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ோ
ȁ
ሺ
߱ǡkሻൌ
ܩ
ߝെ
ଷ
݀
݊
ோ
ൌ
ߑ
p
න
ଵ
ଷ
ሺʹߨሻ
ߝെଶȁଵȁଶȀʹ݉െߤെߑ
Ȼɭɞɟɦ ɢɫɤɚɬɶ ɪɟɲɟɧɢɟ ɷɬɨɝɨ ɭɪɚɜɧɟɧɢɹ ɩɪɢ ߝൌߤ
ଶ
ଶ
ȁkȁ
െߤെߑ
ʹ݉
หݑ൫p െpଵ൯ห
ଶ
ǡ
ோ
Ǥ
ோ
ଶ
ȁpȁଶȀʹ݉
ሺ
Ǥ͵Ͷ
ሺ
Ǥ͵ͷ
, ɢɦɟɧɧɨ ɩɪɢ
ሻ
ሻ
ɷɬɨɦ ɫɨɱɟɬɚɧɢɢ ɚɪɝɭɦɟɧɬɨɜ ߝ ɢ p ɮɭɧɤɰɢɢ Ƚɪɢɧɚ ɛɭɞɭɬ ɜɧɨɫɢɬɶ ɝɥɚɜɧɵɣ ɜɤɥɚɞ ɜ
ɩɪɨɜɨɞɢɦɨɫɬɶ. Ʉɪɨɦɟ ɬɨɝɨ, ɤɚɤ ɢ ɜ ɫɥɭɱɚɟ ɮɟɪɦɢ-ɝɚɡɚ, ɪɟɚɥɶɧɚɹ ɱɚɫɬɶ ɢɧɬɟɝɪɚɥɚ
ɜ (7.35) ɛɭɞɟɬ ɜɧɨɫɢɬɶ ɜɨ ɜɤɥɚɞ ɜ ɫɞɜɢɝ ɯɢɦɢɱɟɫɤɨɝɨ ɩɨɬɟɧɰɢɚɥɚ, ɢ ɞɥɹ ɧɚɫ ɧɟ
ɩɪɟɞɫɬɚɜɥɹɟɬ ɢɧɬɟɪɟɫɚ, ɩɨɷɬɨɦɭ ɫɱɢɬɚɹ, ɱɬɨ ߑ
ȁpȁ
ɫɨɜɩɚɞɚɟɬ ɫ (7.33), ɧɚɩɢɲɟɦ
்
ଷ
݀
ߑ
ோ
ൌെ
݅ߛ
ʹ
ൌെ݅ߨ
݊
ሺpሻ
ɉɟɪɟɯɨɞɹ ɜ ɫɮɟɪɢɱɟɫɤɢɟ ɤɨɨɪɞɢɧɚɬɵ ݀
ɢɧɬɟɝɪɢɪɨɜɚɧɢɟɦ ɩɨ ݀
, ɩɨɥɭɱɢɦ ɞɥɹ ɞɟɤɪɟɦɟɧɬɚ ɡɚɬɭɯɚɧɢɹ
ଵ
ሺpሻ
ߛ
ൌ
න
݊
݉
Ͷߨ
p
ଵ
หݑ൫p െpଵ൯หଶߜቆ
ଷ
ሺʹߨሻ
ଷ
ȁpȁ
න݀ߗଵหݑ൫p െpଵ൯หଶǡ
ଶଷ
اଶȁpȁଶȀʹ݉
ଶ
p
ൌ
݀ଵ݀ߗଵ ɢ ɫɧɢɦɚɹ ߜ-ɮɭɧɤɰɢɸ
ଵ
ଵ
, ɚ ɷɬɨ ɭɫɥɨɜɢɟ ɩɪɢ
ଶ
ଶ
ʹ݉
ȁpȁ
െ
ଶ
ȁȁ
ʹ݉
ଶ
ቇǤሺǤ͵
ሺ
Ǥ͵
ሻ
ሻ
ோ
ɢɦɟɹ ɜɜɢɞɭ ɮɨɪɦɭɥɭ ɞɥɹ ɩɨɥɧɨɝɨ ɫɟɱɟɧɢɹ ɜ ɛɨɪɧɨɜɫɤɨɦ ɩɪɢɛɥɢɠɟɧɢɢ
ଶ
݉
ൌ
න݀ߗଵหݑ൫p െpଵ൯หଶǡ
ଶ
Ͷߨଶ
ሺ
Ǥ͵ͺ
ሻ
ሺpሻ
ߪ
ɩɨɥɭɱɚɟɦ ɜɵɪɚɠɟɧɢɟ ɞɥɹ ɞɟɤɪɟɦɟɧɬɚ ɡɚɬɭɯɚɧɢɹ ɞɵɪɨɤ
݊
ȁpȁ
ሺpሻ
ߛ
Ⱦɥɹ ɨɩɟɪɟɠɚɸɳɟɣ ɮɭɧɤɰɢɢ Ƚɪɢɧɚ ɞɵɪɨɤ ɨɱɟɜɢɞɧɨ ɩɨɥɭɱɢɦ ߑ
ൌ
݉
ሺpሻ
Ǥ
ߪ
ோ
ൌ݅ߛ
ሺ
Ǥ͵ͻ
ሺpሻ
ሻ
Ȁʹ,
ɬɚɤ ɱɬɨ
ோǡ
ሺ
߱ǡkሻൌ
ܩ
ߝെ
ଶ
ʹ݉
ȁk
ଶ
െߤേ
݅ߛ
ሺ
Ǥ
ሺkሻ
ǤͶͲ
ሻ
ʹ
Ⱥɧɚɥɨɝɢɱɧɨɟ ɜɵɱɢɫɥɟɧɢɟ ɞɚɟɬ ɞɥɹ ɷɥɟɤɬɪɨɧɧɨɣ ɮɭɧɤɰɢɢ Ƚɪɢɧɚ ɜɵɪɚɠɟɧɢɟ
ଶ
െ߂ߤേ
՜݉ǡߪ
݅ߛ
ሺ
ǡ
ሺkሻ
ǤͶͳ
ሻ
ʹ
ሺpሻ
՜ߪ
ሺpሻ
.
ɝɞɟ ߛ
ோǡ
ሺ
ܩ
߱ǡkሻൌ
ሺkሻ
ɞɚɟɬɫɹ ɜɵɪɚɠɟɧɢɟɦ (7.39) ɫ ɡɚɦɟɧɨɣ ݉
ߝെ
ଶ
ʹ݉
ȁkȁ
71
7.4. Ɏɨɪɦɭɥɚ Ⱦɪɭɞɟ ɞɥɹ ɩɪɨɜɨɞɢɦɨɫɬɢ ɩɨɥɭɩɪɨɜɨɞɧɢɤɚ
ɉɨɫɤɨɥɶɤɭ ɨɫɧɨɜɧɵɟ ɫɨɨɬɧɨɲɟɧɢɹ ɞɥɹ ɞɢɧɚɦɢɱɟɫɤɨɣ ɜɨɫɩɪɢɢɦɱɢɜɨɫɬɢ «ɬɨɤɬɨɤ» ɛɵɥɢ ɜɵɜɟɞɟɧɵ ɜ ɝɥɚɜɟ 4 ɛɟɡ ɤɚɤɨɝɨ-ɥɢɛɨ ɩɪɟɞɩɨɥɨɠɟɧɢɹ ɨ ɫɬɪɭɤɬɭɪɟ ɨɞɧɨ-
ɱɚɫɬɢɱɧɨɝɨ ɫɩɟɤɬɪɚ, ɬɨ ɮɨɪɦɭɥɵ (4.54), (4.59), (4.65) ɛɭɞɭɬ ɜɟɪɧɵ ɢ ɞɥɹ ɪɚɫɫɦɚɬɪɢɜɚɟɦɨɝɨ ɫɥɭɱɚɹ. ɉɨ ɷɬɨɣ ɩɪɢɱɢɧɟ ɦɵ ɫɪɚɡɭ ɧɚɱɧɟɦ ɫ ɮɨɪɦɭɥɵ (5.40), ɩɪɟɧɟɛɪɟɝɚɹ ɩɪɢ ɷɬɨɦ ɞɢɚɝɪɚɦɦɚɦɢ ɫ ɩɨɩɟɪɟɱɧɵɦɢ ɩɪɢɦɟɫɧɵɦɢ ɥɢɧɢɹɦɢ. Ɋɚɫɩɪɨɫɬɪɚ
