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ɢɥɢ
ʹɎݒ
ൌʹߝி൬െ
ȟ
Ɉɩɪɟɞɟɥɢɦ ɬɟɦɩɟɪɚɬɭɪɭ ɩɟɪɟɯɨɞɚ ܶ
, ɷɬɨ ɬɟɦɩɟɪɚɬɭɪɚ, ɩɪɢ ɤɨɬɨɪɨɣ
൰Ǥ
ଶ
݃
ɳɚɟɬɫɹ ɜ ɧɭɥɶ, ɞɥɹ ɷɬɨɝɨ ɜ ɪɚɜɟɧɫɬɜɟ (11.28) ɩɨɥɚɝɚɟɦ
ఌ
ಷ
݀ߝ
ʹɎݒ
݃
ଶ
ൌන
ߝ
ʹܶ
Ǥ
ߝ
ȁȟȁ
ൌͲ
ሺ
ȁȟȁ
ሺ
ͳͳǤ͵Ͳ
ɨɛɪɚ-
ͳͳǤ͵ͳ
ɂɧɬɟɝɪɚɥ ɜ ɩɪɚɜɨɣ ɱɚɫɬɢ ɩɪɟɨɛɪɚɡɭɟɬɫɹ ɩɨ ɱɚɫɬɹɦ
ఌ
ൌන
ಷ
ଶ்
ݔ
݀ݔൌ൬
ݔ
Ȁʹܶ
ி
ߝ
ி
ʹܶ
ሻ
ൎͳ, ɚ ɩɨɫɤɨɥɶɤɭ ɩɨɫɥɟɞɧɢɣ ɢɧɬɟɝɪɚɥ
൰൬
ߝ
ʹܶ
ி
൰െන
ஶ
ሻ
ሺݔ
݀ݔǡሺͳͳǤ͵ʹ
ଶ
ሺݔሻ
ఌ
ಷ
ߝ
݀ߝ
න
ܶ
ߝ
ɨɱɟɜɢɞɧɨ ɦɨɠɧɨ ɩɨɥɨɠɢɬɶ ሺߝ
ɛɵɫɬɪɨ ɫɯɨɞɢɬɫɹ, ɬɨ ɜɟɪɯɧɢɣ ɩɪɟɞɟɥ ɜ ɧɟɦ ɦɨɠɧɨ ɩɨɥɨɠɢɬɶ ɪɚɜɧɵɦ ɛɟɫɤɨɧɟɱɧɨɫɬɢ. Ɍɚɛɥɢɰɚ ɢɧɬɟɝɪɚɥɨɜ ([13], ɫɬɪ. 533) ɞɚɟɬ
ஶ
ሻ
ሺݔ
න
ଶ
ሺݔሻ
݀ݔൌ൬
ߨ
Ͷߛ
൰ǡ
ሺ
ͳͳǤ͵͵
ɝɞɟ ɀൌܥൌͲǡͷ – ɩɨɫɬɨɹɧɧɚɹ ɗɣɥɟɪɚ. Ɍɨɝɞɚ ɭɪɚɜɧɟɧɢɟ (11.31) ɩɪɢɦɟɬ ɜɢɞ
ሻ
ሻ
ሻ
ሻ
ൌ൬
ʹߛߝ
ߨܶ
ி
൰ǡ
ሺ
ͳͳǤ͵Ͷ
ʹɎݒ
݃
ଶ
ɨɬɤɭɞɚ
ܶ
ߛ
ൌ
ʹߝி൬െ
ߨ
ʹɎݒ
݃
ߛ
൰ൌ
ଶ
ȟൎͲǡͷȟǤ
ߨ
ሺ
ͳͳǤ͵ͷ
Ⱦɥɹ ɨɩɪɟɞɟɥɟɧɢɹ ɬɟɦɩɟɪɚɬɭɪɧɨɣ ɡɚɜɢɫɢɦɨɫɬɢ ɚɦɩɥɢɬɭɞɵ ɮɨɧɨɧɧɨɣ ɦɨɞɵ
ȁȟȁ
ሺܶሻ ɜ ɭɪɚɜɧɟɧɢɢ (11.28), ɜɨ-ɩɟɪɜɵɯ, ɥɟɜɭɸ ɱɚɫɬɶ ɜɵɪɚɡɢɦ ɫ ɩɨɦɨɳɶɸ (11.29)
ɱɟɪɟɡȟ
, ɚ ɜɨ-ɜɬɨɪɵɯ, ɩɪɢɛɚɜɢɦ ɢ ɜɵɱɬɟɦ ɢɡ ɩɪɚɜɨɣ ɱɚɫɬɢ ɢɧɬɟɝɪɚɥ
ఌ
ಷ
݀ߝ
න
ଶ
ඥ
ߝ
ȁȟ
ǡ
ଶ
ȁ
ሺ
ͳͳǤ͵
ɢɧɬɟɝɪɚɥ ɫɨ ɡɧɚɤɨɦ ɩɥɸɫ ɜɵɱɢɫɥɹɟɦ, ɚ ɫ ɩɨɦɨɳɶɸ ɜɬɨɪɨɝɨ ɩɪɟɨɛɪɚɡɭɟɦ ɜɵɪɚɠɟɧɢɟ, ɜ ɢɬɨɝɟ ɩɨɥɭɱɢɦ
ఌ
൬
ȟ
൰ൌʹන
ȁȟȁ
ಷ
ଶ
ඥ
ȁ
ߝ
ȁȟ
ଶ
ቆ
݀ߝ
ቊ
ሺ
Ǥ
ͳͳǤ͵
ଶ
ଶ
ȁȟ
ȁ
ඥ
ߝ
ܶ
ቋ
ͳቇ
ȼɜɢɞɭ ɛɵɫɬɪɨɣ ɫɯɨɞɢɦɨɫɬɢ ɢɧɬɟɝɪɚɥɚ ɜɟɪɯɧɢɣ ɩɪɟɞɟɥ ɦɨɠɧɨ ɩɨɥɨɠɢɬɶ ɪɚɜ-
ɧɵɦ ɛɟɫɤɨɧɟɱɧɨɫɬɢ. ȼ ɢɬɨɝɟ ɩɪɟɞɫɬɚɜɢɦ ɭɪɚɜɧɟɧɢɟ
ሺ
ͳͳǤ͵ሻɜ ɜɢɞɟ
121
ሻ
ሻ
ሻ
ሻ
൬
ȟ
൰ൌܫቆ
ȁȟȁ
ȁȟȁ
ܶ
ቇǡ
ɝɞɟ
ܫሺݑሻൌʹන
ஶ
ଶݑଶ
ݔ
ξ
݀ݔ
൫൛ξݔଶݑଶൟͳ൯
Ǥ
Ɏɭɧɤɰɢɹ ܫሺݑሻ ɢɦɟɟɬ ɫɥɟɞɭɸɳɢɟ ɚɫɢɦɩɬɨɬɢɤɢ ([3], ɫɬɪ. 192): ɩɪɢ ݑ՜λ
ሺ
ͳͳǤ͵ͺ
ሺ
ͳͳǤ͵ͻ
ሻ
ሻ
ܫሺݑሻൎ
ɚ ɪɚɡɥɨɠɟɧɢɟ ܫሺݑሻ ɩɪɢ ݑ՜Ͳ ɢɦɟɟɬ ɜɢɞ
ൌȟ
ߨ
ߛݑ
ͳെ
ܫሺݑሻൎ൬
ɉɟɪɜɚɹ ɮɨɪɦɭɥɚ ɞɚɟɬ ɩɪɢ ܶاȟ
ሺܶሻ
ȁȟȁ
ɫ ɩɨɦɨɳɶɸ ɜɬɨɪɨɣ ɮɨɪɦɭɥɵ ɩɨɥɭɱɚɟɦ
