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Ɋɢɫ. 9.1
ɋɚɦɨɣ ܵ-ɦɚɬɪɢɰɟ ɩɨ ɷɬɨɣ ɫɯɟɦɟ ɧɭɠɧɨ ɩɨɫɬɚɜɢɬɶ ɜ ɫɨɨɬɜɟɬɫɬɜɢɟ ɞɪɨɛɧɨ-
ɥɢɧɟɣɧɨɟ ɩɪɟɨɛɪɚɡɨɜɚɧɢɟ
ݖհܹሺݖሻൌ
መ
ܵ
መ
ܵ
ଵଵ
ଵଶ
Ԝݖܵ
Ԝݖܵ
መ
ଵଶ
ǡ
መ
ଶଶ
ሺ
ͻǤͳ͵
ɝɞɟ ɦɵ ɞɥɹ ɤɪɚɬɤɨɫɬɢ ɨɩɭɫɬɢɥɢ ɚɪɝɭɦɟɧɬɵ ݐ ɢ Ͳ ɭ ܵ-ɦɚɬɪɢɰɵ. ȿɫɥɢ ɜɵɛɪɚɬɶ ɜ
ɤɚɱɟɫɬɜɟ ɧɚɱɚɥɶɧɨɣ ɬɨɱɤɢ ɬɪɚɟɤɬɨɪɢɢ ɫɥɭɱɚɣɧɨɝɨ ɛɥɭɠɞɚɧɢɹ ݖ
ൌͲ, ɬɨ ɤɨɧɟɱ-
ɧɨɣ ɬɨɱɤɨɣ ɛɭɞɟɬ
ܹሺͲሻൌ
ଵଶ
ǡ
መ
ܵ
ଶଶ
ሺ
ͻǤͳͶ
መ
ܵ
ɬɚɤ ɤɚɤ ɜ ɪɟɡɭɥɶɬɚɬɟ ɤɨɦɩɨɡɢɰɢɢ (9.10) ɦɵ ɩɨɥɭɱɚɟɦ ɤɚɤ ɪɚɡ ɪɟɡɭɥɶɬɢɪɭɸɳɟɟ
ɩɪɟɨɛɪɚɡɨɜɚɧɢɟ (9.13). Ⱦɚɥɟɟ ɢɦɟɟɦ
ଶ
ȁܹሺͲ
መ
หܵ
ଵଶ
ଶ
ሻ
ȁ
ൌ
መ
หܵ
ଶଶ
መ
ห
ܵ
ଵଶ
ൌ
ଶ
መ
ܵ
ଶଶ
ห
መ
መ
ܵ
ଶଵ
መ
ܵ
ଵଵ
መ
ܵ
ܵ
െͳ
ଶଶ
ଵଵ
ൌ
መ
ܵ
ଶଶ
መ
ܵ
ଵଵ
ൌͳെൣܵ
መ
መ
ܵ
൧ିଵൌͳെܴሺݐ
ଶଶ
ଵଵ
ሻǡሺ
ͻǤͳͷ
ɝɞɟ ɦɵ ɭɱɥɢ (9.2), ɢ ɩɨɷɬɨɦɭ
ሻ
ܴሺݐ
ൌͳെȁܹሺͲ
ɋɥɟɞɨɜɚɬɟɥɶɧɨ, ɞɥɹ ɜɵɱɢɫɥɟɧɢɹ ɫɪɟɞɧɟɣ ɜɟɥɢɱɢɧɵ ܴሺݐ
ଶ
ሻ
ȁ
Ǥ
ሻ
ɧɟɨɛɯɨɞɢɦɨ ɭɫɪɟɞ-
ሺ
ͻǤͳ
ɧɢɬɶ ɤɜɚɞɪɚɬ ɦɨɞɭɥɹ ɤɨɧɟɱɧɨɣ ɬɨɱɤɢ ɬɪɚɟɤɬɨɪɢɢ ɫɥɭɱɚɣɧɨɝɨ ɛɥɭɠɞɚɧɢɹ ɩɨ ɟɝɨ
ɪɟɚɥɢɡɚɰɢɹɦ.
Ⱦɥɹ ɪɟɲɟɧɢɹ ɷɬɨɣ ɡɚɞɚɱɢ ɦɵ ɜɜɟɞɟɦ ɮɭɧɤɰɢɸ ɪɚɫɩɪɟɞɟɥɟɧɢɹ ɞɥɹ ɤɨɨɪɞɢɧɚɬ
ܰ-ɣ ɬɨɱɤɢ ɬɪɚɟɤɬɨɪɢɢ ɫɥɭɱɚɣɧɨɝɨ ɛɥɭɠɞɚɧɢɹ ɢ ɩɨɥɭɱɢɦ ɭɪɚɜɧɟɧɢɟ ɞɥɹ ɷɬɨɣ
ɮɭɧɤɰɢɢ ɚɧɚɥɨɝɢɱɧɨɟ ɭɪɚɜɧɟɧɢɸ ɞɢɮɮɭɡɢɢ. Ɋɟɲɢɜ ɟɝɨ ɢ ɧɚɣɞɹ ɮɭɧɤɰɢɸ ɪɚɫɩɪɟɞɟɥɟɧɢɹ, ɦɵ ɜɵɱɢɫɥɢɦ ɫɪɟɞɧɟɟ ɨɬ ɜɟɥɢɱɢɧɵ
ܴሺݐ
ሻ
.
ሻ
ሻ
ሻ
ሻ
91
9.2. Ɏɭɧɤɰɢɹ ɪɚɫɩɪɟɞɟɥɟɧɢɹ ɢ ɭɪɚɜɧɟɧɢɟ ɞɢɮɮɭɡɢɢ
ɧɚ ɩɥɨɫɤɨɫɬɢ Ʌɨɛɚɱɟɜɫɤɨɝɨ
Ɍɚɤɢɦ ɨɛɪɚɡɨɦ, ɤɚɤ ɭɠɟ ɭɤɚɡɵɜɚɥɨɫɶ ɜɵɲɟ, ɤɚɠɞɵɣ ɲɚɝ ɫɥɭɱɚɣɧɨɝɨ ɛɥɭɠɞɚ-
ɧɢɹ (9.10) ɟɫɬɶ ɩɪɟɨɛɪɚɡɨɜɚɧɢɟ, ɛɥɢɡɤɨɟ ɤ ɬɨɠɞɟɫɬɜɟɧɧɨɦɭ. ɉɪɢ ɜɵɱɢɫɥɟɧɢɢ
ɫɪɟɞɧɢɯ ɩɪɟɞɩɨɥɚɝɚɟɬɫɹ ɩɪɟɞɟɥɶɧɵɣ ɩɟɪɟɯɨɞ, ɩɪɢ ɤɨɬɨɪɨɦ «ɜɟɥɢɱɢɧɚ ɲɚɝɚ» ߜ ɛɭ-
ɞɟɬ ɫɬɪɟɦɢɬɫɹ ɤ ɧɭɥɸ, ɚ ɱɢɫɥɨ ɲɚɝɨɜ ܰ ɤ ɛɟɫɤɨɧɟɱɧɨɫɬɢ, ɩɪɢ ɭɫɥɨɜɢɢ ܰԜߜൌݐ
ɇɚɲɚ ɡɚɞɚɱɚ ɫɨɫɬɨɢɬ ɜ ɩɨɫɬɪɨɟɧɢɢ ɦɟɬɨɞɚ ɢɫɫɥɟɞɨɜɚɧɢɹ ɚɫɢɦɩɬɨɬɢɱɟɫɤɨɝɨ ɩɨɜɟɞɟɧɢɹ ɜɟɥɢɱɢɧ
.