ɧɹɹ ɷɬɭ ɮɨɪɦɭɥɭ ɧɚ ɷɥɟɤɬɪɨɧɵ ɢ ɞɵɪɤɢ ɜ ɢɦɩɭɥɶɫɧɨɦ ɩɪɟɞɫɬɚɜɥɟɧɢɢ ɞɥɹ ɮɭɧɤɰɢɣ Ƚɪɢɧɚ ɢ ɭɱɢɬɵɜɚɹ ɩɟɪɟɯɨɞ ɜ ɧɢɯ ɨɬ «ɱɚɫɬɨɬ ɤ ɷɧɟɪɝɢɹɦ», ɦɵ ɩɨɥɭɱɢɦ
ଶ
݁
ߪሺߗሻൌെ
͵
ୀǡ
ൈܩ
Ɋɚɫɫɦɨɬɪɢɦ ɨɩɹɬɶ ɩɨɞɪɨɛɧɨ ɞɵɪɨɱɧɵɣ ɜɤɥɚɞ. Ɍɨɝɞɚ ɦɨɠɧɨ ɡɚɩɢɫɚɬɶ
ଶ
ൌെ
ଶ
ʹ݉
ȁpȁ
݁
͵݉
ଶ
െߤ
ൈ
ሺሻ
ሺߗሻ
ߪ
ߝߗെ
ɉɪɟɞɩɨɥɚɝɚɹ, ɱɬɨ ɜɤɥɚɞ ɜ ɢɧɬɟɝɪɚɥ ɩɨ ߝ ɩɪɨɢɫɯɨɞɢɬ ɨɬ ɨɛɥɚɫɬɢ ɲɢɪɢɧɨɣ
ሺሻ
ߛ
ɰɢɸ ݀݊
ɜ ɨɤɪɟɫɬɧɨɫɬɢ ɬɨɱɤɢ
ሺߝሻ
Ȁ݀ߝ ɦɟɞɥɟɧɧɨ ɦɟɧɹɸɳɟɣɫɹ (ɨɧɚ ɦɟɧɹɟɬɫɹ ɧɚ ɦɚɫɲɬɚɛɟ ܶ), ɩɨɥɭɱɢɦ
ி
ɞɥɹ ɢɧɬɟɝɪɚɥɚ ɩɨ ɷɧɟɪɝɢɢ ɜɵɪɚɠɟɧɢɟ
ሺ
݀݊
ߤ
ி
ቆ
ሺߝሻ
ɮɭɧɤɰɢɹ ݊
ɞɚɟɬɫɹ ɜɵɪɚɠɟɧɢɟɦ (7.4), ɱɬɨ ɜ ɩɪɟɞɩɨɥɨɠɟɧɢɢ ߤȀܶبͳ ɞɚɟɬ
ி
ଶ
ሺ
݀݊
ߤ
ி
ȁpȁଶȀʹ݉
݀ߝ
ɢ ɩɨɷɬɨɦɭ ɞɥɹ ɢɧɬɟɝɪɚɥɚ ɩɨ ɷɧɟɪɝɢɢ ɩɨɥɭɱɚɟɦ
ͳ
ିఓȀ்݁ିమȁpȁమȀଶ்
െ
݁
ܶ
ɩɨɞɫɬɚɜɥɹɟɦ ɷɬɨ ɜ (7.41)
ଶଶ
ሺሻ
ሺߗሻ
ߪ
݁
ൌ
͵݉
ିఓȀ்
݁
ଶ
ܶ
න
ஶ
ͳ
݀ߝ
න
ଶ
݉
ோ
ଶ
ʹߨ
ିஶ
ሺ
ߝߗǡpሻܩ
ଷ
݀
p
න
ଷ
ሺʹߨሻ
ሺpሻ
݅ߛ
ʹ
ଶpଶ
Ȁʹ݉ߤ, ɫɱɢɬɚɟɦ ɜ ɷɬɨɣ ɨɤɪɟɫɬɧɨɫɬɢ ɮɭɧɤ-
ଶ
ȁpȁଶȀʹ݉
݀ߝ
ሻ
ൌെ
ଷ
݀
p
ଶ݁ିమȁpȁమȀଶ்
ȁpȁ
ଷ
ሺʹߨሻ
ሺߝሻ
݀݊
ி
Ԝቆ
݀ߝ
ሺ
ߝǡpሻǤ
ஶ
݀ߝ
ଶ
ȁpȁ
න
ʹߨ
ିஶ
ଶ
ߝെ
ʹ݉
ሻ
ቇ
ߗ݅ߛ
ͳ
ିఓȀ்݁ିమȁpȁమȀଶ்
݁
ܶ
݅
ߗ݅ߛ
ቇන
Ԝቆ
ȁpȁ
݅
ሺpሻ
݀݊
ଶ
ሺpሻ
ଷ
݀
p
ሺʹߨሻ
ሺߝሻ
ி
݀ߝ
െߤെ
ǡ
ǡ
ሺpሻ
ߛ
ȁpȁ
ଷ
ቇൈ
݅ߛ
ͳ
െ݅ߗ
ଶ
ൈ
Ǥ
ሺpሻ
ʹ
Ǥ
ሺ
ǤͶʹ
ሺ
ǤͶ͵
ሺ
ǤͶͶ
ሺ
ǤͶͷ
-
ሻ
ሻ
ሻ
ሻ
72
Ⱦɚɥɟɟ ɞɥɹ ɩɪɨɫɬɨɬɵ ɪɚɫɫɦɨɬɪɢɦ ɫɥɭɱɚɣ ߗൌͲ ɢ ɭɱɬɟɦ ɫɨɨɬɧɨɲɟɧɢɟ (7.39)
ሺሻ
ሺ
ߪ
ߗൌͲሻൌ
Ʉɨɧɰɟɧɬɪɚɰɢɹ ɞɵɪɨɤ ݊
݊
݁
͵ܶ݊
ɪɚɜɧɚ
ൌ݁
ଶ
݁
ିఓȀ்
ିఓȀ்
න
න
݀
ሺʹߨሻ
ଷ
݀
ሺʹߨሻ
ଷ
p
ଷ
p
݁
ିమȁpȁమȀଶ்
݁
ଷ
ିమȁpȁమȀଶ்
ȁpȁ
Ǥ
ሺpሻ
݉
ߪ
Ǥ
ȼɜɟɞɟɦ ɭɫɪɟɞɧɟɧɢɟ ɩɨ ɦɚɤɫɜɟɥɥɨɜɫɤɨɦɭ ɪɚɫɩɪɟɞɟɥɟɧɢɸ ɞɵɪɨɤ
మ
௩
ି
ଷ
ଶ்
ݒ݁
ሺǥሻ
ǡ
మ
௩
ି
ଶ்
ۦǥۧ
்ǡ
݀
ݒ݁
ൌ
ଷ
݀
ɬɨɝɞɚ ɩɨɥɭɱɢɦ ɞɥɹ ɩɪɨɜɨɞɢɦɨɫɬɢ ɞɵɪɨɤ
ଶ
ߪ
͵ܶ݊
݁
ሺሻ
ൌ
ݒ
݊
ർ
ሺpሻ
Ԝߪ
Ǥ
்ǡ
Ⱥɧɚɥɨɝɢɱɧɨɟ ɜɵɱɢɫɥɟɧɢɟ, ɜ ɤɨɬɨɪɨɦ ɧɭɠɧɨ ɭɱɟɫɬɶ, ɱɬɨ ɱɢɫɥɨ ɷɥɟɤɬɪɨɧɨɜ ɜ
ɡɨɧɟ ɩɪɨɜɨɞɢɦɨɫɬɢ ɪɚɜɧɨ
మ
ሺ
௱ିఓ
ି
்
ൌ݁
݊
ଷ
ሻ
݀
p
න
ଷ
ሺʹߨሻ
మ
ȁpȁ
ି
ଶ்
݁
ǡ
ሺ
ǤͶ
ሺ
ǤͶ
ሺ
ǤͶͺ
ሺ
ǤͶͻ
ሺ
ǤͷͲ
ሻ
ሻ
ሻ
ሻ
ሻ
(ɩɪɢ ɷɬɨɦ ݊
), ɞɚɟɬ ɪɟɡɭɥɶɬɚɬ
ൌ݊
ଶ
݁
ሺሻ
ൌ
ߪ
͵ܶ݊
ݒ
݊
ർ
Ǥ
ሺpሻ
Ԝߪ
்ǡ
ሺ
Ǥͷͳ
ɇɚ ɩɟɪɜɵɣ ɜɡɝɥɹɞ ɮɨɪɦɭɥɵ (7.48) ɢ (7.51) ɧɟ ɫɨɜɫɟɦ ɩɨɯɨɠɢ ɧɚ ɮɨɪɦɭɥɭ
Ⱦɪɭɞɟ. ɇɨ ɥɟɝɤɨ ɭɛɟɞɢɬɶɫɹ, ɱɬɨ ɢɦ ɦɨɠɧɨ ɩɪɢɞɚɬɶ ɬɚɤɭɸ ɮɨɪɦɭ, ɟɫɥɢ ɨɩɪɟɞɟɥɢɬɶ
ɯɚɪɚɤɬɟɪɧɵɟ ɜɪɟɦɟɧɚ
݉
ൌ