൬
ȟ
൰ൌ൬
ȁȟȁ
Ɏܶ
ߛȁȟ
ɨɬɤɭɞɚ
ܶ
൬
൰ൌെ
ܶ
ɬɨɝɞɚ ɩɪɢ ܶ
െܶاܶ ɜ ɩɟɪɜɨɦ ɩɨɪɹɞɤɟ ɩɨ ܶെܶ ɩɨɥɭɱɢɦ
ሻ
ȁȟ
ሺܶሻ
ȁ
ൌܶ
ߞሺ͵
ቈ
ͺߨ
ଶ
൬ͳെ
ඨ
൰
ʹߨ
ݑ
ඨ
൰
ȁ
ߞሺ͵
ͺߨ
ܶ
൰
ܶ
݁ି௨ǡ
ሻ
ߞሺ͵
ݑଶǤ
ଶ
ͺߨ
ȟ
ߞሺ͵
ሻ
ͺߨ
ȁȟȁ
ܶ
݁
బ
ି
்
ሻ
ȁȟȁ
ଶ
ܶ
ଶ
ǡ
ଶ
ʹߨܶ
ଶ
ଵȀଶ
ൌ͵ǤͲܶඨͳെ
ǡ
ͳͳǤͶͲ
ͳͳǤͶͳ
ͳͳǤͶʹ
ͳͳǤͶ͵
ͳͳǤͶͶ
ͳͳǤͶͷ
ሻ
ሻ
ሻ
ሻ
ሻ
ሻ
ሺ
ሺ
ሺ
ଶ
ǡ
ଶ
ሺ
ሺ
ܶ
ܶ
ሺ
Ǥ
Ɍɚɤɢɦ ɨɛɪɚɡɨɦ, ɩɪɢ ɬɟɦɩɟɪɚɬɭɪɟ ܶ൏ܶ
ɜ ɨɞɧɨɦɟɪɧɨɦ ɜɵɪɨɠɞɟɧɧɨɦ ɷɥɟɤ-
ɬɪɨɧɧɨɦ ɝɚɡɟ ɜɨɡɧɢɤɚɟɬ ɧɟɭɫɬɨɣɱɢɜɨɫɬɶ ɨɬɧɨɫɢɬɟɥɶɧɨ ɨɛɪɚɡɨɜɚɧɢɹ ɫɬɚɰɢɨɧɚɪ-
ɧɨɣ ɞɟɮɨɪɦɚɰɢɢ ɪɟɲɟɬɤɢ ɫ ɜɨɥɧɨɜɵɦ ɜɟɤɬɨɪɨɦ ʹ
. ɉɪɢ ɬɟɦɩɟɪɚɬɭɪɟ ܶൌͲ ɚɦ-
ɩɥɢɬɭɞɚ ɷɬɨɣ ɞɟɮɨɪɦɚɰɢɢ ɨɩɪɟɞɟɥɹɟɬɫɹ ɜɵɪɚɠɟɧɢɟɦ (11.30). ɉɨɞɨɛɧɚɹ ɞɟɮɨɪɦɚɰɢɹ ɩɪɢ ɨɩɪɟɞɟɥɟɧɧɨɦ ɡɚɩɨɥɧɟɧɢɢ ɡɨɧɵ ɩɪɨɜɨɞɢɦɨɫɬɢ ɩɪɢɜɨɞɢɬ ɤ ɷɮɮɟɤɬɢɜɧɨɦɭ ɭɞɜɨɟɧɢɸ ɩɟɪɢɨɞɚ ɪɟɲɟɬɤɢ. ɉɨɹɫɧɢɬɶ ɷɬɨ ɦɨɠɧɨ ɫɥɟɞɭɸɳɢɦ ɨɛɪɚɡɨɦ.
ɋ ɭɱɟɬɨɦ ɫɩɢɧɨɜɨɣ ɫɬɟɩɟɧɢ ɫɜɨɛɨɞɵ, ɫɜɹɡɶ ɢɦɩɭɥɶɫɚ Ɏɟɪɦɢ
ɷɥɟɤɬɪɨɧɨɜ ݊ ɢɦɟɟɬ ɜɢɞ
ൌߨ݊Ȁʹ. ɉɭɫɬɶ ɨɞɧɨɦɟɪɧɚɹ ɰɟɩɨɱɤɟ ɫɨɫɬɨɢɬ ܰ ɚɬɨɦɨɜ,
ɫ ɤɨɧɰɟɧɬɪɚɰɢɟɣ
ɚ ɪɚɫɫɬɨɹɧɢɟ ɦɟɠɞɭ ɫɨɫɟɞɧɢɦɢ ɚɬɨɦɚɦɢ ɟɫɬɶ ܽ (ɬɚɤ ɱɬɨ ɞɥɢɧɚ ɰɟɩɨɱɤɢ ܮൌܰܽሻ.
122
ȿɫɥɢ ɤɚɠɞɵɣ ɚɬɨɦ ɞɚɟɬ ɜ ɡɨɧɭ ɩɪɨɜɨɞɢɦɨɫɬɢ ɨɞɢɧ ɷɥɟɤɬɪɨɧ, ɬɨ ɤɨɧɰɟɧɬɪɚɰɢɹ
ɷɥɟɤɬɪɨɧɨɜ ɛɭɞɟɬ ɪɚɜɧɚ ݊ൌܰȀܮൌͳȀܽ, ɚ ɢɦɩɭɥɶɫ Ɏɟɪɦɢ
ɞɥɢɧɚ ɜɨɥɧɵ ɪɚɫɫɦɚɬɪɢɜɚɟɦɨɣ ɮɨɧɨɧɧɨɣ ɦɨɞɵ ɫ ɜɨɥɧɨɜɵɦ ɜɟɤɬɨɪɨɦ ʹ
ɱɚɟɬɫɹ ɪɚɜɧɨɣ ߣൌʹߨȀʹ
ൌʹܽ. Ɍɨ ɟɫɬɶ ɜ ɞɜɚ ɪɚɡɚ ɛɨɥɶɲɟ ɞɥɢɧɵ ɷɥɟɦɟɧɬɚɪɧɨɣ
ൌߨȀʹܽ. Ɍɨɝɞɚ
ɩɨɥɭ-
ɹɱɟɣɤɢ. ɑɬɨɛɵ ɫɨɡɞɚɬɶ ɬɚɤɭɸ ɮɨɧɨɧɧɭɸ ɦɨɞɭ, ɞɨɫɬɚɬɨɱɧɨ ɫɦɟɫɬɢɬɶ ɤɚɠɞɵɣ ɚɬɨɦ
ɰɟɩɨɱɤɢ ɩɨɨɱɟɪɟɞɧɨ ɧɚ ɧɟɤɨɬɨɪɭɸ ɜɟɥɢɱɢɧɭ ݑ ɢ Ȃݑ, ɝɞɟ ݑȁȟȁǤȼ ɪɟɡɭɥɶɬɚɬɟ ɬɚ-
ɤɨɣ ɦɨɞɭɥɹɰɢɢ ɨɞɧɨɦɟɪɧɨɣ ɪɟɲɟɬɤɢ ɩɪɨɢɡɨɣɞɟɬ ɭɞɜɨɟɧɢɟ ɟɟ ɩɟɪɢɨɞɚ, ɬɚɤ ɤɚɤ
ɷɥɟɦɟɧɬɚɪɧɚɹ ɹɱɟɣɤɚ ɪɟɲɟɬɤɢ ɛɭɞɟɬ ɭɠɟ ɫɨɞɟɪɠɚɬɶ ɧɟ
ɨɞɢɧ, ɚ ɞɜɚ ɚɬɨɦɚ, ɩɪɢ
ɷɬɨɦ ɟɟ ɞɥɢɧɚ ɭɜɟɥɢɱɢɬɫɹ ɫɨɨɬɜɟɬɫɬɜɟɧɧɨ ɜ ɞɜɚ ɪɚɡɚ (ɪɢɫ. 11.1).