ሻ
ߔሺݐ
ɞɥɹ ɩɪɨɢɡɜɨɥɶɧɵɯ ɮɭɧɤɰɢɣ ݂ሺݖ
ൌ
ǡݖ
ఋ՜ǡே՜ஶǡேఋୀ௧
൏݂൫ݓෝேלڮלݓෝ
ಽ
ሻ
ɧɚ ɩɥɨɫɤɨɫɬɢ Ʌɨɛɚɱɟɜɫɤɨɝɨ.
ȼɵɩɢɲɟɦ ɞɪɨɛɧɨ-ɥɢɧɟɣɧɨɟ ɩɪɟɨɛɪɚɡɨɜɚɧɢɟ ɫɨɨɬɜɟɬɫɬɜɭɸɳɟɟ ɦɚɬɪɢɰɟ ݁
ሺ
ሻ
ݖ
൯
ଵ
ሺ
ͻǤͳ
ɫ
ɬɨɱɧɨɫɬɶɸ ɞɨ ߂, ɩɪɢ ɷɬɨɦ ɧɚɞɨ ɢɦɟɬɶ ɜ ɜɢɞɭ, ɱɬɨ ɜ ɧɟɤɨɬɨɪɨɦ ɫɦɵɫɥɟ ɜɟɥɢɱɢɧɵ
ଶ
ȁ
ɢ ߞ
ɢɦɟɸɬ ɩɨɪɹɞɨɤ ɦɚɥɨɫɬɢ
ߟ
ܮ
Ɉɛɨɡɧɚɱɢɦ ݀
ൎͳܮ
ൌെߟ
ͳ
ʹ
ଶ
ɨɬɜɟɬɫɬɜɭɸɳɟɟ ɦɚɬɪɢɰɟ ܮ
ଶ
ܮ
ൌ൬ͳെ
Ȁʹȁߞ
ݓෝ
ξ
ͳ
ଶ
ߟ
ʹ
ଶ
ȁ
Ȁʹ, ɬɨɝɞɚ ɞɪɨɛɧɨ-ɥɢɧɟɣɧɨɟ ɩɪɟɨɛɪɚɡɨɜɚɧɢɟ ɫɨ-
ɟɫɬɶ:
ሺ
ሺݖሻ
ͳ݀
ൌ
כ
ߞ
Ԝݖͳ݀݅ߟ
Ɋɚɡɥɨɠɢɜ ɡɧɚɦɟɧɚɬɟɥɶ ɷɬɨɣ ɞɪɨɛɢ ɜ ɪɹɞ ɩɨ ߟ
ͳ
ଶ
ȁ
ȁ
ߞ
൰െ݅ߟԜߪොଷߞԜߪොାߞ
ʹ
ሻ
Ԝݖߞ
െ݅ߟ
ɢ ߞ
, ɩɨɥɭɱɢɦ ɩɪɟɨɛɪɚɡɨɜɚɧɢɟ ɜ
Ǥ
כ
ȁ
ɢ
ߞ
ԜߪොିǤሺͻǤͳͺ
ሺ
ͻǤͳͻ
ɹɜɧɨɦ ɜɢɞɟ ɛɥɢɡɤɨɟ ɤ ɬɨɠɞɟɫɬɜɟɧɧɨɦɭ:
ݓෝ
ሺݖሻ
ൌݖ൫െ݅ʹߟ
െʹߟ
ଶ
െȁߞ
ଶ
ȁ
൯ݖߞെߞ
כ
ݖଶǤ
ሺ
ͻǤʹͲ
ȼɜɟɞɟɦ ɧɚ ɩɥɨɫɤɨɫɬɢ Ʌɨɛɚɱɟɜɫɤɨɝɨ ɩɨɥɹɪɧɵɟ ɤɨɨɪɞɢɧɚɬɵ ߠ ɢ ߮ ɬɚɤ, ɱɬɨ
ݖൌthሺߠȀʹሻ݁
ఝ
ǡ
ሺ
ͻǤʹͳ
ɜ ɷɬɢɯ ɤɨɨɪɞɢɧɚɬɚɯ ɦɟɬɪɢɤɚ (9.8) ɩɪɢɦɟɬ ɜɢɞ
݀ݏ
ଶ
ൌ݀ߠଶ
ଶ
ሺߠሻ
݀߮
ଶ
Ǥ
ሺ
ͻǤʹʹ
ሻ
ଶ
ሻ
ሻ
ሻ
ሻ
ሻ
ɋɥɭɱɚɣɧɨɦɭ ɛɥɭɠɞɚɧɢɸ ɫɨɨɬɜɟɬɫɬɜɭɟɬ ɩɨɫɥɟɞɨɜɚɬɟɥɶɧɨɫɬɶ ɩɚɪ ɤɨɨɪɞɢɧɚɬ
ǡ߮
ሻǡሺ
ሺߠ
ɩɪɢ ɷɬɨɦ ɩɨɞɥɟɠɚɳɚɹ ɭɫɪɟɞɧɟɧɢɸ ɜɟɥɢɱɢɧɚ ܴሺܶ
ܴሺݐ
ሻ
ൌͳെȁܹሺͲ
ሻ
ȁ
ሻǡǥሺ
ߠ
ଵǡ߮ଵ
ଶ
ൌͳെଶ൬
ߠ
ேିଵǡ߮ேିଵ
ߠ
ே
൰ൌିଶ൬
ʹ
ሻǡሺ
ሻ
, ɫɨɝɥɚɫɧɨ (9.16), ɩɪɢɦɟɬ ɜɢɞ
ߠ
ேǡ߮ே
ߠ
ே
ʹ
൰ൌ
ሻ
ǡ
ሺߠ
ʹ
ሻ
ͳ
ே
92
ሺ
ͻǤʹ͵
ǤሺͻǤʹͶ
ሻ
ሻ
ሺ
ሻ
ݖ
ɋ ɩɨɦɨɳɶɸ ɧɟɫɥɨɠɧɵɯ ɜɵɱɢɫɥɟɧɢɣ, ɢɫɩɨɥɶɡɭɹ (9.20) ɢ ݖ
ɫɨɨɬɧɨɲɟɧɢɟ ɦɟɠɞɭ ɩɨɫɥɟɞɨɜɚɬɟɥɶɧɵɦɢ ɤɨɨɪɞɢɧɚɬɚɦɢ
כ
ߠ
߮
ൌߠିଵ
ൌ߮ିଵ
ఝ
ߜߠǡ᩷ߜߠൌߞ
షభ