߬
͵ܶ݊
ɬɨɝɞɚ ɩɨɥɧɭɸ ɩɪɨɜɨɞɢɦɨɫɬɶ ɧɚ ɧɭɥɟɜɨɣ ɱɚɫɬɨɬɟ ߪ
ݒ
ർ
ǡԜԜԜ߬ൌ
ሺሻ
Ԝߪ
்ǡ
݉
͵ܶ݊
ݒ
ർ
ሺሻ
Ԝߪ
ሺ
ሻൌߪሺሻߪሺሻ
௧௧
ሺ
்ǡ
Ǥͷʹ
ɦɨɠɧɨ ɡɚ-
ɩɢɫɚɬɶ
߬
߬
ߪ
ሺ
௧௧
ሻ
ൌ݁ଶ݊൬
݉
൰Ǥ
݉
ሺ
Ǥͷ͵
ɂɡ ɫɨɨɬɧɨɲɟɧɢɣ (7.52) ɥɟɝɤɨ ɭɜɢɞɟɬɶ ɩɪɢɜɵɱɧɨɟ ɨɩɪɟɞɟɥɟɧɢɟ ɜɪɟɦɟɧɢ ɫɜɨ-
ɛɨɞɧɨɝɨ ɩɪɨɛɟɝɚ. ɉɨɥɚɝɚɹ ɞɥɹ ɩɪɨɫɬɨɬɵ, ɱɬɨ ɫɟɱɟɧɢɟ ߪሺݒ
ɢ ɢɦɟɹ ɜ ɜɢɞɭ ݒ
ඥܶȀ݉, ɩɨɥɭɱɢɦ
்
ሻ
ɧɟ ɡɚɜɢɫɢɬ ɨɬ ɫɤɨɪɨɫɬɢ
73
ሻ
ሻ
ሻ
݉
ൌ
߬
͵ܶ݊
ർ
Ԝߪሺݒ
ሻ
ݒ
்ǡ
ͳ
ݒ
ଶ
ߪ
݊
ݒ
்
ͳ
்
ൌ
Ǥ
݊
ߪ
ݒ்
ሺ
ǤͷͶ
ɍɱɟɬ ɞɢɚɝɪɚɦɦ ɞɥɹ ɩɪɨɜɨɞɢɦɨɫɬɢ ɫ ɧɟɩɟɪɟɫɟɤɚɸɳɢɦɢɫɹ ɩɨɩɟɪɟɱɧɵɦɢ ɩɪɢɦɟɫɧɵɦɢ ɥɢɧɢɹɦɢ ɨɫɭɳɟɫɬɜɥɹɟɬɫɹ ɫɨɜɟɪɲɟɧɧɨ ɚɧɚɥɨɝɢɱɧɨ ɬɨɦɭ, ɤɚɤ ɷɬɨ ɛɵɥɨ
ɫɞɟɥɚɧɨ ɜ ɝɥɚɜɟ 5.
ɂɫɩɨɥɶɡɭɹ ɭɪɚɜɧɟɧɢɹ (5.52), (5.60), (5.61), ɥɟɝɤɨ ɭɛɟɞɢɬɶɫɹ ɜ ɬɨɦ, ɱɬɨ ɫɭɦɦɢ-
ሺݒሻ
ɪɨɜɚɧɢɟ ɷɬɢɯ ɞɢɚɝɪɚɦɦ ɩɪɢɜɨɞɢɬ ɤ ɡɚɦɟɧɟ ߪሺݒሻ՜ߪ
ሺݒሻ
(7.51), ɝɞɟ ߪ
– ɬɪɚɧɫɩɨɪɬɧɨɟ ɫɟɱɟɧɢɟ ɪɚɫɫɟɹɧɢɹ (5.50) ɢ (5.65). ȼɵɩɢɲɟɦ
௧
ɜ ɮɨɪɦɭɥɚɯ (7.49) ɢ
௧
ɨɤɨɧɱɚɬɟɥɶɧɵɣ ɨɬɜɟɬ ɞɥɹ ɩɪɨɜɨɞɢɦɨɫɬɢ ɩɨɥɭɩɪɨɜɨɞɧɢɤɚ
ଶ
ሺ
௧௧
ߪ
݁
ሻ
ൌ
͵ܶ݊
݊ൽ
ݒ
ඁ
ሺݒሻ
Ԝߪ
ǡ௧
݊ൽ
்ǡ
ݒ
ඁ
Ǥ
ሺݒሻ
Ԝߪ
ǡ௧
்ǡ
ሺ
Ǥͷͷ
7.5. Ʉɢɧɟɬɢɱɟɫɤɨɟ ɭɪɚɜɧɟɧɢɟ
ɉɨɥɭɱɢɦ ɬɟɩɟɪɶ ɜɵɪɚɠɟɧɢɟ ɞɥɹ ɩɪɨɜɨɞɢɦɨɫɬɢ ɩɨɥɭɩɪɨɜɨɞɧɢɤɚ, ɢɫɯɨɞɹ ɢɡ
ɤɢɧɟɬɢɱɟɫɤɨɝɨ ɭɪɚɜɧɟɧɢɹ Ȼɨɥɶɰɦɚɧɚ ɞɥɹ ɧɨɫɢɬɟɥɟɣ ɬɨɤɚ. Ɇɵ ɩɪɢɜɟɞɟɦ ɜɵɜɨɞ
ɞɥɹ ɨɞɧɨɝɨ ɬɢɩɚ ɧɨɫɢɬɟɥɟɣ, ɧɟ ɨɬɦɟɱɚɹ ɟɝɨ ɢɧɞɟɤɫɨɦ, ɬɚɤ ɤɚɤ ɪɟɡɭɥɶɬɚɬɵ ɫɨɜɟɪɲɟɧɧɨ ɚɧɚɥɨɝɢɱɧɵ. Ʉɢɧɟɬɢɱɟɫɤɨɟ ɭɪɚɜɧɟɧɢɟ ɨɩɢɫɵɜɚɟɬ ɷɜɨɥɸɰɢɸ ɮɭɧɤɰɢɢ ɪɚɫ-
ɩɪɟɞɟɥɟɧɢɹ ɧɨɫɢɬɟɥɟɣ ɜ ɮɚɡɨɜɨɦ ɩɪɨɫɬɪɚɧɫɬɜɟ ݊ሺݐǡpǡr
ሻ
:
ሻ
ሻ
൫μv
Fp൯݊ሺݐǡpǡrሻൌܫ
r
ሺ
ሾ݊ሿ
ǡpሻǤ
ሺ
Ǥͷ
Ɂɞɟɫɶ F ൌ݁E – ɫɢɥɚ, ɞɟɣɫɬɜɭɸɳɚɹ ɧɚ ɧɨɫɢɬɟɥɢ ɫɨ ɫɬɨɪɨɧɵ ɷɥɟɤɬɪɢɱɟɫɤɨɝɨ
ሺ
ሻ
ɩɨɥɹ, ɚ ܫ
ɧɚɥ ɨɬ ɮɭɧɤɰɢɢ ݊ሺݐǡpǡr
ሾ݊ሿ
ǡp
– ɢɧɬɟɝɪɚɥ ɫɬɨɥɤɧɨɜɟɧɢɣ ɧɨɫɢɬɟɥɟɣ ɫ ɩɪɢɦɟɫɹɦɢ, ɤɚɤ ɮɭɧɤɰɢɨ-
ሻ
ɢ ɮɭɧɤɰɢɹ ɨɬ p. Ɇɵ ɛɭɞɟɦ ɪɚɫɫɦɚɬɪɢɜɚɬɶ ɞɥɹ
ɩɪɨɫɬɨɬɵ
ɫɥɭɱɚɣ ɬɪɚɧɫɩɨɪɬɚ ɧɨɫɢɬɟɥɟɣ ɜ ɩɨɫɬɨɹɧɧɨɦ ɩɨ ɜɪɟɦɟɧɢ ɢ ɨɞɧɨɪɨɞɧɨɦ ɩɨɥɟ E,
ɩɨɷɬɨɦɭ ɮɭɧɤɰɢɹ ɪɚɫɩɪɟɞɟɥɟɧɢɹ ɛɭɞɟɬ ɡɚɜɢɫɟɬɶ ɬɨɥɶɤɨ ɨɬ ɢɦɩɭɥɶɫɚ, ɢ ɭɪɚɜɧɟɧɢɟ
ɩɪɢɦɟɬ ɜɢɞ
݁E
ሻ
ሾ
p
ሿ
ሺ
ሺ
ሻ
ሺ
ȼ ɫɥɭɱɚɟ ɭɩɪɭɝɨɝɨ ɪɚɫɫɟɹɧɢɹ ɧɚ ɩɪɢɦɟɫɹɯ ɢɧɬɟɝɪɚɥ ɫɬɨɥɤɧɨɜɟɧɢɹ ɢɦɟɟɬ ɨɱɟɧɶ