Ɋɢɫ. 11.1
11.3. ɉɥɨɬɧɨɫɬɶ ɫɨɫɬɨɹɧɢɣ ɢ ɷɧɟɪɝɟɬɢɱɟɫɤɚɹ ɳɟɥɶ
ɜ ɫɩɟɤɬɪɟ ɷɥɟɤɬɪɨɧɨɜ
Ɇɚɰɭɛɚɪɨɜɫɤɚɹ ɮɭɧɤɰɢɹ Ƚɪɢɧɚ ɷɥɟɤɬɪɨɧɨɜ ɞɚɟɬɫɹ ɜɵɪɚɠɟɧɢɟɦ (11.24), ɢ ɦɨ-
ɠɟɬ ɛɵɬɶ ɩɪɟɞɫɬɚɜɥɟɧɚ ɜ ɜɢɞɟ
ሺାሻ
ǡ݇ሻൌ
௦
݅߱
ெ
ሺ
ܩ
݅߱
݇ߪොଷȟԜߪො
௦ݒ
൫݅߱௦െ߳ሺ݇ሻ൯൫݅߱௦߳ሺ݇ሻ൯
ȟכԜߪො
ሺିሻ
ǡ
ሺ
ͳͳǤͶ
ሻ
ଶ
ɝɞɟ ߳ሺ݇ሻൌ
ሺ
ሻ
ݒ
݇
ඥ
ଶ
ȁ
ȁȟ
. ɗɬɨ ɜɵɪɚɠɟɧɢɟ ɩɨɡɜɨɥɹɟɬ ɥɟɝɤɨ ɜɵɩɢɫɚɬɶ ɜɵɪɚɠɟ-
ɧɢɟ ɞɥɹ ɡɚɩɚɡɞɵɜɚɸɳɟɣ ɮɭɧɤɰɢɢ Ƚɪɢɧɚ, ɨɧɚ ɩɨɥɭɱɚɟɬɫɹ ɡɚɦɟɧɨɣ ɦɨɰɭɛɚɪɨɜɫɤɨɣ
ɱɚɫɬɨɬɵ ɩɨ ɩɪɚɜɢɥɭ ݅߱
ܩ
՜߱݅ߜ. ɉɨɥɭɱɚɟɦ
௦
ோ
ሺ
߱ǡ݇ሻൌ
߱ݒ
ሺ
߱െ߳ሺ݇ሻ݅ߜሻሺ߱߳ሺ݇ሻ݅ߜ
߱ݒ
ൌ
ሺ
߱݅ߜ
݇ߪොଷȟԜߪො
݇ߪොଷȟԜߪො
ଶ
ሻ
െሺݒ݇
ሺାሻ
ሺାሻ
ȟכԜߪො
ଶ
ሻ
െȁȟ
ȟכԜߪො
ሺିሻ
ଶ
ȁ
ሺିሻ
ൌ
ሻ
Ǥ
ሺ
ͳͳǤͶ
ȼɵɪɚɠɟɧɢɟ ɞɥɹ ɡɚɩɚɡɞɵɜɚɸɳɟɣ ɮɭɧɤɰɢɢ Ƚɪɢɧɚ ɩɨɡɜɨɥɹɟɬ ɜɵɱɢɫɥɢɬɶ ɮɭɧɤ-
ɰɢɸ ɩɥɨɬɧɨɫɬɢ ɫɨɫɬɨɹɧɢɣ ߥሺ߱ሻ. ɗɬɨ ɮɭɧɤɰɢɹ ɨɩɪɟɞɟɥɹɟɬ ɱɢɫɥɨ ɫɨɫɬɨɹɧɢɣ ɷɥɟɤɬɪɨɧɨɜ ɜ ɢɧɬɟɪɜɚɥɟ ɷɧɟɪɝɢɣ ሺ߱ǡ߱݀߱ሻ, ɩɪɢɯɨɞɹɳɢɯɫɹ ɧɚ ɟɞɢɧɢɰɭ ɞɥɢɧɵ ɩɪɨ-
ɜɨɞɧɢɤɚ.
Ⱦɥɹ ɬɨɝɨ ɱɬɨɛɵ ɫɜɹɡɚɬɶ ɮɭɧɤɰɢɸ ɩɥɨɬɧɨɫɬɢ ɫɨɫɬɨɹɧɢɣ ɫ ɩɪɢɱɢɧɧɨɣ ɮɭɧɤ-
ɰɢɟɣ Ƚɪɢɧɚ, ɪɚɫɫɦɨɬɪɢɦ ɨɛɳɟɟ ɜɵɪɚɠɟɧɢɟ ɞɥɹ ɧɟɟ, ɞɚɜɚɟɦɨɟ ɮɨɪɦɭɥɨɣ
(3.22)
123
ሻ
כ
ሺ
ሻ
ሺ
ோ
ሺ
߱ǡx
ܩ
ଵǡxଶ
ɦɧɢɦɚɹ ɱɚɫɬɶ ɷɬɨɣ ɮɭɧɤɰɢɢ ɩɪɢ x
ோ
ܩ
ሺ
߱ǡǡሻൌെߨ
ሻ
ൌ
ଵൌxଶ
߮
߱െߦ
ൌ ɟɫɬɶ
ȁ
σ
߮
x
ሺሻ
ଵ
߮
x
݅ߜ
ଶ
ȁ
ߜሺ߱െߦ
ሻ
ଶ
ǡ
ሻ
ǤሺͳͳǤͶͻ
ሺ
ͳͳǤͶͺ
ɋ ɞɪɭɝɨɣ ɫɬɨɪɨɧɵ ɞɥɹ ɮɭɧɤɰɢɢ ɩɥɨɬɧɨɫɬɢ ɫɨɫɬɨɹɧɢɣ ɧɚ ɟɞɢɧɢɰɭ ɨɛɴɟɦɚ ߥሺ߱ሻ
ɢɦɟɟɦ, ɩɨ ɨɩɪɟɞɟɥɟɧɢɸ
ߥሺ߱ሻൌ
ͳ
ߜሺ߱െߦ
ܸ
cɪɚɜɧɢɜɚɹ ɫ (11.49) ɢ ɭɱɢɬɵɜɚɹ ɬɨɬ ɮɚɤɬ, ɱɬɨ ɜɨɥɧɨɜɵɟ ɮɭɧɤɰɢɢ ߮
ሻ
ǡ
ሺሻ
ሺ
ͳͳǤͷͲ
ɧɨɪɦɢɪɨ-
ɜɚɧɵ, ɢɦɟɟɦ
ߥሺ߱ሻൌെ
ͳ
ߨܸ
නܩ
ோ
ሺ
߱ǡǡሻ݀ǡ
ሺ
ͳͳǤͷͳ
ɷɬɨ ɨɛɳɚɹ ɮɨɪɦɭɥɚ. ȼ ɧɚɲɟɦ ɫɥɭɱɚɟ ɤɨɨɪɞɢɧɚɬɚ ɡɚɦɟɧɹɟɬɫɹ ɧɚ ݖ – ɨɞɧɨɦɟɪɧɭɸ
ሺሻ
ɤɨɨɪɞɢɧɚɬɭ ɷɥɟɤɬɪɨɧɚ, ɚ ɫɨɫɬɨɹɧɢɹ ߮
ɢɦɟɸɬ ɜɢɞ
ሻ
ሻ
ሻ
ሻ
ሺሻ
ൌሺ݅
߮
– ɷɧɟɪɝɢɹ ɷɥɟɤɬɪɨɧɚ, ɨɬɫɱɢɬɚɧɧɚɹ ɨɬ ɷɧɟɪɝɢɢ Ɏɟɪɦɢ. ȼɨɥɧɨɜɵɟ ɮɭɧɤɰɢɢ
ɚ ߦ
߰
ሺݖሻ
ଵ
ɢ െ
ሺݖሻ
ɢ ߰
ɫɨɨɬɜɟɬɫɬɜɟɧɧɨ. Ɏɭɧɤɰɢɹ Ƚɪɢɧɚ (11.47) ɜ ɤɨɨɪɞɢɧɚɬɧɨɦ ɩɪɟɞɫɬɚɜɥɟɧɢɢ
ɨɩɢɫɵɜɚɸɬ ɫɨɫɬɨɹɧɢɹ ɷɥɟɤɬɪɨɧɚ, ɧɚɯɨɞɹɳɟɝɨɫɹ ɜ ɨɤɪɟɫɬɧɨɫɬɢ
ଶ
ݖሻ߰
ଵ
ሺݖሻ
ሺെ݅
ݖሻ߰
ଶ
ሺݖሻ
ǡ
ɟɫɬɶ ɮɚɤɬɢɱɟɫɤɢ
כ
ሺ
ሻ
ሺ
߰
z
ఈ
ோ
ሺ
߱ǡݖ
ܩ
ఈఉ
ଵݖଶ
ሻ
ൌ
ଵ
߱െߦ
߰
ఉ
݅ߜ
ሻ
ݖ
ଶ
ǡ
ɢ ɨɧɚ ɫɜɹɡɚɧɚ ɫ ɮɭɧɤɰɢɟɣ (11.48) ɫɨɨɬɧɨɲɟɧɢɟɦ
ఈఉ
ோ
ఈఉ
ሺ
߱ǡݖ
