟɫɬɜɨ, ɤɨɬɨɪɨɟ ɹɜɥɹɟɬɫɹ ɚɧɚɥɨɝɨɦ ɬɨɠɞɟɫɬɜɭ ɍɨɪɞɚ
ɜ ɤɜɚɧɬɨɜɨɣ ɷɥɟɤɬɪɨɞɢɧɚɦɢɤɟ, ɝɚɪɚɧɬɢɪɭɸɳɟɦɭ ɤɚɥɢɛɪɨɜɨɱɧɭɸ ɢɧɜɚɪɢɚɧɬɧɨɫɬɶ ɩɨɥɹɪɢɡɚɰɢɨɧɧɨɝɨ ɨɩɟɪɚɬɨɪɚ. ɉɨɞɨɛɧɨ ɬɨɠɞɟɫɬɜɭ ɍɨɪɞɚ, ɜ ɧɚɲɟɦ ɫɥɭɱɚɟ
ɪɚɜɟɧɫɬɜɚ (4.53) ɨɛɟɫɩɟɱɢɬ ɤɚɥɢɛɪɨɜɨɱɧɭɸ ɢɧɜɚɪɢɚɧɬɧɨɫɬɶ ɢ ɤɨɧɟɱɧɨɫɬɶ ɩɪɨ-
ɜɨɞɢɦɨɫɬɢ.
4.5. Ɉɛɳɚɹ ɮɨɪɦɭɥɚ ɞɥɹ ɩɪɨɜɨɞɢɦɨɫɬɢ
ɫɢɫɬɟɦɵ ɷɥɟɤɬɪɨɧɨɜ
ɉɪɨɢɧɬɟɝɪɢɪɭɟɦ ɪɚɜɟɧɫɬɜɨ (4.45) ɩɨ x
ɨɬɞɟɥɶɧɨ ɞɥɹ ɨɩɟɪɟɠɚɸɳɟɣ ɢ ɡɚɩɚɡɞɵɜɚɸɳɟɣ ɮɭɧɤɰɢɣ, ɜɵɪɚɡɢɦ ɩɪɨɢɡɜɟɞɟɧɢɹ
ோܩோ
ܩ
ˋܩܩ ɱɟɪɟɡ, ɫɨɨɬɜɟɬɫɬɜɟɧɧɨ, ܩோ ɢ ܩ
, ɜɨɫɩɨɥɶɡɨɜɚɜɲɢɫɶ ɬɨɠɞɟɫɬɜɨɦ (4.54)
ଵ
, ɩɨɥɭɱɢɦ
ሻ
ሻ
ሻ
ሻ
ሺ
ሻ
௦௧௧
ൌߜ
ఈఉ
ௗ
x
න݀
ଵ
ஶ
ଶ
݀߱
݁
݉
න
ିஶ
Ͷߨ݅
Ԝth ൬
ሺ
ఈఉ
ߚ߱
ʹ
ߗ՜Ͳǡxǡx
ோ
ሺ
߱ǡxǡxሻെܩ
൰ሾܩ
ߎ
ሻ
ൌ
ଵ
Ɍɟɩɟɪɶ ɜɨɫɩɨɥɶɡɭɟɦɫɹ ɫɨɨɬɧɨɲɟɧɢɟɦ (3.82), ɩɨɥɭɱɢɦ
ଶ
න݀
ሺ
ሻ
௦௧௧
ௗ
x
ଵ
ሺ
ఈఉ
ߗ՜Ͳǡxǡx
ߎ
ሻ
ଵ
ൌߜ
ఈఉ
݁
݉
40
ሺ
߱ǡxǡx
ሺxሻ
݊
ሻ
ሿ
Ǥ
Ǥ
ሺ
ͶǤͷͶ
ሺ
ͶǤͷͷ
ሻ
ሻ
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