ɩɪɨɫɬɨɣ ɜɢɞ
ɝɞɟ หp
ଵ
ሺ
ሾ݊ሿ
ܫ
ǡpሻൌ݊
ȁpȁ
หൌ
ൌ݉ݒ, ݊ – ɤɨɧɰɟɧɬɪɚɰɢɹ ɩɪɢɦɟɫɟɣ, ߯ሺݒǡߴሻൌሺ݀ߪȀ݀ߗሻሺݒǡߴ
ݒන߯ሺݒǡߴሻൣ݊൫pଵ൯െ݊
ሺpሻ
൧݀ߗଵǡ
ሺ
Ǥͷͺ
ሻ
–
ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɨɟ ɫɟɱɟɧɢɟ ɪɚɫɫɟɹɧɢɹ ɧɨɫɢɬɟɥɹ ɧɚ ɩɪɢɦɟɫɢ, ɤɚɤ ɮɭɧɤɰɢɹ ɫɤɨɪɨ-
ɫɬɢ ɢ ɭɝɥɚ ɪɚɫɫɟɹɧɢɹ ߴ. Ɍɚɤɢɦ ɨɛɪɚɡɨɦ, ɤɢɧɟɬɢɱɟɫɤɨɟ ɭɪɚɜɧɟɧɢɟ ɢɦɟɟɬ ɜɢɞ
݁E
ሺpሻ
ԜԜ݊
ൌ݊ݒන߯ሺݒǡߴሻൣ݊൫pଵ൯െ݊
p
ሺpሻ
൧݀ߗଵǤ
ሺ
Ǥͷͻ
74
ሻ
ሻ
ሻ
ሻ
Ȼɭɞɟɦ ɢɫɤɚɬɶ ɪɟɲɟɧɢɟ ɷɬɨɝɨ ɭɪɚɜɧɟɧɢɹ ɜ ɜɢɞɟ
ɝɞɟ ݊
݊
ൌ݊
ሺሻ
ሺ
ȁpȁሻߜ݊
ఓ
ൌ݁
ሺpሻ
݊
ሺሻ
– ɦɚɤɫɜɟɥɥɨɜɫɤɚɹ ɮɭɧɤɰɢɹ ɪɚɫɩɪɟɞɟɥɟɧɢɹ ɫ ɬɟɦɩɟɪɚɬɭɪɨɣ ܶ
ሺpሻ
ǡ
మ
ି
ଶ்
்
݁
ǡ
ሺ
ǤͲ
ሺ
Ǥͳ
ɬɨɝɞɚ ɥɢɧɟɚɪɢɡɨɜɚɧɧɨɟ ɭɪɚɜɧɟɧɢɟ ɩɪɢɦɟɬ ɜɢɞ
ሺሻ
ൌ݊
ԜԜ݊
݁E
p
ɝɞɟ ൌȁpȁ,ݒൌȀ݉ ɬɟɩɟɪɶ
ݒන߯ሺݒǡߴሻൣߜ݊൫pଵ൯െߜ݊
ሺሻ
ൌെv݊
ԜԜ݊
p
ሺሻ
ሺpሻ
൧݀ߗଵǡ
ሺ
Ǥʹ
Ȁܶ, ɢ ɛɭɞɟɦ ɢɫɤɚɬɶ ɪɟɲɟɧɢɟ ɜ
ɜɢɞɟ
ሺሻ
݊
ߜ݊
ሺpሻ
ൌ݁Ev
݃ሺݒሻǡ
ܶ
ሺ
Ǥ͵
ɩɨɞɫɬɚɜɢɦ ɷɬɨ ɜ (7.62), ɩɨɥɭɱɢɦ
Ɂɞɟɫɶ ߠ ɢ ߠ
ሻ
െߠൌ݊
– ɭɝɥɵ ɦɟɠɞɭ ɜɟɤɬɨɪɚɦɢ E ɢ p, ɢ E ɢ pଵ ɫɨɨɬɜɟɬɫɬɜɟɧɧɨ. ɋɨɝɥɚɫɧɨ
ଵ
ݒ݃ሺݒሻන߯ሺݒǡߴ
ሾ
ߠଵെߠሿ݀ߗଵǤ
ሺ
ǤͶ
ɫɮɟɪɢɱɟɫɤɨɣ ɬɟɨɪɟɦɟ ɤɨɫɢɧɭɫɨɜ ɢɦɟɟɦ ɫɨɨɬɧɨɲɟɧɢɟ
ߠ
ൌߠߴെߠߴሺ߮െ߶ሻǡ
ଵ
ሺ
Ǥͷ
ɩɨɞɫɬɚɜɥɹɟɦ ɷɬɨ ɜɵɪɚɠɟɧɢɟ ɜ (7.64) ɢ ɭɱɢɬɵɜɚɟɦ, ɱɬɨ ɜɬɨɪɨɟ ɫɥɚɝɚɟɦɨɟ ɞɚɟɬ ɧɭ-
ɥɟɜɨɣ ɜɤɥɚɞ ɢɡ-ɡɚ ɢɧɬɟɝɪɢɪɨɜɚɧɢɹ ɩɨ ɭɝɥɭ ߶, ɬɨɝɞɚ
ሻ
ሻ
ሻ
ሻ
ሻ
ሻ
ሻ
െߠൌെ݊
ݒ݃ሺݒሻߠන߯ሺݒǡߴ
ሾ
ͳെߴሿ݀ߗଵǤ
ɍɱɢɬɵɜɚɹ ɨɩɪɟɞɟɥɟɧɢɟ ɬɪɚɧɫɩɨɪɬɧɨɝɨ ɫɟɱɟɧɢɹ ɪɚɫɫɟɹɧɢɹ
ሺݒሻ
ൌන߯ሺݒǡߴ
ߪ
௧
ɩɨɥɭɱɢɦ ɞɥɹ ɢɫɤɨɦɨɣ ɮɭɧɤɰɢɢ ݃ሺݒ
݃ሺݒሻൌ
ሻ
ሾ
ͳെߴሿ݀ߗ
ሻ
ͳ
Ǥ
ሺݒሻ
݊
ݒߪ௧
ǡ
ଵ
Ɍɟɩɟɪɶ ɜɵɱɢɫɥɢɦ ɩɪɨɜɨɞɢɦɨɫɬɶ. Ɉɱɟɜɢɞɧɨ, ɱɬɨ ɞɥɹ ɩɥɨɬɧɨɫɬɢ ɬɨɤɚ ɢɦɟɟɦ
ଷ
p
݀
ൌ݁නݒఈߜ݊
݆
ఈ
ሺpሻ
ሺʹߨሻ
ǡ
ଷ
75
ሺ
Ǥ
ሺ
Ǥ
ሺ
Ǥͺ
ሺ
Ǥͻ
ሻ
ሻ
ሻ
ሻ
ɩɨɞɫɬɚɜɥɹɟɦ (7.63) ɢ (7.68)
ଶ
݆
ఈൌܧఉ
݁
ܶ
නݒఈݒఉ݊
ሺሻ
݊
ݒߪ௧
ଷ
݀
ሺʹߨሻ
p
ൌߪఈఉܧఉǤ
ଷ
ሺ
ǤͲ
ሻ
ͳ
ሺݒሻ
ɍɫɪɟɞɧɟɧɢɟ ɩɨ ɬɟɥɟɫɧɨɦɭ ɭɝɥɭ ɞɚɟɬ ൏ݒ
ɰɢɢ ɧɨɫɢɬɟɥɟɣ ݊
ɩɨɷɬɨɦɭ ɢɡ (7.70) ɩɨɥɭɱɚɟɦ ߪ
ۦǥۧ
ɝɞɟ
– ɭɫɪɟɞɧɟɧɢɟ ɩɨ ɦɚɤɫɜɟɥɥɨɜɫɤɨɦɭ ɪɚɫɩɪɟɞɟɥɟɧɢɸ. ɗɬɨ ɜɵɪɚɠɟɧɢɟ ɜ
்
ɢɦɟɟɦ
݊
ఈఉ
ߪൌ
ൌන݊
ൌߪߜ
ଶ
݁
ܶ݊
ሺሻ
ɢ
ఈఉ
݊
ർ
͵Ԝߪ
ൌߜఈఉݒଶȀ͵, ɚ ɞɥɹ ɤɨɧɰɟɧɬɪɚ-
ఈݒఉ
ଷ
p
݀
ǡ
ଷ
ሺʹߨሻ
ݒ
்ǡ
ሺݒሻ
௧
ሺ
Ǥͳ
ሺ
Ǥʹ
ሻ
ሻ
ɬɨɱɧɨɫɬɢ ɫɨɜɩɚɞɚɟɬ ɫ ɜɵɪɚɠɟɧɢɹɦɢ (7.49) ɢ (7.51).