ଵݖଶ
ሻ
ൣ݅൫ݖ
ሺ
ߪ
ଵ
ଷሻఈఈെݖଶ
ሺ
ߪ
ଷሻఉఉ
൯൧ǡሺͳͳǤͷͶ
ܩ
ோ
ሺ
߱ǡz
ଵǡzଶ
ሻ
ൌܩ
ɩɨɞɫɬɚɜɥɹɹ ɷɬɨ ɜ (11.51), ɩɨɥɭɱɢɦ
ߥሺ߱ሻൌെ
ͳ
නܩ
ߨܮ
ோ
ሺ
ሻ
߱ǡǡ
݀ǡ
ɡɞɟɫɶ ܮ – ɞɥɢɧɚ ɨɛɪɚɡɰɚ. ɉɨɞɫɬɚɜɥɹɹ ɫɸɞɚ (11.47), ɩɨɥɭɱɢɦ
ߥሺ߱ሻൌെ
ൌെ
ͳ
ߨ
ߨݒ
න
ʹ
න
݀݇
ʹߨ
݀ݔ
ʹߨ
߱ݒ
ሺ
߱݅ߜ
ሺ
߱݅ߜ
݇ߪොଷȟԜߪො
ଶ
ሻ
െሺݒ݇
߱
ଶെݔଶ
ሻ
ሺାሻ
ሻ
െȁȟ
ȟכԜߪො
ଶ
െȁȟ
ǡ
ଶ
ȁ
ȁ
ሺିሻ
ଶ
ൌ
124
ሺ
ͳͳǤͷʹ
ሺ
ͳͳǤͷ͵
ሺ
ͳͳǤͷͷ
ሺ
ͳͳǤͷ
ሻ
ሻ
ሻ
ሻ
ሻ
ɩɨɞɵɧɬɟɝɪɚɥɶɧɨɟ ɜɵɪɚɠɟɧɢɟ ɜ ɮɨɪɦɭɥɟ (11.56) ɢɦɟɟɬ ɩɨɥɸɫɵ ɜ ɬɨɱɤɚɯ
ଶ
െȁȟ
െȁȟ
ଶ
ȁ
Ǥ
Ǥ
ଶ
ȁ
ሺ
ͳͳǤͷ
ȁ
, ɬɨ ɩɨɥɸɫɚ
ሺ
ͳͳǤͷͺ
ȿɫɥɢ
ȁ߱ȁ൏ȁȟȁ
ሺ
ඥ
ൌേ
ݔ
߱݅ߜ
, ɬɨ ɨɛɚ ɩɨɥɸɫɚ ɪɚɫɩɨɥɨɠɟɧɵ ɧɚ ɦɧɢɦɨɣ ɨɫɢ, ɢ ɢɧɬɟɝɪɚɥ ɛɭɞɟɬ
ሻ
ɜɟɳɟɫɬɜɟɧɧɨɣ ɜɟɥɢɱɢɧɨɣ, ɢ ɫɥɟɞɨɜɚɬɟɥɶɧɨ ߥሺ߱ሻൌͲǤ ȿɫɥɢ ȁ߱ȁȁȟ
ଶ
ɟɫɬɶ ݔ
ൌേඥ߱ଶെȁȟ
ȁ
േ݅ߜሺ߱ሻ ɢ ɜɵɱɢɫɥɟɧɢɟ ɞɚɟɬ
ȁ߱ȁ
ߥሺ߱ሻൌ
ߨݒ
ͳ
ଶ
ඥ
߱
Ɍɚɤɢɦ ɨɛɪɚɡɨɦ, ɞɥɹ ɩɥɨɬɧɨɫɬɢ ɫɨɫɬɨɹɧɢɣ ɩɨɥɭɱɚɟɦ
Ͳ˒˓ˋȁ߱ȁ൏ȁȟȁǡ
ߥሺ߱ሻൌቐ
ߨݒ
ͳ
ଶ
ඥ
߱
െȁȟ
ଶ
ȁ
˒˓ˋȁ߱ȁȁȟȁǤ
ሺ
ͳͳǤͷͻ
ȁ߱ȁ
ɂɡ ɜɵɪɚɠɟɧɢɹ (11.59) ɫɥɟɞɭɟɬ, ɱɬɨ ɜ ɫɩɟɤɬɪɟ ɷɥɟɤɬɪɨɧɨɜ ɩɨɹɜɥɹɟɬɫɹ ɡɚɩɪɟ-
ɳɟɧɧɚɹ ɳɟɥɶ ɜɟɥɢɱɢɧɨɣ ʹȁȟ
ȁ
. ȼ ɧɚɥɢɱɢɢ ɳɟɥɢ ɦɨɠɧɨ ɭɛɟɞɢɬɶɫɹ ɬɚɤɠɟ ɫ ɩɨɦɨɳɶɸ ɫɥɟɞɭɸɳɢɯ ɫɨɨɛɪɚɠɟɧɢɣ. ȼ ɝɥɚɜɟ 4 ɛɵɥɨ ɭɫɬɚɧɨɜɥɟɧɨ, ɱɬɨ ɩɨɥɸɫɚ ɡɚɩɚɡɞɵɜɚɸɳɟɣ ɮɭɧɤɰɢɢ Ƚɪɢɧɚ ɨɩɪɟɞɟɥɹɸɬ ɫɩɟɤɬɪ ɨɞɧɨɱɚɫɬɢɱɧɵɯ ɜɨɡɛɭɠɞɟɧɢɣ.
ɂɡ (11.47) ɦɨɠɧɨ ɜɢɞɟɬɶ, ɱɬɨ
ɫɩɟɤɬɪ ɷɥɟɤɬɪɨɧɨɜ ɜ ɨɤɪɟɫɬɧɨɫɬɢ ɮɟɪɦɢ-
ɢɦɩɭɥɶɫɨɜ ɞɚɟɬɫɹ ɜɵɪɚɠɟɧɢɟɦ
ଶ
ȁȟ
ଶ
ȁ
ǡ
ሺ
ȁ
.
ͳͳǤͲ
ሺ
ሻ
ඥ
ݒ
߳ሺ݇ሻൌേ
݇
ɨɬɫɸɞɚ ɜɢɞɧɨ, ɱɬɨ ɜ ɫɩɟɤɬɪɟ ɷɥɟɤɬɪɨɧɨɜ ɜɨɡɧɢɤɚɟɬ ɳɟɥɶ ɜɟɥɢɱɢɧɨɣ ʹȁȟ
ሻ
ሻ
ሻ
ሻ
Ɋɢɫ. 11.2
ɂɦɟɧɧɨ ɫɭɦɦɚɪɧɨɟ ɩɨɧɢɠɟɧɢɟ ɷɧɟɪɝɢɢ ɷɥɟɤɬɪɨɧɨɜ ɫ ɷɧɟɪɝɢɹɦɢ
ଶ
ሻ
ሺ
݇
െ
ݒ
ඥ
ଶ
ȁ
ȁȟ
(ɪɢɫ. 11.2) ɞɟɥɚɟɬ ɷɧɟɪɝɟɬɢɱɟɫɤɢ ɜɵɝɨɞɧɵɦ ɜɨɡɧɢɤɧɨɜɟɧɢɟ ɫɬɚɰɢɨɧɚɪɧɨɣ ɞɟɮɨɪɦɚɰɢɢ (11.1), ɧɟɫɦɨɬɪɹ ɧɚ ɡɚɬɪɚɬɵ ɷɧɟɪɝɢɢ ɧɚ ɭɩɪɭɝɭɸ ɞɟɮɨɪɦɚɰɢɸ, ɤɨɬɨɪɵɟ ɪɚɜɧɵ (11.5).
125
Ʌɢɬɟɪɚɬɭɪɚ
Ɍɟɨɪɢɹ ɩɟɪɟɯɨɞɚ ɉɚɣɟɪɥɫɚ ɜ ɨɞɧɨɦɟɪɧɨɣ ɫɢɫɬɟɦɟ ɢɡɥɨɠɟɧɚ ɜ [9], ɜ ɷɬɨɦ ɨɛɡɨɪɟ ɩɪɢɜɟɞɟɧɵ ɫɫɵɥɤɢ ɧɚ ɨɛɲɢɪɧɭɸ ɥɢɬɟɪɚɬɭɪɭ ɩɨ ɩɚɣɟɪɥɫɨɜɫɤɨɦɭ ɩɟɪɟɯɨɞɭ ɜ
ɨɞɧɨɦɟɪɧɵɯ ɢ ɤɜɚɡɢ-ɨɞɧɨɦɟɪɧɵɯ ɫɢɫɬɟɦɚɯ.
Ʉɨɧɬɪɨɥɶɧɵɟ ɜɨɩɪɨɫɵ
1. Ɂɚ ɫɱɟɬ ɱɟɝɨ ɞɟɮɨɪɦɚɰɢɹ ɪɟɲɟɬɤɢ ɫ ɭɞɜɨɟɧɢɟɦ ɩɟɪɢɨɞɚ ɨɤɚɡɵɜɚɟɬɫɹ ɷɧɟɪ-
ɝɟɬɢɱɟɫɤɢ ɜ ɜɵɝɨɞɧɨɣ?