݁
ߜ߮ǡ᩷ߜ߮ൌെʹߟ݅൫ߞ
ߞ݁
כ
݁
ఝ
ିఝ
షభ
షభ
െߞ݁
ȁߞ
ሺ
ߠ
ିఝ
ǡ߮
ଶ
ȁ
ሺߠ
షభ
ൌݓෝ
ሻ
ɢ
൯ሺߠ
ିଵ
ሺ
ߠ
ିଵǡ߮ିଵ
ሻ
ିଵ
ିଵ
, ɧɚɣɞɟɦ
Ǣ
ሻǤሺ
ሺ
ͻǤʹͷ
ͻǤʹ
ሻ
:
9.2.1. Ɉɩɪɟɞɟɥɟɧɢɟ ɮɭɧɤɰɢɢ ɪɚɫɩɪɟɞɟɥɟɧɢɹ
ɢ ɪɟɤɭɪɪɟɧɬɧɨɟ ɫɨɨɬɧɨɲɟɧɢɟ
ሺ
Ⱦɥɹ ɥɸɛɨɝɨ ݇ ɨɩɪɟɞɟɥɢɦ ɮɭɧɤɰɢɸ ɪɚɫɩɪɟɞɟɥɟɧɢɹ ࣪
ɬɨɱɤɢ ɫɥɭɱɚɣɧɨɝɨ ɛɥɭɠɞɚɧɢɹ ɫ ɧɚɱɚɥɶɧɨɣ ɬɨɱɤɨɣ ݖ
ሺ
ሻ
ݖ
൯ൿൌන Ԝ݀ߤሺݖ
ଵ
Ͷ݀ݔ݀ݕ
ሺ
ͳെȁݖ
ȁ௭ȁ
ழଵ
ൌሺɅሻɅɔ
ଶ
ሻ
ȁ
ɝɞɟ ݂ሺݖ
לڮלݓෝ
ൻ݂൫ݓෝ
ሻ
– ɩɪɨɢɡɜɨɥɶɧɚɹ ɮɭɧɤɰɢɹ, ɚ
݀ߤሺݖሻൌ
ɫɥɟɞɭɸɳɢɦ ɫɨɨɬɧɨɲɟɧɢɟɦ
ሻ
݂ሺݖሻ࣪
ሻ
ݖǡݖ
ɤɨɨɪɞɢɧɚɬ ݇-ɣ
ሺ
ሻ
ݖǡݖ
ǡ
ሺ
ͻǤʹ
ሺ
ͻǤʹͺ
– ɢɧɜɚɪɢɚɧɬɧɚɹ ɦɟɪɚ ɩɨ ɩɥɨɫɤɨɫɬɢ Ʌɨɛɚɱɟɜɫɤɨɝɨ, ɫɨɨɬɜɟɬɫɬɜɭɸɳɚɹ ɦɟɬɪɢɤɟ
(9.8). Ɏɭɧɤɰɢɢ ࣪
ሻ
ݖǡݖ
ɨɩɪɟɞɟɥɹɟɬ ɮɭɧɤɰɢɸ ɪɚɫɩɪɟɞɟɥɟɧɢɹ
௧
ሺ
ߠǡ߮
ሻ
, ɡɚɜɢɫɹ-
ሺ
ɳɭɸ ɨɬ ɧɟɩɪɟɪɵɜɧɨɝɨ ɜɪɟɦɟɧɢ ݐ ɫ ɩɨɦɨɳɶɸ ɩɪɟɞɟɥɶɧɨɝɨ ɩɟɪɟɯɨɞɚ
ሺݐǡߠǡ߮ሻൌ
ఋ՜ǡԜே՜ஶǡԜேఋୀ௧
᩷࣪ே൫thሺߠȀʹሻ݁ఝǡͲ൯Ǥ
Ɏɭɧɤɰɢɢ ሺݐǡߠǡ߮ሻ ɩɨɡɜɨɥɹɸɬ ɩɪɨɜɟɫɬɢ ɭɫɪɟɞɧɟɧɢɟ ɜɟɥɢɱɢɧɵ ܴሺܶ
ሺ
ͻǤʹͻ
ሻ
ɩɪɨɫɬɵɦ
ɢɧɬɟɝɪɢɪɨɜɚɧɢɟɦ:
ஶ
ۃ
ܴሺݐ
ሻ
ۄ
ൌන݀ߠ
ʹሺߠ
ሺߠሻͳ
ɉɨɤɚɠɟɦ ɬɟɩɟɪɶ, ɱɬɨ ɮɭɧɤɰɢɹ ɪɚɫɩɪɟɞɟɥɟɧɢɹ ࣪
ଶగ
ሻ
නሺݐǡߠǡ߮ሻ݀߮
ሺ
ݖǡݖ
Ǥ
ሻ
ɞɥɹ ɥɸɛɨɝɨ ݇ ɭɞɨɜɥɟ-
ሺ
ͻǤ͵Ͳ
ɬɜɨɪɹɟɬ ɫɥɟɞɭɸɳɟɦɭ ɪɟɤɭɪɪɟɧɬɧɨɦɭ ɫɨɨɬɧɨɲɟɧɢɸ
ିଵ
ିଵ
ሺ
ሺݖሻ
ݓෝ
ɝɞɟ ɫɤɨɛɤɢ
ሺ
࣪
ۦ
ۧ
ɨɡɧɚɱɚɸɬ ɭɫɪɟɞɧɟɧɢɟ ɬɨɥɶɤɨ ɩɨ ɫɥɭɱɚɣɧɵɦ ɩɟɪɟɦɟɧɧɵɦ ߟǡԜߞ
ǤǤǤ
ሻ
ݖǡݖ
ൌۦ࣪
Ⱦɥɹ ɷɬɨɝɨ ɨɛɨɡɧɚɱɢɦ ɞɥɹ ɩɪɨɢɡɜɨɥɶɧɨɣ ɮɭɧɤɰɢɢ ݂ሺݖ
ሺሻ
᪄
ൌൻ݂൫ݓෝלڮלݓෝ
݂
ଵ
ሻ
ۧ
ǡݖ
ሺ
ݖ
ǡ
ሻ
ሻ
൯ൿ
ǡ
ሺ
ͻǤ͵ͳ
ሺ
ͻǤ͵ʹ
ɪɚɫɫɦɨɬɪɢɦ ɬɚɤɠɟ ɜɫɩɨɦɨɝɚɬɟɥɶɧɭɸ ɮɭɧɤɰɢɸ
ሻ
ሻ
ሻ
ሻ
ሻ
ሻ
ሻ
.