Ɍɚɤɢɦ ɨɛɪɚɡɨɦ, ɦɟɬɨɞ ɮɭɧɤɰɢɣ Ƚɪɢɧɚ ɩɪɢɜɨɞɢɬ ɤ ɬɨɦɭ ɠɟ ɪɟɡɭɥɶɬɚɬɭ, ɱɬɨ ɢ
ɤɢɧɟɬɢɱɟɫɤɨɟ ɭɪɚɜɧɟɧɢɟ ɢ ɜ ɫɥɭɱɚɟ ɧɟɜɵɪɨɠɞɟɧɧɵɯ ɧɨɫɢɬɟɥɟɣ ɜ ɩɨɥɭɩɪɨɜɨɞɧɢɤɟ.
Ʌɢɬɟɪɚɬɭɪɚ
ɗɥɟɦɟɧɬɚɪɧɵɟ ɫɜɟɞɟɧɢɹ ɢɡ ɮɢɡɢɤɢ ɩɨɥɭɩɪɨɜɨɞɧɢɤɨɜ ɦɨɠɧɨ ɩɨɥɭɱɢɬɶ ɢɡ
ɦɧɨɝɨɱɢɫɥɟɧɧɵɯ ɭɱɟɛɧɢɤɨɜ ɩɨ ɷɬɨɦɭ ɩɪɟɞɦɟɬɭ. ȼ ɱɚɫɬɧɨɫɬɢ ɫɨɨɬɧɨɲɟɧɢɹ (7.11)
ɢ (7.15) ɩɪɢɜɟɞɟɧɵ ɜ ɤɧɢɝɟ Ʉ. ȼ. ɒɚɥɢɦɨɜɨɣ «Ɏɢɡɢɤɚ ɩɨɥɭɩɪɨɜɨɞɧɢɤɨɜ»
Ɇ.: ɗɧɟɪɝɨɚɬɨɦɢɡɞɚɬ, 1985, ɮɨɪɦɭɥɵ (4.88), (4.89) ɫɬɪ. 110.
Ʉɨɧɬɪɨɥɶɧɵɟ ɜɨɩɪɨɫɵ
1. Ʉɚɤɨɣ ɦɚɥɵɣ ɩɚɪɚɦɟɬɪ ɩɨɡɜɨɥɹɟɬ ɩɪɟɧɟɛɪɟɱɶ ɜɤɥɚɞɨɦ ɞɢɚɝɪɚɦɦ ɫ ɩɟɪɟɫɟ-
ɱɟɧɢɟɦ ɩɪɢɦɟɫɧɵɯ ɥɢɧɢɣ ɜ ɫɥɭɱɚɟ ɷɥɟɤɬɪɨɧɨɜ ɢɥɢ ɞɵɪɨɤ, ɢɦɟɸɳɢɯ ɦɚɤɫɜɟɥɥɨɜ-
ɫɤɨɟ ɪɚɫɩɪɟɞɟɥɟɧɢɟ ɫ ɬɟɦɩɟɪɚɬɭɪɨɣ ܶ?
2. ɋɱɢɬɚɹ ɩɨɥɭɩɪɨɜɨɞɧɢɤ ɞɜɭɦɟɪɧɵɦ, ɜɵɱɢɫɥɢɬɶ ɜɤɥɚɞ ɜ ɩɪɨɜɨɞɢɦɨɫɬɶ ɜɟɟɪ-
ɧɵɯ ɞɢɚɝɪɚɦɦ. ɉɪɢɜɨɞɢɬ ɥɢ ɷɬɨɬ ɜɤɥɚɞ ɤ ɨɫɨɛɟɧɧɨɫɬɢ ɜ ɩɪɨɜɨɞɢɦɨɫɬɢ ɩɨɥɭɩɪɨɜɨɞɧɢɤɚ?
76
Ƚɥɚɜɚ 8
ɈȾɇɈɆȿɊɇɕɃ ɆȿɌȺɅɅ
ȼ ɉɈɌȿɇɐɂȺɅȿ ɉɊɂɆȿɋȿɃ.
S-ɆȺɌɊɂɐȺ ɂ ɈȻɓɂȿ ɋɈɈɌɇɈɒȿɇɂə
ȼ ɷɬɨɣ ɝɥɚɜɟ ɦɵ ɪɚɫɫɦɨɬɪɢɦ ɨɞɧɨɦɟɪɧɵɣ ɮɟɪɦɢ-ɝɚɡ. Ʉɚɤ ɭɠɟ ɨɬɦɟɱɚɥɨɫɶ ɜ
ɝɥɚɜɟ 5 ɜ ɤɨɧɰɟ ɪɚɡɞɟɥɚ 5.1, ɜ ɨɞɧɨɦɟɪɧɨɦ ɫɥɭɱɚɟ ɧɟɥɶɡɹ ɩɪɟɧɟɛɪɟɱɶ ɜɤɥɚɞɚɦɢ ɨɬ
ɞɢɚɝɪɚɦɦ ɫ ɩɟɪɟɫɟɤɚɸɳɢɦɢɫɹ ɩɪɢɦɟɫɧɵɦɢ ɥɢɧɢɹɦɢ. ɗɬɨ ɨɡɧɚɱɚɟɬ, ɱɬɨ ɜ ɨɞɧɨɦɟɪɧɨɦ ɫɥɭɱɚɟ ɢɧɬɟɪɮɟɪɟɧɰɢɨɧɧɵɟ ɷɮɮɟɤɬɵ ɫɭɳɟɫɬɜɟɧɧɵ ɢ ɨɩɪɟɞɟɥɹɸɬ ɤɚɪɬɢɧɭ ɷɥɟɤɬɪɨɧɧɵɯ ɫɨɫɬɨɹɧɢɣ. ɉɨ ɷɬɨɣ ɩɪɢɱɢɧɟ ɞɥɹ ɨɞɧɨɦɟɪɧɨɝɨ
ɷɥɟɤɬɪɨɧɧɨɝɨ
ɝɚɡɚ ɤɢɧɟɬɢɱɟɫɤɨɟ ɭɪɚɜɧɟɧɢɟ ɧɟɩɪɢɦɟɧɢɦɨ ɜɨɜɫɟ.
ɂɡ-ɡɚ ɧɟɜɨɡɦɨɠɧɨɫɬɢ ɜɵɞɟɥɢɬɶ ɩɪɨɫɬɭɸ ɩɨɫɥɟɞɨɜɚɬɟɥɶɧɨɫɬɶ ɝɥɚɜɧɵɯ ɞɢɚ-
ɝɪɚɦɦ ɢ ɩɪɨɫɭɦɦɢɪɨɜɚɬɶ ɢɯ, ɞɢɚɝɪɚɦɦɧɵɣ ɦɟɬɨɞ ɨɤɚɡɵɜɚɟɬɫɹ ɧɟɩɪɢɦɟɧɢɦ ɤ ɡɚɞɚɱɟ ɩɪɨɜɨɞɢɦɨɫɬɢ ɨɞɧɨɦɟɪɧɨɝɨ ɦɟɬɚɥɥɚ. ɇɟɨɛɯɨɞɢɦɨ ɢɫɩɨɥɶɡɨɜɚɬɶ ɞɪɭɝɨɣ ɩɨɞ-
ɯɨɞ ɤ ɷɬɨɣ ɩɪɨɛɥɟɦɟ. Ɇɵ ɪɚɫɫɦɨɬɪɢɦ ɡɞɟɫɶ ɬɚɤ ɧɚɡɵɜɚɟɦɵɣ «ɦɟɬɨɞ ܵ-ɦɚɬɪɢɰɵ»,
ɪɚɡɪɚɛɨɬɚɧɧɵɣ Ⱥɛɪɢɤɨɫɨɜɵɦ ɢ Ɋɵɠɤɢɧɵɦ [5].
ɗɬɨɬ ɦɟɬɨɞ ɩɨɡɜɨɥɹɟɬ ɩɪɨɢɡɜɟɫɬɢ
ɬɨɱɧɨɟ ɭɫɪɟɞɧɟɧɢɟ ɩɨ ɛɟɫɩɨɪɹɞɤɭ ɜ ɨɞɧɨɦɟɪɧɨɦ ɫɥɭɱɚɟ, ɚ ɬɚɤɠɟ ɩɨɡɜɨɥɹɟɬ ɪɚɫɩɪɨɫɬɪɚɧɢɬɶ ɷɬɨɬ ɩɨɞɯɨɞ ɧɚ ɫɥɭɱɚɣ ɤɜɚɡɢɨɞɧɨɦɟɪɧɵɯ ɫɢɫɬɟɦ.