2. Ʉɚɤɨɜ ɷɧɟɪɝɟɬɢɱɟɫɤɢɣ ɫɩɟɤɬɪ ɷɥɟɤɬɪɨɧɨɜ ɜ ɨɤɪɟɫɬɧɨɫɬɢ ɭɪɨɜɧɹ ɮɟɪɦɢ ɩɨ-
ɫɥɟ ɩɚɣɟɪɥɫɨɜɫɤɨɝɨ ɩɟɪɟɯɨɞɚ?
3. Ɉɩɪɟɞɟɥɹɟɬɫɹ ɥɢ ɮɚɡɚ ɜɨɥɧɵ ɡɚɪɹɞɨɜɨɣ ɩɥɨɬɧɨɫɬɢ ɭɫɥɨɜɢɟɦ ɪɚɜɧɨɜɟɫɢɹ?
4. Ʉɚɤ ɫɜɹɡɚɧɚ ɬɟɦɩɟɪɚɬɭɪɚ ɩɚɣɟɪɥɫɨɜɫɤɨɝɨ ɩɟɪɟɯɨɞɚ ɫ ɪɚɜɧɨɜɟɫɧɨɣ ɚɦɩɥɢɬɭ-
ɞɨɣ ɮɨɧɨɧɧɨɣ ɦɨɞɵ?
126
Ƚɥɚɜɚ 12
ɉȺɃȿɊɅɋɈȼɋɄɂɃ ɉȿɊȿɏɈȾ
ȼ ɋɅɍɑȺɃɇɈɆ ɉɈɌȿɇɐɂȺɅȿ ɉɊɂɆȿɋȿɃ
ȼ ɷɬɨ ɝɥɚɜɟ ɦɵ ɪɚɫɫɦɨɬɪɢɦ ɜɥɢɹɧɢɟ ɫɥɭɱɚɣɧɵɯ ɩɪɢɦɟɫɟɣ ɧɚ ɨɞɧɨɦɟɪɧɵɣ ɦɟɬɚɥɥ, ɩɪɟɬɟɪɩɟɜɚɸɳɢɣ ɩɟɪɟɯɨɞ ɉɚɣɟɪɥɫɚ. Ɇɵ ɪɚɫɫɦɨɬɪɢɦ ɜɥɢɹɧɢɟ ɫɥɭɱɚɣɧɨɝɨ
ɩɨɬɟɧɰɢɚɥɚ ɩɪɢɦɟɫɟɣ ɧɚ ɫɥɟɞɭɸɳɢɟ ɫɜɨɣɫɬɜɚ ɩɚɣɟɪɥɫɨɜɫɤɨɝɨ ɩɟɪɟɯɨɞɚ: ɩɚɪɚɦɟɬɪ ɩɨɪɹɞɤɚ, ɬɟɦɩɟɪɚɬɭɪɚ ɩɟɪɟɯɨɞɚ, ɩɥɨɬɧɨɫɬɶ ɫɨɫɬɨɹɧɢɣ. Ʉɪɨɦɟ ɷɬɨɝɨ ɦɵ ɪɚɫɫɦɨɬɪɢɦ ɜɨɩɪɨɫ ɨ ɬɨɦ, ɤɚɤ ɜɨɥɧɚ ɡɚɪɹɞɨɜɨɣ ɩɥɨɬɧɨɫɬɢ ɜɥɢɹɟɬ ɧɚ ɹɜɥɟɧɢɟ ɥɨɤɚɥɢɡɚɰɢɢ
ɷɥɟɤɬɪɨɧɨɜ ɫɥɭɱɚɣɧɵɦ ɩɨɬɟɧɰɢɚɥɨɦ, ɚ ɬɚɤɠɟ ɜɵɱɢɫɥɢɦ ɩɪɨɜɨɞɢɦɨɫɬɶ
ɩɪɢ ɪɚɡɧɵɯ ɬɟɦɩɟɪɚɬɭɪɚɯ.
12.1. ȼɥɢɹɧɢɟ ɩɪɢɦɟɫɟɣ ɧɚ ɬɟɦɩɟɪɚɬɭɪɭ ɩɟɪɟɯɨɞɚ
ɢ ɩɚɪɚɦɟɬɪ ɩɨɪɹɞɤɚ
12.1.1. Ƚɚɦɢɥɶɬɨɧɢɚɧ ɷɥɟɤɬɪɨɧɨɜ ɜ ɩɪɢɫɭɬɫɬɜɢɢ ɩɪɢɦɟɫɟɣ.
Ɍɚɤ ɠɟ ɤɚɤ ɢ ɜ ɝɥɚɜɟ 8 ɦɵ ɛɭɞɟɦ ɦɨɞɟɥɢɪɨɜɚɬɶ ɜɡɚɢɦɨɞɟɣɫɬɜɢɟ ɷɥɟɤɬɪɨɧɨɜ ɫ
ɩɪɢɦɟɫɹɦɢ ɫɥɭɱɚɣɧɵɦɢ ɩɨɥɹɦɢ ߟሺݖሻ ɢ ߞሺݖ
(11.13), ɨɩɢɫɵɜɚɸɳɟɦɭ ɷɥɟɤɬɪɨɧɵ ɜ ɩɪɢɫɭɬɫɬɜɢɢ ɮɨɧɨɧɧɨɣ ɦɨɞɵ
ɍɫɥɨɜɢɟ ɪɚɜɧɨɜɟɫɢɹ ɢ ɟɝɨ ɭɫɪɟɞɧɟɧɢɟ
ሻ
. Ɍɚɤɢɦ ɨɛɪɚɡɨɦ, ɤ ɝɚɦɢɥɶɬɨɧɢɚɧɭ
ܪൌ݀ݖ߰
ା
ሺݖሻ
൬െ݅ݒ
ߪොଷ
μ
μݖ
ȟԜߪො
ሺାሻ
ȟכԜߪො
ሺିሻ
൰߰
ሺݖሻ
൨ǡሺͳʹǤͳ
ɧɟɨɛɯɨɞɢɦɨ ɞɨɛɚɜɢɬɶ ɫɥɚɝɚɟɦɵɟ ɢɡ ɝɚɦɢɥɶɬɨɧɢɚɧɚ ɷɥɟɤɬɪɨɧɨɜ ɜ ɩɨɥɟ ɩɪɢɦɟɫɟɣ
(ɫɦ. (8.22))
ା
ሺݖሻ
නൣ߰
൫ߟሺݖሻߞሺݖሻԜߪො
Ɇɵ ɜɢɞɢɦ, ɱɬɨ ɧɚɥɢɱɢɟ ɜɡɚɢɦɨɞɟɣɫɬɜɢɹ ɫ ɩɪɢɦɟɫɹɦɢ ɦɨɠɧɨ ɭɱɟɫɬɶ ɜ ɝɚɦɢɥɶ-
ɬɨɧɢɚɧɟ ɡɚɦɟɧɨɣ ȟ՜ȟߞሺݖሻ ɢ ȟ
ሺାሻ
ߞ
כ՜ȟכߞכ
Ʉɨɪɪɟɥɹɰɢɨɧɧɵɟ ɫɜɨɣɫɬɜɚ ɫɥɭɱɚɣɧɵɯ ɩɨɥɟɣ ߟሺݖሻ ɢ ߞሺݖ
ሺିሻ
כ
ሺݖሻ
ሺݖሻ
ɢ ɞɨɛɚɜɥɟɧɢɟɦ ɫɥɚɝɚɟɦɨɝɨ ߟሺݖ
Ԝߪො
ሺݖሻ
൯߰
൧݀ݖǤ
ሻ
ɞɚɸɬɫɹ ɮɨɪɦɭɥɚɦɢ
ሺ
ͳʹǤʹ
ሻ
(8.17), (8.20), ɤɨɬɨɪɵɟ ɦɵ ɡɞɟɫɶ ɧɚɩɢɲɟɦ ɟɳɟ ɪɚɡ
ݒ
ൌ
Ԝߜሺݖെݖ
߬
ଵ
ᇱ
ሻ
ǡ᩷ۦߞሺݖሻߞሺݖ
ᇱ
ሻ
ǡ
ᇱ
ሻ
ۧ
ሺ
ͳʹǤ͵
ൌͲǤሺͳʹǤͶ
ۦߞሺݖ
ሻ
ۧ
ൌͲǡ᩷ۦߞሺݖሻߞ
ۦ
ߟሺݖ
ሻ
ۧ
ൌͲǡ᩷ۦߟሺݖሻߟሺݖ
כ
ሺ
ݖ
ᇱ
ሻ
ۧ
ݒ
ൌ
Ԝߜሺݖെݖ
߬
ଶ
ᇱ
ሻ
ۧ
Ɂɞɟɫɶ ɢ ɞɚɥɟɟ ɭɫɪɟɞɧɟɧɢɟ ɩɨ ɪɟɚɥɢɡɚɰɢɢ ɫɥɭɱɚɣɧɵɯ ɩɨɥɟɣ ɨɛɨɡɧɚɱɟɧɨ ɫɤɨɛ-
ۃǥۄ
ɤɚɦɢ
.