ሻ
݃ሺݖሻൌ݂൫ݓ
ɡɞɟɫɶ ɩɨ ɫɥɭɱɚɣɧɵɦ ɩɟɪɟɦɟɧɧɵɦ ߟ
(9.32) ɢ (9.27)
ሺݖሻ
൯ǡ
ǡԜߞ
ɭɫɪɟɞɧɟɧɢɹ ɧɟɬ, ɬɨɝɞɚ ɩɨ ɨɩɪɟɞɟɥɟɧɢɹɦ
93
ሺ
ͻǤ͵͵
ሻ
݃᪄
ିଵ
ൌൻ݃൫ݓෝ
ିଵ
לڮלݓෝ
ൌන Ԝ݀ߤሺݖ
ȁ௭ȁ
ழଵ
ሺ
ሻ
ݖ
൯ൿ
ሻ
݂൫ݓෝ
ൌන Ԝ݀ߤሺݖ
ȁ௭ȁ
ሺݖሻ
൯࣪
ழଵ
ଵ
ିଵ
ሺ
ݖǡݖ
ሻ
݃ሺݖሻ࣪
ሻ
ǡ
ିଵ
ሺ
ሻ
ɫ ɞɪɭɝɨɣ ɫɬɨɪɨɧɵ ɩɨ ɨɩɪɟɞɟɥɟɧɢɸ
ൻ݂൫ݓෝ
לڮלݓෝ
ሻ
ݖ
൯ൿ
ଵ
ൌൻ݃൫ݓෝ
ൌන Ԝ݀ߤሺݖ
ȁ௭ȁ
לڮלݓෝ
ିଵ
ழଵ
ሻ
ሺ
ሻ
ݖ
൯ൿ
ଵ
݂ሺݖሻ࣪
ሺ
ݖǡݖ
ሺ
ɩɨɷɬɨɦɭ
න Ԝ݀ߤሺݖ
ȁ௭ȁ
ழଵ
ሻ
݂ሺݖሻ࣪
ሻ
ݖǡݖ
ൌൽන Ԝ݀ߤሺݖ
ȁ௭ȁ
ழଵ
ሻ
݂൫ݓෝ
ሺݖሻ
൯࣪
ିଵ
ሺ
ɍɱɢɬɵɜɚɹ, ɱɬɨ ɦɟɪɚ ɢɧɜɚɪɢɚɧɬɧɚ ɩɪɢ ɞɜɢɠɟɧɢɹɯ ɦɟɬɪɢɤɢ ݀ߤቀݓෝ
ሻ
ൌ݀ߤሺݖ
, ɩɨɥɭɱɢɦ
ሺ
ሻ
ݖǡݖ
ሺ
ൌ
ݖǡݖ
ሻ
ሻ
ඁ
ൌ
ሺ
ͻǤ͵Ͷ
ሺ
ͻǤ͵ͷ
ǤሺͻǤ͵
ିଵ
ሺݖሻ
ሻ
ሻ
ሻ
ቁൌ
න Ԝ݀ߤሺݖ
ȁ௭ȁ
ழଵ
ሻ
ɬɚɤ ɤɚɤ ɮɭɧɤɰɢɹ ݂ሺݖ
ሺ
ሻ
ݖǡݖ
݂ሺݖሻ࣪
ሻ
ɩɪɨɢɡɜɨɥɶɧɚɹ ɩɨɥɭɱɚɟɦ (9.31).
ൌන Ԝ݀ߤሺݖ
ȁ௭ȁ
ழଵ
ሻ
݂ሺݖ
ିଵ
ሺ
ሻ
ۦ
࣪
ିଵ
ሺݖሻ
ݓෝ
ǡݖ
ሻ
ۧ
ǡሺͻǤ͵
9.2.2. ȼɵɜɨɞ ɭɪɚɜɧɟɧɢɹ ɞɢɮɮɭɡɢɢ
ɂɫɩɨɥɶɡɭɹ ɮɨɪɦɭɥɭ (9.31) ɞɥɹ ݇ൌܰ, ɩɨɥɭɱɢɦ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɨɟ ɭɪɚɜɧɟɧɢɟ
ɧɚ ɮɭɧɤɰɢɸ ɪɚɫɩɪɟɞɟɥɟɧɢɹ
ሺ
ߠǡ߮ሻ. Ⱦɥɹ ɷɬɨɝɨ ɪɚɫɩɢɲɟɦ ɩɪɟɨɛɪɚɡɨɜɚɧɢɟ ݓෝ
௧
ே
ିଵ
ɩɨɥɹɪɧɵɯ ɤɨɨɪɞɢɧɚɬɚɯ, ɫɨɝɥɚɫɧɨ (9.25) (9.26), ɫ ɭɱɟɬɨɦ ɬɨɝɨ, ɱɬɨ ɨɛɪɚɬɧɨɟ ɩɪɟ-
ɨɛɪɚɡɨɜɚɧɢɟ ɩɨɥɭɱɚɟɬɫɹ ɢɡ ݓෝ
ɡɚɦɟɧɨɣ ɡɧɚɤɚ ɩɪɢ ߟேǡߞ
ே
ɟɬɫɹ (ɚɪɝɭɦɟɧɬ ɧɚɱɚɥɶɧɨɣ ɬɨɱɤɢ ݖ
࣪
ሺ
ே
ߠ
ேǡ߮ே
ሻ
ൌۦ࣪
ɨɩɭɫɤɚɟɦ ɞɥɹ ɤɪɚɬɤɨɫɬɢ):
ሺ
ߠ
ேିଵ
ߜߠேǡ߮ேߜ߮
ே
. Ɍɨɝɞɚ ɢɡ (9.31) ɩɨɥɭɱɚ-
ே
ሻ
ۧ
ǡ
ே
ே
ሺ
ͻǤ͵ͺ
ɝɞɟ ɜ ɞɚɧɧɨɦ ɫɥɭɱɚɟ
ߜߠ
ߜ߮
ே
ே
ఝ
ൌെߞ
ಿ
݁
ே
ൌʹߟே݅൫െߞ
െߞே݁
כ
݁
ே
ఝ
ିఝ
ಿ
ߞே݁
ಿ
ȁߞ
ே
ିఝ
ଶ
ȁ
ሺߠ
ಿ
൯ሺߠ
ሻ
ǡ
ே
ሻ
Ǥ
ே
ሺ
ͻǤ͵ͻ
ሺ
ͻǤͶͲ
כ