8.1. Ɏɭɧɤɰɢɢ Ƚɪɢɧɚ ɢ ɫɥɭɱɚɣɧɵɟ ɩɨɥɹ
Ɋɚɫɫɦɨɬɪɢɦ ɫɢɫɬɟɦɭ ɧɟɜɡɚɢɦɨɞɟɣɫɬɜɭɸɳɢɯ ɷɥɟɤɬɪɨɧɨɜ ɫ ɷɧɟɪɝɟɬɢɱɟɫɤɢɦ
ɫɩɟɤɬɪɨɦ ߝሺሻൌ
ɢɦɩɭɥɶɫ Ɏɟɪɦɢ ɡɚɜɢɫɢɬ ɨɬ ɤɨɧɰɟɧɬɪɚɰɢɢ ɷɥɟɤɬɪɨɧɨɜ ɫɥɟɞɭɸɳɢɦ ɨɛɪɚɡɨɦ:
(ɦɵ ɛɭɞɟɦ ɪɚɫɫɦɚɬɪɢɜɚɬɶ ɷɥɟɤɬɪɨɧɵ ɫ ɨɞɧɨɣ ɩɪɨɟɤɰɢɟɣ ɫɩɢɧɚ, ɤɪɨɦɟ ɬɨɝɨ ɜ ɷɬɨɣ
ɝɥɚɜɟ ɦɵ ɛɭɞɟɦ ɩɨɥɶɡɨɜɚɬɶɫɹ ɫɢɫɬɟɦɨɣ ɟɞɢɧɢɰ, ɜ ɤɨɬɨɪɨɣ ൌͳ, ɩɪɢ ɷɬɨɦ ɢɦ-
ɩɭɥɶɫɵ ɛɭɞɭɬ ɫɨɜɩɚɞɚɬɶ ɫ ɜɨɥɧɨɜɵɦɢ ɱɢɫɥɚɦɢ, ɚ ɷɧɟɪɝɢɢ ɫ ɱɚɫɬɨɬɚɦɢ).
ɗɥɟɤɬɪɨɧɵ ɪɚɫɫɟɢɜɚɸɬɫɹ ɧɚ ɫɥɭɱɚɣɧɨ ɪɚɫɩɨɥɨɠɟɧɧɵɯ ɩɪɢɦɟɫɹɯ, ɢɦɟɸɳɢɯ
ɤɨɧɰɟɧɬɪɚɰɢɸ ݊
ɨɩɪɟɞɟɥɹɬɶɫɹ ɷɥɟɤɬɪɨɧɚɦɢ, ɢɦɟɸɳɢɦɢ ɢɦɩɭɥɶɫɵ ɜ ɧɟɩɨɫɪɟɞɫɬɜɟɧɧɨɣ ɛɥɢɡɨɫɬɢ
ɨɬ ൌ
, ɦɨɠɟɬ ɨɫɬɚɜɢɬɶ ɟɝɨ ɜ ɷɬɨɣ ɠɟ ɨɤɪɟɫɬɧɨɫɬɢ ɢɥɢ ɩɟɪɟɛɪɨɫɢɬɶ ɟɝɨ ɜ ɨɤɪɟɫɬ-
ɧɨɫɬɶ െ
ɜɬɨɪɚɹ – ݑ
ɢ ˓ൌെ
. ɉɟɪɜɵɣ ɩɪɨɰɟɫɫ ɨɩɢɫɵɜɚɟɬɫɹ ɛɨɪɧɨɜɫɤɨɣ ɚɦɩɥɢɬɭɞɨɣ ݑଵൌݑොሺͲ
ଶ
Ⱦɟɣɫɬɜɢɬɟɥɶɧɨ ɪɚɫɫɦɨɬɪɢɦ ɞɥɹ ɩɪɢɦɟɪɚ ɡɚɩɚɡɞɵɜɚɸɳɭɸ ɮɭɧɤɰɢɸ Ƚɪɢɧɚ
ɭɫɪɟɞɧɟɧɧɭɸ ɩɨ ɩɪɢɦɟɫɹɦ
. ȿɫɥɢ ݊
ൌݑሺʹ
ଶ
Ȁʹ݉. ȼ ɪɚɜɧɨɜɟɫɧɨɦ ɫɨɫɬɨɹɧɢɢ ɷɥɟɤɬɪɨɧɵ ɜɵɪɨɠɞɟɧɵ, ɢ ɢɯ
ʹ
ൌ
݊
ʹߨ
ɧɟ ɫɥɢɲɤɨɦ ɜɟɥɢɤɨ, ɜɫɟ ɮɢɡɢɱɟɫɤɢɟ ɹɜɥɟɧɢɹ ɛɭɞɭɬ
. ɉɪɨɰɟɫɫ ɪɚɫɫɟɹɧɢɹ ɷɥɟɤɬɪɨɧɚ ɫ ɢɦɩɭɥɶɫɨɦ, ɛɥɢɡɤɢɦ ɞɨ
ሻ
, ɝɞɟ ݑሺ
ܩ
ሻ
ɮɭɪɶɟ-ɤɨɦɩɨɧɟɧɬɚ ɩɪɢɦɟɫɧɨɝɨ ɩɨɬɟɧɰɢɚɥɚ.
ʹ݉
ͳ
ଶ
ଶ
ʹ݉
െߑ
Ǥ
ோ
ோ
ሺ
߱ǡሻൌ
߱െ
ሺ
ሺ
ͺǤͳ
ͺǤʹ
77
ሻ
, ɚ
ሻ
ሻ
ோ
ɜ ɩɟɪɜɨɦ ɩɪɢɛɥɢɠɟɧɢɢ ɩɨ ɩɪɢɦɟɫɧɨɣ ɥɢɧɢɢ ɢɦɟɟɦ ɜɵɪɚɠɟɧɢɟ (5.18),
Ⱦɥɹ ߑ
ɪɢɫ. 8.1
ଶ
ோ
ߑ
ൌ݊න
݀
ʹߨ
ଵ
߱െ
ȁ
ݑሺെ
ʹ݉
ሻ
ȁ
ଵ
ଶ
ଶ
݅ߜ
ʹ݉
ൌߑ
ோ
ଵ
ߑ
ோ
ǡ
ଶ
ሺ
ͺǤ͵
Ɋɢɫ. 8.1
ɝɞɟ
ଶ
ሻ
ȁ
ȁ
ோ
ൌ݊න
ߑ
ଵ
ோ
ߑ
ൌ݊න
ଶ
ோ
ɋɥɚɝɚɟɦɨɟ ߑ
ɠɟ ɨɤɪɟɫɬɧɨɫɬɢ
ଶ
െ
Ȁʹ݉ൎݒ݇ǡԜԜԜݒൌȀ݉. ȼɬɨɪɨɟ ɫɥɚɝɚɟɦɨɟ ߑ
ɫɨɨɬɜɟɬɫɬɜɭɟɬ ɪɚɫɫɟɹɧɢɸ ɜɩɟɪɟɞ, ɤɨɝɞɚ ɷɥɟɤɬɪɨɧ ɨɫɬɚɟɬɫɹ ɜ ɬɨɣ
ଵ
. ȼ ɷɬɨɦ ɫɥɭɱɚɟ ଵൌ݇, ɩɪɢ ɷɬɨɦ ȁ݇ȁا ɢ ଶȀʹ݉െ
ݑሺͲ
߱െݒ݇݅ߜ
ȁ
ݑሺʹ
߱ݒ݇݅ߜ
ɧɚɡɚɞ, ɤɨɝɞɚ ɷɥɟɤɬɪɨɧ ɩɟɪɟɯɨɞɢɬ ɜ ɨɤɪɟɫɬɧɨɫɬɶ
ȁ݇ȁ
ا
ɢ ଶȀʹ݉െ
Ɂɞɟɫɶ ɦɵ ɜɜɟɥɢ ɯɚɪɚɤɬɟɪɧɵɟ ɜɪɟɦɟɧɚ ߬
ɉɭɫɬɶ ܽො
ା
, ɢ ܽො
ଶ
Ȁʹ݉ൎെݒ݇. ɂɧɬɟɝɪɢɪɭɹ ɜ (8.4) ɢ (8.5), ɩɨɥɭɱɢɦ
ଶ
ȁ
ȁ
݊
ݑ
ோ
ൌെ݅
ߑ
ଵ
ோ
ߑ
ൌെ݅
ଶ
, ɮɟɪɦɢɟɜɫɤɢɟ ɨɩɟɪɚɬɨɪɵ ɪɨɠɞɟɧɢɹ ɢ ɭɧɢɱɬɨɠɟɧɢɹ ɷɥɟɤɬɪɨɧɚ
ଵ
ʹݒ
ଶ
ȁ
ȁ
݊
ݑ
ଶ
ʹݒ
ଵ
ଶ
ሻ
ȁ
ൌെ݅
ൌെ݅
ɢ ߬
ଶ
݀݇
ǡ
ʹߨ
݀݇
Ǥ
ʹߨ
ோ
ɫɨɨɬɜɟɬɫɬɜɭɟɬ ɪɚɫɫɟɹɧɢɸ
ଶ
, ɜ ɷɬɨɦ ɫɥɭɱɚɟ ଵൌെ݇,
ͳ
ǡ
ʹ߬
ଵ
ͳ
Ǥ
ʹ߬
ଶ
.