ȼɨɨɛɳɟ ɝɨɜɨɪɹ, ɜ ɩɪɢɫɭɬɫɬɜɢɢ ɩɨɥɹ ɩɪɢɦɟɫɟɣ ɜɟɥɢɱɢɧɚ ȟ ɞɨɥɠɧɚ ɡɚɜɢɫɟɬɶ
ɨɬ ɤɨɨɪɞɢɧɚɬɵ ݖ, ɚ ɟɟ ɪɚɜɧɨɜɟɫɧɚɹ ɤɨɧɮɢɝɭɪɚɰɢɹ ɞɨɥɠɧɚ ɨɩɪɟɞɟɥɹɬɶɫɹ ɭɪɚɜɧɟ-
ɧɢɟɦ (ɫɦ. (11.20)):
127
ሻ
ሻ
.
ሻ
ሻ
ሻ
ȟሺݖ
ൌെܶൣߪො
ଶ
݃
ɝɞɟ ɦɚɰɭɛɚɪɨɜɫɤɚɹ ɮɭɧɤɰɢɹ ܩ
݅߱
݅ݒߪො
௦
μ
െߟሺݖሻെሺȟሺݖሻߞሺݖሻሻԜߪො
ଷ
μݖ
ൈܩ
ఠ
ெ
ሺ
݅߱
ெ
ሺ
݅߱
ሺିሻ
ெ
ሺ
݅߱
ܩ
ೞ
ሻ
ǡݖǡݖ
ɨɩɪɟɞɟɥɹɟɬɫɹ ɭɪɚɜɧɟɧɢɟɦ
௦
ሺାሻ
ǡݖǡݖሻൌͳߜሺݖെݖ
௦
ǡݖǡݖሻ൧
௦
െሺȟ
ᇱ
ሻ
ሺିሻ
൨ൈ
ሺ
ͳʹǤͷ
ሺ
ͳʹǤ
ǡ
כ
כ
ሺݖሻ
ሺݖሻ
ߞ
ሻԜߪො
Ǥ
ɉɪɨɢɡɜɟɞɟɦ ɭɫɪɟɞɧɟɧɢɟ ɭɪɚɜɧɟɧɢɹ (12.5) ɩɨ ɩɪɢɦɟɫɧɵɦ ɩɨɥɹɦ, ɩɨɥɭɱɢɦ
ȟ
ൌെܶൣߪො
ଶ
݃
ఠ
ɝɞɟ
ȟൌۃȟሺݖ
Ⱦɚɥɟɟ ɦɵ ɛɭɞɟɦ ɩɪɟɞɩɨɥɚɝɚɬɶ, ɱɬɨ ɜ ɝɚɦɢɥɶɬɨɧɢɚɧɟ (12.1) ɮɢɝɭɪɢɪɭɸɬ
ɢɦɟɧɧɨ ɩɨɫɬɨɹɧɧɵɟ ɜɟɥɢɱɢɧɵ ȟ ɢ ȟ
ɷɬɨɝɨ ɩɪɟɞɩɨɥɨɠɟɧɢɹ ɟɫɬɶ ɧɟɤɨɬɨɪɵɟ ɫɨɦɧɟɧɢɹ. Ɉɧɢ ɫɜɹɡɚɧɵ ɫ ɬɟɦ, ɱɬɨ ɜɨɨɛɳɟ
ɝɨɜɨɪɹ, ɩɪɢ ɭɫɪɟɞɧɟɧɢɢ ɩɨ ɩɪɢɦɟɫɹɦ
ɦɨɠɧɨ ɫɭɳɟɫɬɜɟɧɧɵɦ ɨɤɚɠɟɬɫɹ ɜɥɢɹɧɢɟ ɮɥɭɤɬɭɚɰɢɣ ȟሺݖ
ɧɨɜ. Ɉɞɧɚɤɨ ɦɨɠɧɨ ɩɨɤɚɡɚɬɶ, ɜ ɫɢɥɭ ɦɚɥɨɫɬɢ ɩɚɪɚɦɟɬɪɚ
ɬɭɚɰɢɢ ɨɤɚɡɵɜɚɸɬɫɹ ɧɟɫɭɳɟɫɬɜɟɧɧɵ, ɢ ɩɨɫɬɨɹɧɫɬɜɨ ȟ ɢ ȟ
ೞ
ሺିሻ
ெ
ሺ
ۃ
ܩ
݅߱
ሻ
ۄ
ǡȟכൌۃȟ
כ
ɢɡ ɭɪɚɜɧɟɧɢɹ (12.8). ȼ ɫɩɪɚɜɟɞɥɢɜɨɫɬɢ
ଶ
ሺݖሻ
ۄ
ۃ
ȟ
௦
כ
ሺݖሻ
്ۃȟሺݖ
ǡݖǡݖ
ۄ
ሻ
ۄ
൧ǡ
Ǥ
ଶ
ሻ
ۄ
ሻ
ሺ
ሺߝ
ଶ
ൌȟ
ɧɚ ɫɨɫɬɨɹɧɢɹ ɷɥɟɤɬɪɨ-
כ
, ɩɨɷɬɨɦɭ ɜɨɡ-
ିଵ
ሻ
ሻ
ிȀܶ
ɜ ɝɚɦɢɥɶɬɨɧɢɚɧɟ ɹɜ-
ሺ
ͳʹǤ
ሺ
ͳʹǤͺ
ɷɬɢ ɮɥɭɤ-
ɥɹɟɬɫɹ ɨɛɨɫɧɨɜɚɧɧɵɦ.
Ⱦɚɥɟɟ ɞɥɹ ɩɪɨɜɟɞɟɧɢɹ ɭɫɪɟɞɧɟɧɢɹ ɜ ɭɪɚɜɧɟɧɢɢ (11.67) ɦɵ ɩɪɢɦɟɧɢɦ ɬɟɯɧɢɤɭ
ܵ-ɦɚɬɪɢɰɵ, ɢɡɥɨɠɟɧɧɭɸ ɜ ɝɥɚɜɟ 8. Ⱦɥɹ ɷɬɨɝɨ ɜɨɫɩɨɥɶɡɭɟɦɫɹ ɫɨɨɬɧɨɲɟɧɢɹɦɢ
(3.59), (3.60), ɫɜɹɡɵɜɚɸɳɢɟ ɦɚɰɭɛɚɪɨɜɫɤɭɸ ɮɭɧɤɰɢɸ Ƚɪɢɧɚ ɫ ɨɩɟɪɟɠɚɸɳɟɣ ɢ
ɡɚɩɚɡɞɵɜɚɸɳɟɣ ɮɭɧɤɰɢɹɦɢ. ɗɬɨ ɩɨɡɜɨɥɹɟɬ ɩɟɪɟɩɢɫɚɬɶ (11.67) ɜ ɜɢɞɟ
ȟ
ൌെܶ൛ൣߪො
ଶ
݃
ఠೞவ
Ⱦɚɥɟɟ ɜɨɫɩɨɥɶɡɭɟɦɫɹ ɜɫɩɨɦɨɝɚɬɟɥɶɧɵɦɢ ɮɭɧɤɰɢɹɦɢ ݃ො
ሺିሻ
ோ
ሺ
ۃ
ܩ
݅߱
ǡݖǡݖ
௦
ሻ
ۄ
൧ൣߪො
ሺିሻ
ሺ
ۃ
ܩ
െ݅߱
௦
ோ
ɢ ݃ො
ሻ
ۄ
ǡݖǡݖ
ɢɡ (8.35), ɢ
൧ൟ
ǤሺͳʹǤͻ
ɫɪɚɡɭ ɜɵɱɢɫɥɢɦ ɫɥɟɞɵ ɦɚɬɪɢɰ, ɩɨɥɭɱɢɦ
ଶ
ൌെ
݅ܶ
ݒ
ఠೞவ
ሼۃ݃ො
ଵଶ
ோ
ሺ
݅߱
ǡݖȀݒǡݖȀݒ
௦
ۃ݃ො
ଵଶ
ሺ
െ݅߱
ǡݖȀݒǡݖȀݒ
௦
ሻ
ۄ
ሻ
ۄ
Ɍɚɤɢɦ ɨɛɪɚɡɨɦ, ɧɚɦ ɧɟɨɛɯɨɞɢɦɨ ɜɵɱɢɫɥɢɬɶ ɭɫɪɟɞɧɟɧɧɵɟ ɮɭɧɤɰɢɢ
ۄ
. ɇɚɱɧɟɦ ɫ
ோ
ۄ
ۃ
݃ො
.