Ɋɚɫɤɥɚɞɵɜɚɟɦ ɜ ɪɹɞ ɩɨ ɩɪɢɪɚɳɟɧɢɹɦ ɢ ɨɩɭɫɤɚɟɦ ɢɧɞɟɤɫ ܰ ɭ ɩɟɪɟɦɟɧɧɵɯ ߟ ɢ ߞ
μ࣪
ሺ
ߠǡ߮ሻൌ࣪
࣪
ே
ͳ
ሺ
ۦ
ቈ
ߜߠ
ʹ
μଶ࣪
ଶ
ሻ
ۧ
ே
ே
ேିଵ
ேିଵ
μߠ
ሺ
ߠǡ߮ሻۦߜߠ
ଶ
ଶ
ሻ
ሺ
ۦ
ߜ߮
ۧ
ே
ேۧே
ே
μଶ࣪
μ߮
ேିଵ
μߠ
ேିଵ
ʹۦߜߠேߜ߮
ଶ
ۦߜ߮
ேۧே
μ࣪
ேۧே
ேିଵ
μ߮
μଶ࣪
μߠμ߮
ேିଵ
ǡሺͻǤͶͳ
ɜ
ሻ
ሻ
ሻ
ሻ
ሻ
94
ɭɱɬɟɦ, ɱɬɨ ɫɨɝɥɚɫɧɨ (8.53)
Ɂ
ே
ۄ
ۃ
ߜߠ
ൌ
ሻ
ே
ሺɅሻǡۦߜ߮
߬
ଶ
ே
ଶ
ۄ
ൌ
ͶɁ
߬
ଵ
ே
ሺ
ۃ
ߜ߮
ʹɁ
߬
ே
ۧ
ே
ଶ
ଶ
ൌͲǡ
ሺɅሻ
ۃ
ǡۃߜߠ
ሺ
ߜߠ
ே
ேߜ߮ே
ே
ଶ
ሻ
ۄ
ே
ۄ
ɬɨɝɞɚ ɭɪɚɜɧɟɧɢɟ ɩɪɢɦɟɬ ɜɢɞ
ሺ
࣪
ߠǡ߮ሻെ࣪
ே
ሺ
ሻ
ߠǡ߮
ேିଵ
Ɂ
ʹ
ቆ
߬
ͳ
ൌ
߬
ଶ
ͳ
߬
ଵ
ଶ
ሺɅ
ଶ
ሺɅሻ
μ࣪
ேିଵ
ሻ
μߠ
ଶ
μ
࣪
ቇ
μ߮
ேିଵ
ଶ
ͳ
߬
ଶ
ɂɡ ɷɬɨɝɨ ɭɪɚɜɧɟɧɢɹ ɩɨɫɥɟ ɩɪɟɞɟɥɶɧɨɝɨ ɩɟɪɟɯɨɞɚ (9.29) ɩɨɥɭɱɢɦ
ൌ
ൌͲǡ
ଶ
μ
μߠ
࣪
ʹɁ
߬
ேିଵ
ሺ
ሻ
ǡ
ଶ
ଶ
ͻǤͶʹ
ሺ
ͻǤͶ͵
ሺ
ͻǤͶͶ
ሻ
ሻ
ɝɞɟ ߬
ିଵ
ൌʹ߬
ଵ
ିଵ
μሺݐǡߠǡ߮
μݐ
ିଵ
߬
ɢ
ଶ
ᶭ
ሻ
ͳ
ൌ
ᶭ
Ԝሺݐǡߠǡ߮ሻ
߬
ଶ
ଶ
μ
ൌ
μߠ
ሺɅ
ଶ
ሻ
μߠ
ͳ
μଶሺݐǡߠǡ߮
߬
μ
ͳ
ଶ
ሺߠሻ
μ߮
μ߮
μ
ሻ
ǡ
ଶ
ଶ
ଶ
ሺ
ͻǤͶͷ
ሺ
ͻǤͶ
– ɨɩɟɪɚɬɨɪ Ʌɚɩɥɚɫɚ – Ȼɟɥɶɬɪɚɦɢ ɧɚ ɩɥɨɫɤɨɫɬɢ Ʌɨɛɚɱɟɜɫɤɨɝɨ. ɍɪɚɜɧɟɧɢɟ (9.45)
ɟɫɬɶ ɭɪɚɜɧɟɧɢɟ ɞɢɮɮɭɡɢɢ ɧɚ ɩɥɨɫɤɨɫɬɢ Ʌɨɛɚɱɟɜɫɤɨɝɨ.
9.2.3. Ɋɟɲɟɧɢɟ ɭɪɚɜɧɟɧɢɹ ɞɢɮɮɭɡɢɢ
ɇɚɦ ɧɟɨɛɯɨɞɢɦɨ ɧɚɣɬɢ ɪɟɲɟɧɢɟ ɭɪɚɜɧɟɧɢɹ (9.45), ɫɨɨɬɜɟɬɫɬɜɭɸɳɟɟ ɫɥɭɱɚɣ-
ɧɨɦɭ ɛɥɭɠɞɚɧɢɸ ɫ ɧɚɱɚɥɶɧɨɣ ɬɨɱɤɨɣ ɧɚ ɤɨɦɩɥɟɤɫɧɨɣ ɩɥɨɫɤɨɫɬɢ ݖ
ൌͲ. ɉɨ-
ɫɤɨɥɶɤɭ ɭɫɪɟɞɧɟɧɢɟ ɩɨ ɮɭɧɤɰɢɢ ɪɚɫɩɪɟɞɟɥɟɧɢɹ ɨɫɭɳɟɫɬɜɥɹɟɬɫɹ ɩɨ ɮɨɪɦɭɥɟ
ۦ
݂ሺݖ
ሻۧሺ
ݐǡݖ
ሻ
ൌන ݂
ȁ௭ȁ
ழଵ
ሺݖሻሺ
ݐǡݖǡݖ
ሻ݀ߤሺݖሻ
ǡ
ሺ
ͻǤͶ
(ɡɞɟɫɶ ɦɵ ɧɚ ɤɨɪɨɬɤɨɟ ɜɪɟɦɹ ɜɟɪɧɭɥɢɫɶ ɤ ɤɨɦɩɥɟɤɫɧɨɣ ɤɨɨɪɞɢɧɚɬɟ ݖ ɢ ɭɤɚɡɚɥɢ
ɡɚɜɢɫɢɦɨɫɬɶ ɨɬ ɧɚɱɚɥɶɧɨɣ ɬɨɱɤɢ ɭ ɮɭɧɤɰɢɢ ), ɬɨ ɷɬɨ ɨɡɧɚɱɚɟɬ ɫɥɟɞɭɸɳɟɟ
ɧɚɱɚɥɶɧɨɟ ɭɫɥɨɜɢɟ ɞɥɹ ɮɭɧɤɰɢɢ ɪɚɫɩɪɟɞɟɥɟɧɢɹ