ሺ
ሺ
ሺ
ሺ
ͺǤͶ
ͺǤͷ
ͺǤ
ͺǤ
ɜ ɫɨɫɬɨɹɧɢɢ ɫ ɢɦɩɭɥɶɫɨɦ . Ɍɚɤ ɤɚɤ ɫɭɳɟɫɬɜɟɧɧɵɟ ɷɥɟɤɬɪɨɧɧɵɟ ɫɨɫɬɨɹɧɢɹ ɥɟɠɚɬ
ɢ െ
ɜ ɨɤɪɟɫɬɧɨɫɬɹɯ
, ɬɨ ɝɚɦɢɥɶɬɨɧɢɚɧ, ɭɱɢɬɵɜɚɸɳɢɣ ɬɨɥɶɤɨ ɷɬɢ ɫɨɫɬɨɹɧɢɹ,
ɦɨɠɧɨ ɡɚɩɢɫɚɬɶ ɬɚɤ
ሻ
ሻ
ሻ
ሻ
ሻ
ൌൣݒ݇Ԝܽො
ܪ
ȁȁ
ا
ା
ܽො
బାെݒ
బା
బ
݇Ԝܽො
ା
ିబା
(ɦɵ ɤɚɤ ɨɛɵɱɧɨ ɜɵɱɥɢ ɢɡ ɝɚɦɢɥɶɬɨɧɢɚɧɚ ɨɩɟɪɚɬɨɪ ߤܰ
ଶ
ɜɵɪɨɠɞɟɧɧɨɝɨ ɷɥɟɤɬɪɨɧɧɨɝɨ ɝɚɡɚ ߤൌ
Ȁʹ݉). ȼɜɟɞɟɦ ɬɟɩɟɪɶ ɨɩɟɪɚɬɨɪɵ ɭɧɢɱɬɨ-
78
ܽො
൧
ǡ
ିబା
ɭɱɬɹ ɩɪɢ ɷɬɨɦ, ɱɬɨ ɞɥɹ
ሺ
ͺǤͺ
ሻ
ɠɟɧɢɹ ɜ ɤɨɨɪɞɢɧɚɬɧɨɦ ɩɪɟɞɫɬɚɜɥɟɧɢɢ ɷɥɟɤɬɪɨɧɨɜ ɜ ɫɨɫɬɨɹɧɢɹɯ, ɛɥɢɡɤɢɯ ɤ
(ɜ ɷɬɨɣ ɝɥɚɜɟ ɤɨɨɪɞɢɧɚɬɭ ɷɥɟɤɬɪɨɧɚ ɛɭɞɟɦ ɨɛɨɡɧɚɱɚɬɶ ݖ)
ɢ െ
߰
ଵ
ሺݖሻ
ͳ
ൌ
ξ
ܮ
݁
௭
ܽො
బା
ǡ᩷᩷᩷᩷߰
ଶ
ሺݖሻ
ͳ
ൌ
ξ
ܮ
݁
௭
ܽො
ିబା
ǡሺͺǤͻ
ɢ ɫɨɨɬɜɟɬɫɬɜɭɸɳɢɟ ɨɩɟɪɚɬɨɪɵ ɪɨɠɞɟɧɢɹ
߰
ሺݖሻ
ͳ
ൌ
݁
ܮ
ξ
ା
ଵ
௭
ܽො
ା
బା
ǡ᩷᩷᩷᩷߰
ା
ଶ
ሺݖሻ
ͳ
ൌ
݁
ܮ
ξ
௭
ା
ܽො
ିబା
ǡሺͺǤͳͲ
ɬɨɝɞɚ ɝɚɦɢɥɶɬɨɧɢɚɧ (8.8) ɜ ɬɟɪɦɢɧɚɯ ɨɩɟɪɚɬɨɪɨɜ (8.9), (8.10) ɡɚɩɢɲɟɬɫɹ ɜ ɜɢɞɟ
ܪ
ൌ݀ݖ߰
ା
ሺݖሻ
ଵ
൬െ݅ݒ
μ
ሺݖሻ
൰߰
μݖ
߰
ଵ
ȼɜɟɞɟɦ ɬɟɩɟɪɶ ɞɜɭɯɤɨɦɩɨɧɟɧɬɧɵɟ ɨɩɟɪɚɬɨɪɵ ߰
ሺݖሻ
߰
ൌ൬
ା
ଵ
߰
ሺݖሻ
ଵ
ଶ
ሺݖሻ
ǡԜ߰
൰ǡ
ା
ଶ
ሺݖሻ
ሺݖሻ
߰
ା
ሺݖሻ
߰
ൌ൫߰
ା
ሺݖሻ
ଶ
ሺݖሻ
൯ǡ
൬݅ݒ
ɢ ߰
μ
ሺݖሻ
ଶ
൨ǤሺͺǤͳͳ
ሺ
ͺǤͳʹ
ሺ
ͺǤͳ͵
൰߰
μݖ
ା
ሺݖሻ
ɬɨɝɞɚ ɨɩɟɪɚɬɨɪ (8.11) ɦɨɠɧɨ ɡɚɩɢɫɚɬɶ ɜ ɜɢɞɟ
ܪ
ൌන݀ݖ߰
ା
ሺݖሻ
൬െ݅ݒ
ߪොଷ
μݖ
μ
൰߰
ሺݖሻ
൨ǡ
ሺ
ͺǤͳͶ
ɝɞɟ
ߪො
ଷ
ͳͲ
ൌቀ
Ͳെͳ
ቁǤ
ሺ
ͺǤͳͷ
ȼɦɟɫɬɨ ɜɡɚɢɦɨɞɟɣɫɬɜɢɹ ɫ ɩɪɢɦɟɫɹɦɢ ɦɵ ɜɜɟɞɟɦ ɫɥɭɱɚɣɧɵɟ ɩɨɥɹ ߟሺݖሻ ɢ ߞሺݖ
ȼɡɚɢɦɨɞɟɣɫɬɜɢɟ ɫ ɩɨɥɟɦ ߟሺݖሻ ɨɫɬɚɜɥɹɟɬ ɷɥɟɤɬɪɨɧ ɜ ɨɤɪɟɫɬɧɨɫɬɢ
, ɬɨ ɟɫɬɶ ɷɬɨ
ɜɡɚɢɦɨɞɟɣɫɬɜɢɟ ɞɢɚɝɨɧɚɥɶɧɨ ɩɨ ɢɧɞɟɤɫɚɦ ͳ ɢ ʹ ɢ, ɫɥɟɞɨɜɚɬɟɥɶɧɨ, ɢɦɟɟɬ ɜɢɞ
ା
߰
ሺݖሻߟሺݖሻ
ሺݖሻ
߰
Ǥ
ሺ
ͺǤͳ
ሻ
ሻ
ሻ
ሻ
ሻ
ሻ
ሻ
ሻ
.
ሻ
ɉɨɥɟ ߟሺݖ
ɝɞɟ ɫɤɨɛɤɢ
ȼɬɨɪɨɟ ɩɨɥɟ ߞሺݖሻ ɩɟɪɟɤɢɞɵɜɚɟɬ ɷɥɟɤɬɪɨɧ ɢɡ ɨɤɪɟɫɬɧɨɫɬɢ
. ɗɬɨ ɩɨɥɟ ɫ ɧɟɨɛɯɨɞɢɦɨɫɬɶɸ ɞɨɥɠɧɨ ɛɵɬɶ ɤɨɦɩɥɟɤɫɧɨ, ɬɚɤ ɤɚɤ «ɤɜɚɧɬ»
െ
ሻ
ɜɟɳɟɫɬɜɟɧɧɨ ɢ ɢɦɟɟɬ ɫɥɟɞɭɸɳɢɟ ɫɬɚɬɢɫɬɢɱɟɫɤɢɟ ɫɜɨɣɫɬɜɚ
ݒ
ൌ
Ԝߜሺݖെݖ
߬
ଵ
ᇱ
ሻ
ǡ
ɜ ɨɤɪɟɫɬɧɨɫɬɶ
ᇱ
ሻ
ۧ
ۦߟሺݖ
ۦ
ۧ
ǤǤǤ
ɨɡɧɚɱɚɸɬ ɭɫɪɟɞɧɟɧɢɟ ɩɨ ɪɟɚɥɢɡɚɰɢɢ ɩɨɥɹ.