ଵଶ
ሽ
ۃ
ሺ
Ǥ
ͳʹǤͳͲ
ோ
݃ො
ଵଶ
ۄ
ɢ
ۃ
ȟ
݃
݃ො
ଵଶ
12.1.2. ɍɫɪɟɞɧɟɧɢɟ ɮɭɧɤɰɢɣ Ƚɪɢɧɚ
ȼɨɫɩɨɥɶɡɭɟɦɫɹ ɮɨɪɦɭɥɚɦɢ (8.67) ɢ (8.69) ɜɵɪɚɠɚɸɳɢɦɢ ɮɭɧɤɰɢɢ ݃ො
ɱɟɪɟɡ ɷɥɟɦɟɧɬɵ S-ɦɚɬɢɰɵ. Ɍɨɝɞɚ ɞɥɹ ݃ො
ோ
ɢɦɟɟɦ (ɡɞɟɫɶ ݐൌݖȀݒ
):
ோ
ɢ ݃ො
128
ሻ
ሻ
ሻ
ሻ
ሻ
ሻ
መ
ܵ
ோ
ሺ
݅߱
ଵଶ
ǡݐǡݐሻൌെ
௦
݃ො
ఠೞԜଶଶ
ܵ
ሺ
λǡݐሻܵ
መ
ఠೞԜଶଶ
መ
ఠೞԜଵଶ
ሺ
λǡെλ
ሺ
ݐǡെλሻԜ
ሻ
Ǥ
ሺ
ͳʹǤͳͳ
Ɇɚɬɪɢɱɧɵɣ ɷɥɟɦɟɧɬ ɜ ɡɧɚɦɟɧɚɬɟɥɟ ɩɪɟɨɛɪɚɡɭɟɦ, ɩɨɥɶɡɭɹɫɶ ɝɪɭɩɩɨɜɵɦ ɫɜɨɣɫɬɜɨɦ S-ɦɚɬɢɰɵ (8.45). Ⱦɟɣɫɬɜɢɬɟɥɶɧɨ ɦɨɠɧɨ ɧɚɩɢɫɚɬɶ (ɱɚɫɬɨɬɵ ɞɥɹ ɩɪɨɫɬɨɬɵ
ɮɨɪɦɭɥ ɨɩɭɫɤɚɟɦ)
መ
ሺ
ܵ
λǡെλሻൌܵ
መ
ሺ
λǡݐሻܵ
መ
ሺ
ݐǡെλሻǡ
ሺ
ͳʹǤͳʹ
ɨɬɤɭɞɚ
ܵ
መ
ሺ
λǡെλሻൌܵ
ଶଶ
መ
ଶଵ
ሺ
λǡݐሻܵ
መ
ሺ
ݐǡെλሻܵ
ଵଶ
መ
ଶଶ
ሺ
λǡݐሻܵ
መ
ଶଶ
ሺ
ݐǡെλሻǡ
ሺ
ͳʹǤͳ͵
ɞɚɥɟɟ
ሻ
ሻ
ሻ
ൣܵ
መ
ሺ
λǡെλሻ൧
ଶଶ
ିଵ
ൌൣܵ
መ
ሺ
λǡݐሻܵ
ଶଶ
ஶ
ൌሺെͳ
ୀ
ሻ
መ
ଶଶ
ൣܵ
ሺ
ݐǡെλሻ൧
መ
ሺ
ൣܵ
ଶଵ
መ
ሺ
λǡݐሻܵ
ଶଶ
ିଵ
λǡݐሻܵ
መ
ଵଶ
መ
ଶଶ
ቆͳ
ሺ
ݐǡെλሻ൧
ሺ
ݐǡെλሻ൧
መ
ܵ
ଶଵ
መ
ܵ
ଶଶ
ሺ
λǡݐሻܵ
ሺ
λǡݐሻܵ
ାଵ
መ
ሺ
ሻ
ݐǡെλ
ଵଶ
መ
ሺ
ଶଶ
Ǥ
ݐǡെλ
ቇିଵൌ
ሻ
ሺ
ͳʹǤͳͶ
ɂɫɩɨɥɶɡɭɹ ɷɬɨ ɪɚɡɥɨɠɟɧɢɟ, ɮɭɧɤɰɢɸ Ƚɪɢɧɚ ɜ (12.11) ɦɨɠɧɨ ɡɚɩɢɫɚɬɶ ɜ ɜɢɞɟ
ஶ
݃ො
ோ
ሺ
݅߱
ǡݐǡݐሻൌെሺെͳ
௦
ଵଶ
ୀ
ሺ
ሻ
݅߱
ܽ
ǡݐሻܾ
௦
ାଵ
ሺ
݅߱
௦
ǡݐሻǡ
ሺ
ͳʹǤͳͷ
ɝɞɟ
መ
ሺ
ܵ
ܽሺ݅߱
ɉɪɢ ɩɪɨɜɟɞɟɧɢɢ ɭɫɪɟɞɧɟɧɢɹ ɜ (12.15) ɩɨ ɩɪɢɦɟɫɹɦ, ɜɟɥɢɱɢɧɵ ܽ
ାଵ
ሺ
݅߱
ܾ
௦
௦
ሻ
ǡݐ
ɭɫɪɟɞɧɹɸɬɫɹ ɧɟɡɚɜɢɫɢɦɨ, ɬɚɤ ɤɚɤ ɩɨɥɹ ߟ ɢ ߞ, ɜɯɨɞɹɳɢɟ ɜ ɧɢɯ, ɫɨɫɪɟ-
ǡݐሻൌ
ఠೞଶଵ
መ
ܵ
ఠೞଶଶ
ɞɨɬɨɱɟɧɵ ɧɚ ɧɟɩɟɪɟɫɟɤɚɸɳɢɯɫɹ ɢɧɬɟɪɜɚɥɚɯ
ሻ
λǡݐ
ǡܾሺ݅߱௦ǡݐሻൌ
ሺ
ሻ
λǡݐ
ሺ
λǡݐ
መ
ሺ
ܵ
ఠೞଵଶ
መ
ܵ
ఠೞଶଶ
ሻ
ሺ
ɢ
ݐǡെλ
ሻ
ݐǡെλ
ሺ
ሻ
ݐǡെλ
ሻ
, ɚ ɤɨɪɪɟɥɹɬɨɪɵ ɩɨ-
ሺ
Ǥ
ͳʹǤͳ
ሺ
݅߱
ሻ
ǡݐ
௦
ɥɟɣ ɩɪɨɩɨɪɰɢɨɧɚɥɶɧɵ ߜ-ɮɭɧɤɰɢɢ (12.3), (12.4), ɩɨɷɬɨɦɭ ɫɪɟɞɧɟɟ ɨɬ ɩɪɨɢɡɜɟɞɟ-
ɢ ܤ
ɧɢɹ ɩɨɥɟɣ ɜ ɪɚɡɧɵɯ ɬɨɱɤɚɯ ɪɚɧɨ ɧɭɥɸ. ȼɜɟɞɟɦ ɬɟɩɟɪɶ ɜɟɥɢɱɢɧɵ ܣ
ܣ
ሺ
݅߱
ɬɨɝɞɚ ɭɫɪɟɞɧɹɹ ɪɚɜɟɧɫɬɜɨ
ۃ݃ො
ଵଶ
ோ
Ⱦɥɹ ɜɵɱɢɫɥɟɧɢɹ ɜɟɥɢɱɢɧ ܣ
ɬɨɪɵɦ ɩɨɞɱɢɧɹɸɬɫɹ ܽ
Ⱦɥɹ ɷɬɨɝɨ ɦɵ ɜɨɫɩɨɥɶɡɭɟɦɫɹ ɩɪɟɞɫɬɚɜɥɟɧɢɟɦ ܵ-ɦɚɬɪɢɰɵ ɜ ɜɢɞɟ ɩɪɟɞɟɥɚ
ɩɨɥɹɦ.