ሻ
ሺ
ɝɞɟ ߜ
ሺͲǡݖǡݖ
ሺ
ሻ
ݖǡݖ
ఓ
ԟߜ-ɮɭɧɤɰɢɹ ɩɨ ɦɟɪɟ ߤ ɫɨ ɫɜɨɣɫɬɜɨɦ
ൌߜ
ሻ
ݖǡݖ
ఓ
ǡ
ሺ
ͻǤͶͺ
ሻ
ሻ
ሻ
ሻ
ሺݖሻ
න݂
ȁ௭ȁ
ழଵ
ሺ
ݖǡݖ
ఓ
ሻ݀ߤሺݖሻൌ݂ሺ
ߜ
ሻ
ݖ
Ǥ
ሺ
ͻǤͶͻ
ሻ
95
ȼ ɩɨɥɹɪɧɵɯ ɤɨɨɪɞɢɧɚɬɚɯ
ɨɞɧɚɤɨ, ɟɫɥɢ ݖ
ൌͲ, ɬɨ ɢɡ-ɡɚ ɨɫɨɛɟɧɧɨɫɬɢ ɩɨɥɹɪɧɵɯ ɤɨɨɪɞɢɧɚɬ ɩɪɢ ߠൌͲ ɜ ɷɬɨɣ
ߜ
ఓ
ሺ
ߠǡ߮ሻ, ɫɨɝɥɚɫɧɨ (9.28), ݀ߤሺݖሻൌሺɅሻɅɔ, ɩɨɷɬɨɦɭ
ሺ
ሻ
ݖǡݖ
ൌߜ൫ሺߠሻെሺߠ
ሻ൯ߜሺ
߮െ߮
ሻሺ
ͻǤͷͲ
ɬɨɱɤɟ ɢɦɟɟɦ
ሺ
ݖǡݖ
ߜ
ൌͲሻൌ
ఓ
ͳ
ߜሺሺߠሻെͳሻǡ
ߨ
ሺ
ͻǤͷͳ
ɝɞɟ ɭɱɬɟɧɨ ɢɡɜɟɫɬɧɨɟ ɫɜɨɣɫɬɜɨ ߜ-ɮɭɧɤɰɢɢ
௫
భ
ሺݔሻߜሺݔሻ
න݂
Ɍɚɤɢɦ ɨɛɪɚɡɨɦ, ɧɚɱɚɥɶɧɨɟ ɭɫɥɨɜɢɟ ɞɥɹ ሺݐǡߠǡ߮
ሺݐ՜Ͳǡߠǡ߮ሻൌ
݀ݔൌ
ͳ
݂ሺͲሻǡ᩷ɩɪɢ ɥɸɛɨɦ᩷ݔଵͲǤ
ʹ
ሻ
ɜ ɧɚɲɟɦ ɫɥɭɱɚɟ ɢɦɟɟɬ ɜɢɞ
ͳ
ߜሺሺߠሻെͳ
ߨ
ሻሺ
ሺ
ͻǤͷʹ
ͻǤͷ͵
ɩɪɢ ɷɬɨɦ ɜɵɩɨɥɧɹɟɬɫɹ ɭɫɥɨɜɢɟ ɧɨɪɦɢɪɨɜɤɢ ɮɭɧɤɰɢɢ ɪɚɫɩɪɟɞɟɥɟɧɢɹ
ஶ
න݀ߠሺߠ
ଶగ
ሻ
නሺݐǡߠǡ߮ሻ݀߮ൌͳǤ
ሺ
ͻǤͷͶ
ɉɨɫɬɨɥɶɤɭ ɧɚɱɚɥɶɧɨɟ ɭɫɥɨɜɢɟ ɧɟ ɡɚɜɢɫɢɬ ɨɬ ߮, ɬɨ ɢ ɪɟɲɟɧɢɟ ɭɪɚɜɧɟɧɢɹ (9.45)
ɦɨɠɧɨ ɢɫɤɚɬɶ ɧɟ ɡɚɜɢɫɹɳɢɦ ɨɬ ߮, ɜ ɷɬɨɦ ɫɥɭɱɚɟ ɨɧɨ ɩɪɢɨɛɪɟɬɚɟɬ ɜɢɞ
ଶ
ሻ
ͳ
μሺݐǡߠ
μݐ
μ
ൌ
ቆ
߬
ଶ
ɉɟɪɟɣɞɟɦ ɤ ɩɟɪɟɦɟɧɧɵɦ ݑൌሺߠሻ ɢ ݏൌݐȀ߬
ሻ
μሺݏǡݑ
μݏ
ൌԜሺݑଶെͳ
μ
ൌԜ
μݑ
ሺݐǡߠ
μߠ
ሺ
ݑ
ሻ
ଶ
ଶ
ଶ
μ
െͳ
ሻ
ሺɅ
ሺݐǡݑ
ଶ
μݑ
μሺݐǡݑ
ሻ
ሻ
ʹݑ
μݑ
ሻ
ଶ
ሻ
μሺݐǡߠ
μߠ
μሺݐǡݑ
μݑ
ǡ
ሻ
ቇǤ
ሻ
ൌ
ሺ
ͻǤͷͷ
ሺ
ͻǤͷ
ɢ ɛɭɞɟɦ ɢɫɤɚɬɶ ɪɟɲɟɧɢɟ ɷɬɨɝɨ ɭɪɚɜɧɟɧɢɹ ɪɚɡɥɨɠɟɧɢɟɦ ɪɟɲɟɧɢɹ ɩɨ ɫɨɛɫɬɜɟɧɧɵɦ
ሺݑሻ
ɮɭɧɤɰɢɹɦ ݓ
ɨɩɟɪɚɬɨɪɚ ɜ ɩɪɚɜɨɟ ɱɚɫɬɢ ɭɪɚɜɧɟɧɢɹ:
ଶ
ሺݑሻ
ݓ
μ
ଶ
ሺ
െͳ
ݑ
ሻ
ଶ
μݑ
ʹݑ
μݓ
μݑ
ሺݑሻ
ൌെ݇ଶԜݓ
ሺݑሻ
ǡ
ሺ
ͻǤͷ
ɨɛɨɡɧɚɱɢɦ ɩɪɨɢɡɜɨɞɧɵɟ ɩɨ ݑ ɲɬɪɢɯɨɦ, ɬɨɝɞɚ ɷɬɨ ɭɪɚɜɧɟɧɢɟ ɩɪɢɦɟɬ ɜɢɞ
ሺ
ͳെݑ
ሻ
ݓ
െʹݑԜݓ
ᇳ
ଶ
ɝɞɟ ɜɜɟɞɟɧ ɩɚɪɚɦɟɬɪ ߥ ɫɨɝɥɚɫɧɨ െ݇
ᇱ
ߥሺߥͳሻݓൌͲǡ
ଶ
ൌߥሺߥͳ
ሺ
ͻǤͷͺ
ሻ
.