ൌͲǡ᩷ۦߟሺݖሻߟሺݖ
ሻ
ۧ
ሺ
ͺǤͳ
ɷɬɨɝɨ ɩɨɥɹ, «ɪɨɞɢɜɲɢɣɫɹ» ɩɪɢ ɩɟɪɟɯɨɞɟ ͳ՜ʹ, ɞɨɥɠɟɧ ɛɵɬɶ «ɩɨɝɥɨɳɟɧ» ɩɪɢ
ɨɛɪɚɬɧɨɦ ɩɟɪɟɯɨɞɟ. ɗɬɨ ɩɪɟɞɫɬɚɜɥɟɧɢɟ ɫɨɨɬɜɟɬɫɬɜɭɟɬ ɬɨɦɭ ɮɚɤɬɭ, ɤɨɝɞɚ ɦɵ ɪɚɫɫɦɚɬɪɢɜɚɟɦ ɪɟɚɥɶɧɨɟ ɪɚɫɫɟɹɧɢɟ, ɬɨ ɢɦɩɭɥɶɫɵ, ɩɟɪɟɞɚɧɧɵɟ ɜ ɞɜɭɯ ɤɪɟɫɬɚɯ ɧɚ
ɪɢɫɭɧɤɟ 7.1 ɞɨɥɠɧɵ ɜ ɫɭɦɦɟ ɞɚɬɶ ɧɨɥɶ. ȼɡɚɢɦɨɞɟɣɫɬɜɢɟ ɫ
ɷɬɢɦ ɩɨɥɟɦ ɢɦɟɟɬ
79
ሻ
ା
ሺݖሻ൫ߞሺݖሻ
߰
ɝɞɟ ߪ
ሺേሻ
ൌሺߪଵേ݅ߪ
ሻ
Ȁʹ, ɬɨ ɟɫɬɶ
ଶ
ሺାሻ
ߪො
ൌቀ
ɉɪɚɜɢɥɨ ɭɫɪɟɞɧɟɧɢɹ ɞɥɹ ɩɨɥɹ ߞሺݖ
ሻ
ۧ
ۦߞሺݖ
ൌͲǡ᩷ۦߞሺݖሻߞ
Ͳͳ
ͲͲ
כ
ᇱ
ሺ
ݖ
ሺାሻ
ߪො
ቁǡ᩷᩷ߪො
ሻ
ۧ
ߞ
ሻ
ɢɦɟɟɬ ɜɢɞ
ݒ
ൌ
߬
ଶ
כ
ሺݖሻ
ሺିሻ
Ԝߜሺݖെݖ
ߪො
ൌቀ
ሺିሻ
ᇱ
൯߰
ͲͲ
ͳͲ
ሻ
ሺݖሻ
ǡ
ቁǤ
ǡ᩷ۦߞሺݖሻߞሺݖ
Ɇɵ ɛɭɞɟɦ ɩɪɟɞɩɨɥɚɝɚɬɶ ɬɚɤɠɟ, ɱɬɨ ɩɨɥɹ ߟሺݖሻ ɢ ߞሺݖ
(8.17) ɢ (8.20) ɫɨɨɬɜɟɬɫɬɜɭɸɬ ɭɫɪɟɞɧɟɧɢɸ ɩɨ ɮɭɧɤɰɢɨɧɚɥɭ
ሺ
ͺǤͳͺ
ሺ
ͺǤͳͻ
ᇱ
ሻ
ۧ
ൌͲǤሺͺǤʹͲ
ሻ
ɝɚɭɫɫɨɜɵ. ɍɪɚɜɧɟɧɢɹ
ሻ
ሻ
ሻ
ߗሺߟǡߞሻൌെ൭
߬
ଵ
ߟ
ݒ
߬
ሺݖሻ
ଶ
ଶ
ݒ
൱݀ݖ൩Ǥ
ሺݖሻ
ȁߞȁ
ሺ
ͺǤʹͳ
ଶ
ȼ ɢɬɨɝɟ ɝɚɦɢɥɶɬɨɧɢɚɧ ɷɥɟɤɬɪɨɧɨɜ ɜ ɫɥɭɱɚɣɧɵɯ ɩɨɥɹɯ, ɦɨɞɟɥɢɪɭɸɳɢɯ ɜɡɚɢ-
ɦɨɞɟɣɫɬɜɢɟ ɷɥɟɤɬɪɨɧɚ ɫ ɩɪɢɦɟɫɹɦɢ, ɩɪɢɦɟɬ ɜɢɞ
ܪൌ݀ݖ߰
ା
ሺݖሻ
൬െ݅ݒ
μ
ߟሺݖሻߞሺݖሻԜߪො
ߪොଷ
μݖ
ሺାሻ
ߞ
כ
ሺݖሻ
ሺିሻ
൰߰
ሺݖሻ൨Ǥሺ
ͺǤʹʹ
Ԝߪො
Ɉɩɪɟɞɟɥɢɦ ɬɟɩɟɪɶ ɨɛɵɱɧɵɦ ɨɛɪɚɡɨɦ ɨɩɟɪɟɠɚɸɳɭɸ ɢ ɡɚɩɚɡɞɵɜɚɸɳɭɸ ɮɭɧɤ-
ɰɢɢ Ƚɪɢɧɚ. ɗɬɨ ɛɭɞɭɬ ɦɚɬɪɢɰɵ ɩɨ ɢɧɞɟɤɫɚɦ ͳ ɢ ʹ
ஶ
ൌ݅න ݀
ିஶ
ݐԜ݁
ݐԜ݁
ఠ௧
ఠ௧
Ԝൻ߰
Ԝൻ߰
ఓ
ఓ
ሺ
ݐǡݖሻ߰
ሺ
ݐǡݖሻ߰
ା
ሺ
Ͳǡݖ
ఔ
ା
ሺ
Ͳǡݖ
ఔ
ᇱ
ᇱ
ሻ
ሻ
߰
߰
ା
ᇱ
ఔ
ା
ఔ
ሺ
ሺ
Ͳǡݖ
Ͳǡݖ
ሻ
ሺ
߰
ݐǡݖሻൿ
ݐǡݖሻൿ
ǡሺͺǤʹ͵
்
ǤሺͺǤʹͶ
்
ఓ
ᇱ
ሻ
ሺ
߰
ఓ
ൣܩ
ൣܩ
ோ
ᇱ
ሺ
ሻ
ݖǡݖ
൧
ఠ
ሺ
ݖǡݖ
ఠ
ൌെ݅න݀
ఓఔ
ᇱ
ሻ
൧
ఓఔ
Ʌɟɝɤɨ ɭɛɟɞɢɬɶɫɹ, ɱɬɨ ɨɛɟ ɷɬɢ ɮɭɧɤɰɢɢ ɭɞɨɜɥɟɬɜɨɪɹɸɬ ɭɪɚɜɧɟɧɢɸ
߱݅Ԝݒ
μ
Ԝߪො
ଷ
μݖ
െߟሺݖሻെߞሺݖሻԜߪො
ሺାሻ
െߞ
כ
ሺݖሻ
ሺିሻ
ோ
ᇱ
ሺ
Ԝߪො
൨ܩ
ሻൌߜሺ
ݖǡݖ
ఠ
ݖെݖ
ᇱ
ሻǤሺ
ͺǤʹͷ
ɉɨɥɨɠɢɜ ߟൌߞൌͲ, ɩɨɥɭɱɢɦ ɭɪɚɜɧɟɧɢɟ ɧɚ ɮɭɧɤɰɢɢ Ƚɪɢɧɚ ɜ ɨɬɫɭɬɫɬɜɢɢ ɩɨɥɹ
ɩɪɢɦɟɫɟɣ. ɗɬɢ ɮɭɧɤɰɢɢ ɛɭɞɭɬ ɞɢɚɝɨɧɚɥɶɧɵ ɢ ɥɟɝɤɨ ɧɚɯɨɞɹɬɫɹ ɱɟɪɟɡ ɢɦɩɭɥɶɫɧɨɟ
ɩɪɟɞɫɬɚɜɥɟɧɢɟ, ɧɚɩɪɢɦɟɪ
ܩ
ோ
ఠԜଵଵ
ሺ݇ሻ
ൌ
߱െݒ
ͳ
݇݅ߜ
ǡ᩷᩷ܩ
ோ
ఠԜଶଶ
ሺ݇ሻ
ൌ
߱ݒ
ͳ
݇݅ߜ
ሺ
ǡ
ͺǤʹ
ɨɬɫɸɞɚ ɥɟɝɤɨ ɩɨɥɭɱɢɦ ɜ ɤɨɨɪɞɢɧɚɬɧɨɦ ɩɪɟɞɫɬɚɜɥɟɧɢɢ
ܩ
ɝɞɟ ߠሺݖ
ߠሺݖሻ݁
ఠ௭
௩
బ
ǡ᩷᩷ܩ
ோ
ఠԜଶଶ
ሺݖሻ
ൌെ
݅
ߠሺെݖሻ݁
ݒ
ோ
ሺݖሻ
ఠԜଵଵ
ሻ
ɮɭɧɤɰɢɹ ɏɟɜɢɫɚɣɞɚ. Ⱥɧɚɥɨɝɢɱɧɨ ɩɨɥɭɱɚɸɬɫɹ ɫɨɨɬɧɨɲɟɧɢɹ
ൌെ
݅
ݒ
ఠ௭
ି
௩
బ
ሺ
ǡ
ͺǤʹ
ሻ
ሻ
ሻ
ሻ
ሻ
ሻ
ሻ
80
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