ሻ
௦
ሺ
݅߱
ൌۃܽ
௦
ሺ
ሺ
ሻ
݅߱
ۄ
ͳʹǤͳͷ
ሺ
݅߱
௦
ሻ
, ɩɨɥɭɱɢɦ
ൌሺെͳ
ୀ
ɢ ܤ
ǡݐሻɢ ܾ
௦
ሻ
ǡݐ
ஶ
ɜɵɜɟɞɟɦ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɵɟ ɭɪɚɜɧɟɧɢɹ, ɤɨ-
ሺ
ۄ
ሺ
݅߱
ǡܤ
ାଵ
ሻ
ܣ
ሻ
݅߱
ǡݐ
, ɡɚɬɟɦ ɭɫɪɟɞɧɢɦ ɢɯ ɩɨ ɫɥɭɱɚɣɧɵɦ
௦
ሻ
௦
ሺ
ൌۃܾ
݅߱
ሺ
ሻ
ۄ
݅߱
ǡݐ
௦
ሻ
ሺ
݅߱
ሻ
௦
ܤ
௦
ାଵ
Ǥ
ǡ
ɫɨɝɥɚɫɧɨ
ሺ
ͳʹǤͳ
ሺ
ͳʹǤͳͺ
ɤɨɧɟɱɧɨɝɨ ɩɪɨɢɡɜɟɞɟɧɢɹ ɦɚɬɪɢɰ (ɮɨɪɦɭɥɚ (8.54)):
ሻ
ሻ
ሻ
ɢ
ሻ
ሻ
129
ே
ᇱ
ɩɪɢ ɷɬɨɦ ɨɬɪɟɡɨɤ
መ
ሺ
ݐǡݐ
ܵ
ఠ
ᇱ
ሾ
ǡݐሿ ɪɚɡɛɢɬ ɧɚ ܰ ɪɚɜɧɵɯ ɱɚɫɬɟɣ ݐᇱൌݐ൏ݐଵ൏ڮ൏ݐ
ݐ
ሻ
ൌ
௧
ෑ݁
ୀଵ
ൌݐ, ɝɞɟ ɲɚɝ ɪɚɡɛɢɟɧɢɹ ɟɫɬɶ ߜൌݐ
ɦɚɬɪɢɰɚ ܮ
ܮ
ሺ
ఠ
ೞ
ሻ
ɢɦɟɟɬ ɜɢɞ
ሺ
ሻ
ఠ
ೞ
ൌെ߱௦ߜߪොଷെ݅ȟɁߪො
ሺାሻ
ഘೖ
ൌ݁
ାଵെݐ
݅ȟכɁߪො
ഘಿ
ഘಿషభ
݁
ᇱ
ൌሺݐെݐ
ሺିሻ
ሻ
െ݅ߟߪොଷߞߪො
ഘమ
ڮ݁
ഘభ
݁
ǡ
ሺ
ͳʹǤͳͻ
ேିଵ൏ݐே
Ȁܰ. ɋ ɭɱɟɬɨɦ ɜɟɥɢɱɢɧ ȟ ɢ ȟ
ሺାሻ
ߞ
ሺିሻ
כ
ߪො
ǡሺͳʹǤʹͲ
ሻ
ൌ
כ
ሻ
ɝɞɟ ɫɥɭɱɚɣɧɵɟ ɩɟɪɟɦɟɧɧɵɟ ߟ
ൌߟሺݐ
ሻ
ߜǡߞ
ൌߞሺݐ
ሻ
ߜ ɨɛɥɚɞɚɸɬ ɤɨɪɪɟɥɹɬɨɪɚɦɢ
ɫɥɟɞɭɸɳɢɦɢ ɢɡ (11.62) ɢ (11.63)
ۦߟ
ۧ
ൌۦߞ
ۧ
ൌͲǡ᩷ۦߟߟ
ۧൌߜ
ߜ
ǡ᩷ۦߞߞ
߬
ଵ
כ
ۧ
ൌߜ
ߜ
ǤሺͳʹǤʹͳ
߬
ଶ
Ɋɚɡɥɨɠɢɦ ɫɨɦɧɨɠɢɬɟɥɶ ɜ (11.83) ɜ ɪɹɞ ɩɨ ߜ ɞɨ ɩɟɪɜɨɝɨ ɩɨɪɹɞɤɚ. ɉɪɢ ɷɬɨɦ
ɩɨɥɹ ߟ
ɢ ߞ
ɧɭɠɧɨ ɫɱɢɬɚɬɶ ɜɟɥɢɱɢɧɚɦɢ ɩɨɪɹɞɤɚ
ߜ ɢ ɭɞɟɪɠɢɜɚɬɶ ɢɯ ɧɟɢɫɱɟɡɚɸ-
ξ
ɳɢɟ ɩɪɢ ɭɫɪɟɞɧɟɧɢɢ ɩɪɨɢɡɜɟɞɟɧɢɹ, ɬɨɝɞɚ
ͳ
ఋ
ഘೞೖ
ൌ݁
െ݅ȟɁߞ
ͳ߱௦ߜ݅ߟ
ൌͳܮ
൰
ఠೞ
ͳ
െɄ
ቆ
ʹ
ͲെɄ
ൌ൬
ͳെ߱
ߜെ݅ߟ
௦
݅ȟכߜߞ
መ
ܵ
כ
ɂɫɩɨɥɶɡɭɹ ɷɬɨ ɜɵɪɚɠɟɧɢɹ ɞɥɹ ɦɚɬɪɢɰɵ «ɨɞɧɨɝɨ ɲɚɝɚ» ܵ
ɦɟɧɹɟɬɫɹ ɜɟɥɢɱɢɧɚ ܾ
ሺݐሻ
ɩɪɢ ɩɪɢɪɚɳɟɧɢɢ ɤɨɨɪɞɢɧɚɬɵ ɧɚ ɜɟɥɢɱɢɧɭ ߜ (ɱɚɫɬɨɬɵ
ଶ
ଶ
ܮ
ఠೞ
ʹ
ȁߞȁ
ൌ
ଶ
ଶ
መ
Ͳ
ቇǤሺͳʹǤʹʹ
ଶ
ȁߞȁ
ఋ
, ɪɚɫɫɦɨɬɪɢɦ ɤɚɤ
ɨɩɭɫɬɢɦ ɞɥɹ ɭɩɪɨɳɟɧɢɹ ɮɨɪɦɭɥ)
ൌ
ൌܾ
ሺ
ݐߜሻൌܵ
ܾ
ఋ
መ
መ
൯ଵଵܵ
ቂ൫ܵ
ൌ
ఋ
መ
൯ଶଶܵ
መ
ൣሺͳെ߱
ቂ൫ܵ
ߜെ݅ߟെɄ
௦
ൣሺͳ߱௦ߜ݅ߟെɄ
ሺݐሻ
൬ͳെ߱
ߜെ݅ߟെ
௦
൬ͳ߱௦ߜ݅ߟെ
መ
ሺ
ݐߜǡെλሻȀܵ
ଵଶ
ሺ
ݐǡെλሻ൫ܵ
ଵଶ
ሺ
ݐǡെλሻ൫ܵ
ଶଶ
ଶ
ȀʹȁߞȁଶȀʹሻܵ
ଶ
ȀʹȁߞȁଶȀʹሻܵ
ଶ
Ʉ
ʹ
ଶ
Ʉ
ʹ
መ
ଶଶ
ఋ
መ
መ
൯ଵଶܵ
ଶଶ
ఋ
መ
መ
൯ଶଵܵ
ଵଶ
መ
ሺെ݅ȟɁߞ
ଵଶ
መ
ଶଶ
ଶ
ȁߞ
ȁ
൰ሺെ݅ȟɁߞ
ʹ
ଶ
ȁ
ȁߞ
൰ሺ݅ȟכߜߞ
ʹ
ሺ
ݐߜǡെλሻൌ
ሺ
ݐǡെλሻቃ
ሺ
ݐǡെλሻቃ
ሺ݅ȟכߜߞ
ൌ
መ
ሻ
ܵ
൧
ଶଶ
ൌ
כ
መ
ሻ
ܵ
൧
ଵଶ
መ
ܵ
ଶଶ
ሻ
൬
൰൨
መ
ܵ
ଵଶ
ǤሺͳʹǤʹ͵
መ
ܵ
ଵଶ
כ
ሻ
൬
൰൨
መ
ܵ
ଶଶ
ȼɨɫɩɨɥɶɡɭɟɦɫɹ ɮɨɪɦɭɥɨɣ ɩɪɢ ݔǡݕ՜Ͳ
ଶ
݊
ଶݕଶ
ሺ
ݔ
ʹ
݊
ሻ
ଶെݕଶ
ሺ
െ
ݔ
ʹ
ሻǡሺ
൬
ͳݔ
൰ൎͳ݊ሺݔെݕሻെ݊ଶݔݕ
ͳݕ
ͳʹǤʹͶ
ሻ
ሻ
ሻ
ሻ
130
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