ሻ
ሻ
ሻ
ሻ
ሻ
ሻ
ሻ
ሻ
ሻ
96
Ʉɚɤ ɢɡɜɟɫɬɧɨ ɪɟɲɟɧɢɟɦ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɨɝɨ ɭɪɚɜɧɟɧɢɹ (9.58) ɹɜɥɹɸɬɫɹ ɬɚɤ
ሺݑሻ
ɧɚɡɵɜɚɟɦɵɟ ɮɭɧɤɰɢɢ Ʌɟɠɚɧɞɪɚ ܲ
. ȼ ɧɚɲɟɦ ɫɥɭɱɚɟ ɷɬɢ ɮɭɧɤɰɢɢ ɧɭɠɧɨ ɜɡɹɬɶ
ఔ
ɩɪɢ ɜɟɳɟɫɬɜɟɧɧɨɦ ɡɧɚɱɟɧɢɢ ɚɪɝɭɦɟɧɬɚ ɧɚ ɩɨɥɭɢɧɬɟɪɜɚɥɟ ሾͳǡλሻ, ɬɚɤ ɤɚɤ
ݑൌሺߠሻ. ɍɫɥɨɜɢɟ ݇
ଶ
ൌെߥሺߥͳሻͲ ɚɜɬɨɦɚɬɢɱɟɫɤɢ ɜɵɩɨɥɧɹɟɬɫɹ, ɟɫɥɢ ɩɨɥɨ-
ɠɢɬɶ ߥൌെͳȀʹ݅ߣ, ɜ ɷɬɨɦ ɫɥɭɱɚɟ ɢɦɟɟɦ:
ଶ
݇
ൌെߥሺߥͳሻൌͳȀͶߣଶǤ
ሺ
ͻǤͷͻ
Ɍɚɤɢɦ ɨɛɪɚɡɨɦ, ɜ ɡɚɞɚɱɟ ɫɥɭɱɚɣɧɵɯ ɛɥɭɠɞɚɧɢɣ ɧɚ ɩɥɨɫɤɨɫɬɢ Ʌɨɛɚɱɟɜɫɤɨɝɨ
ሺݑሻ
ɜɚɠɧɭɸ ɪɨɥɶ ɛɭɞɭɬ ɢɝɪɚɬɶ ɮɭɧɤɰɢɢ Ʌɟɠɚɧɞɪɚ ܲ
ɫ ɤɨɦɩɥɟɤɫɧɵɦ ɢɧɞɟɤɫɨɦ
ఔ
ߥൌെͳȀʹ݅ߣ.
ɉɪɢɜɟɞɟɦ ɩɨɥɟɡɧɵɟ ɜ ɞɚɥɶɧɟɣɲɟɦ ɪɚɫɫɦɨɬɪɟɧɢɢ ɫɜɨɣɫɬɜɚ ɮɭɧɤɰɢɣ Ʌɟɠɚɧɞɪɚ:
ሻ
ɂɡ (9.60) ɫɪɚɡɭ ɩɨɥɭɱɚɟɦ ܲ
ሺݑሻ
ܲ
ఔ
ିଵȀଶାఒ
ൌܲ
ሺݑሻ
ିఔିଵ
ൌܲ
ሺݑሻ
Ǥ
ିଵȀଶିఒ
ሺݑሻ
, ɬɨ ɟɫɬɶ ܲ
ିଵȀଶାఒ
ሺݑሻ
ሺ
ͻǤͲ
ɱɟɬ-
ɧɚɹ ɮɭɧɤɰɢɹ ɨɬ ߣ. Ⱦɚɥɟɟ, ɢɦɟɟɬ ɦɟɫɬɨ ɫɥɟɞɭɸɳɟɟ ɢɧɬɟɝɪɚɥɶɧɨɟ ɩɪɟɞɫɬɚɜɥɟɧɢɟ
ሺݑሻ
ɞɥɹ ܲ
:
ఔ
ஶ
ሺ
ܲ
൫ሺߠሻ൯ൌെ
ఔ
ʹ
ξ
ሺߥߨሻන
ߨ
ሾ
ඥ
ሺݐሻ݄ܿሺߠ
ߥͳȀʹሻݐ
ሿ
݀ݐ
ሺ
ͻǤͳ
ሻ
ɢɡ ɤɨɬɨɪɨɝɨ ɩɨɫɥɟ ɩɨɞɫɬɚɧɨɜɤɢ ߥൌെͳȀʹ݅ߣ ɥɟɝɤɨ ɩɨɥɭɱɢɬɶ
ஶ
ඥ
ሺݐሻ݄ܿሺߠ
ܲ
ିଵȀଶାఒ
൫ሺߠሻ൯ൌെ
ʹ
ξ
ሺߣߨሻන
ߨ
ɂɡ ɷɬɨɣ ɮɨɪɦɭɥɵ ɜɟɳɟɫɬɜɟɧɧɨɫɬɶ ɢ ɱɟɬɧɨɫɬɶ ɩɨ ߣ ɞɥɹ ɮɭɧɤɰɢɢ ܲ
ሺɉݐ
ሻ
݀ݐǤ
ሺ
ͻǤʹ
ሻ
൫ሺߠሻ൯
ିଵȀଶାఒ
ɫɨɜɫɟɦ ɨɱɟɜɢɞɧɚ.
Ɏɭɧɤɰɢɹ Ʌɚɝɪɚɧɠɚ ɫ ɤɨɦɩɥɟɤɫɧɵɦ ɩɚɪɚɦɟɬɪɨɦ ܲ
ିଵȀଶିఒ
ሺݑሻ
ɩɨɡɜɨɥɹɸɬ ɪɚɡɥɨ-
ɠɢɬɶ ɥɸɛɭɸ ɮɭɧɤɰɢɸ ɧɚ ɩɨɥɭɢɧɬɟɪɜɚɥɟ ሾͳǡλሻ ɜ ɢɧɬɟɝɪɚɥ ɬɢɩɚ Ɏɭɪɶɟ. Ɍɨɱɧɟɟ
ɢɦɟɟɬ ɦɟɫɬɨ ɬɟɨɪɟɦɚ Ɇɟɥɟɪɚ – Ɏɨɤɚ ɭɬɜɟɪɠɞɚɸɳɚɹ, ɱɬɨ ɮɭɧɤɰɢɹ ݃ሺݑ
ሻ
, ɡɚɞɚɧɧɚɹ
ɧɚ ɩɨɥɭɢɧɬɟɪɜɚɥɟ ሾͳǡλሻ ɦɨɠɟɬ ɛɵɬɶ ɩɪɟɞɫɬɚɜɥɟɧɚ ɜ ɜɢɞɟ ɢɧɬɟɝɪɚɥɚ
ஶ
݃ሺݑሻൌනܲ
ିଵȀଶାఒ
ሺݑሻԜ݂ሺߣሻ
ԜߣԜሺߨߣሻԜ݀ߣǡ
ሺ
ͻǤ͵
ɝɞɟ
ஶ
݂ሺߣሻൌනܲ
ɉɪɟɨɛɪɚɡɨɜɚɧɢɟ ɮɭɧɤɰɢɢ ݃ሺݑ
ሺݑሻ݃ሺݑሻ
ିଵȀଶାఒ
ଵ
ሻ
ɞɚɜɚɟɦɨɟ ɮɨɪɦɭɥɨɣ (9.64) ɧɚɡɵɜɚɟɬɫɹ ɩɪɟɨɛ-
݀ݑǤ
ሺ
ͻǤͶ
ɪɚɡɨɜɚɧɢɟɦ Ɇɟɥɟɪɚ – Ɏɨɤɚ. Ɏɨɪɦɭɥɵ (9.63) ɢ (9.64) ɷɤɜɢɜɚɥɟɧɬɧɵ ɞɜɭɦ ɫɥɟɞɭɸ-
ሺݑሻ
ሺݑሻ
ൌ
ͳ
ߣԜሺߨߣ
ߜሺߣെߤሻǡ
ሻ
ɳɢɦ ɭɫɥɨɜɢɹɦ ɩɨɥɧɨɬɵ ɮɭɧɤɰɢɣ ܲ
ஶ
ሺݑሻ
න݀
ଵ
ݑԜܲ
ିଵȀଶାఒ
Ԝܲ
ିଵȀଶାఒ
ିଵȀଶାఒ
ሻ
ሻ
ሻ
ሻ
ሻ
97
ஶ
ሺݑሻ
ߣԜߣԜሺߨߣሻԜܲ
න݀
Ɍɚɤɢɦ ɨɛɪɚɡɨɦ, ɫɨɛɫɬɜɟɧɧɵɟ ɮɭɧɤɰɢɢ ɨɩɟɪɚɬɨɪɚ ɞɢɮɮɭɡɢɢ ɫ ɫɨɛɫɬɜɟɧɧɵɦ
ɡɧɚɱɟɧɢɟɦ െ݇
ଶ
, ɝɞɟ ݇
ଶ
ɞɚɟɬɫɹ (9.59), ɢɦɟɸɬ ɜɢɞ
ିଵȀଶାఒ
ܲ
ିଵȀଶାఒ
ሺݒሻൌߜሺ
ݑെݒሻǤ
ሺ
ͻǤͷ
ሻ
ሺߠሻ
ൌܲ
ݓ
ఒ
Ɍɟɩɟɪɶ ɪɟɲɟɧɢɟ ɭɪɚɜɧɟɧɢɹ (9.55) ɦɨɠɧɨ ɡɚɩɢɫɚɬɶ ɜ ɜɢɞɟ ɪɚɡɥɨɠɟɧɢɹ ɩɨ
ሺߠሻ
ɫ ɤɨɷɮɮɢɰɢɟɧɬɚɦɢ ߩ
ݓ
ఒ
ȼɟɫɨɜɭɸ ɮɭɧɤɰɢɸ ߩ
ሺݏǡߠሻൌන݀ߣߩ
ሾെሺͳȀͶߣ
ఒ
ஶ
݁
ఒ
ɥɟɝɤɨ ɭɫɬɚɧɨɜɢɬɶ ɢɡ ɧɚɱɚɥɶɧɨɝɨ ɭɫɥɨɜɢɹ (9.53), ɢɫɩɨɥɶ-
ఒ
ିଵȀଶାఒ
ିሺଵȀସାఒ
൫ሺߠሻ൯Ǥ
ଶ
ሻ
ሿ
ݏ
మ
ሻ
௦
ܲ
ିଵȀଶାఒ
൫ሺߠሻ൯Ǥ
ሺ
ͻǤ
ሺ
ͻǤ
ɡɭɹ ɩɪɟɨɛɪɚɡɨɜɚɧɢɹ Ɇɟɥɟɪɚ – Ɏɨɤɚ (9.64) ɢ ɨɛɪɚɬɧɨɟ ɟɦɭ (9.63). Ⱦɟɣɫɬɜɢɬɟɥɶɧɨ,
ɩɪɢ ݐൌͲ ɢɦɟɟɦ (ɩɪɢɧɹɥɢ ݑൌሺߠሻ)
ͳ
ߜሺݑെͳሻൌන݀
ߨ
ɫɨɩɨɫɬɚɜɥɹɹ ɷɬɨ ɫ (9.63), ɩɨɥɭɱɢɦ ݂ሺߣሻൌߩ
ߣԜሺߨߣ
(ɦɵ ɭɱɥɢ ܲ
ߩ
ఒ
ିଵȀଶାఒ
ൌනܲ
ሻ
ሺͳሻ
ஶ
ଵ
ିଵȀଶାఒ
ൌͳ).
ሺݑሻ
ஶ
ߣԜߩఒԜܲ
Ȁ൫ߣԜሺߨߣሻ൯, ɬɨɝɞɚ (9.64) ɞɚɟɬ
ఒ
ͳ
ߜሺݑെͳሻ݀ݑൌ
ߨ
ିଵȀଶାఒ
ͳ
ʹߨ
ሺݑሻ
ܲ
ିଵȀଶାఒ
ͳ
ǡሺͻǤͻ
ʹߨ
ሺ
ͻǤͺ
ǡ
ሺͳሻ
ൌ
Ɍɚɤɢɦ ɨɛɪɚɡɨɦ, ɪɟɲɟɧɢɟ ɭɪɚɜɧɟɧɢɹ ɞɢɮɮɭɡɢɢ (9.55) ɫ ɧɚɱɚɥɶɧɵɦ ɭɫɥɨɜɢɟɦ
(9.53) ɢɦɟɟɬ ɜɢɞ (ɜɨɡɜɪɚɳɚɟɦɫɹ ɤ ɩɟɪɟɦɟɧɧɨɣ ݐൌ߬
ஶ
ሺݏǡߠሻൌ
ͳ
න ݀ߣߣԜሺߨߣሻ݁
ʹߨ
ିሺଵȀସାఒ
ݏ)
ଶ
మ
ሻ
ሺ௧Ȁఛ
ሻ
మ
ܲ
ିଵȀଶାఒ
൫ሺߠሻ൯
ሺ
Ǥ
ͻǤͲ
9.3. ȼɵɱɢɫɥɟɧɢɟ ɫɬɚɬɢɱɟɫɤɨɣ ɩɪɨɜɨɞɢɦɨɫɬɢ
ɂɫɯɨɞɢɦ ɢɡ ɮɨɪɦɭɥɵ (9.30). ɉɨɞɫɬɚɜɥɹɟɦ (9.70) ɢ ɢɧɬɟɝɪɢɪɭɹ ɩɨ ߮
ஶ
ሻ
ۄ
ۃ
ஶ
ൌʹන ݀ߣߣԜሺߨߣሻ݁
ஶ
ൌʹԜන݀
ൌͶߨන݀ߠ
ܴሺݐ
ߣԜߣԜሺߨߣሻԜ݁
ଵ
మ
ሺ
ሻ
ଵȀସାఒ
ି
ఛ
మ
௧
ଵ
ሺ
ଵȀସାఒ
ି
ఛ
మ
ɦɵ ɫɞɟɥɚɥɢ ɜɨ ɜɬɨɪɨɦ ɢɧɬɟɝɪɚɥɟ ɡɚɦɟɧɭ ɩɟɪɟɦɟɧɧɨɣ ݑൌሺߠሻ. ɂɫɩɨɥɶɡɭɹ ɢɧɬɟ-
ɝɪɚɥɶɧɨɟ ɩɪɟɞɫɬɚɜɥɟɧɢɟ (9.62) ɞɥɹ ܲ
ିଵȀଶାఒ
98
ሻ
ሺߠ
ሺߠሻͳ
ಽ
න ݀ߠ
ଵ
ሺݑሻ
ሺݐǡߠሻൌ
ஶ
ሺߠሻܲ
ሺߠሻͳ
ஶ
మ
ሻ
௧
ಽ
න݀
ଵ
ܲ
ݑ
, ɦɨɠɧɨ ɭɛɟɞɢɬɶɫɹ, ɱɬɨ
ିଵȀଶାఒ
ିଵȀଶାఒ
ݑͳ
൫ሺߠሻ൯
ሺݑሻ
ǡ
ൌ
ሺ
ͻǤͳ
ሻ
ሻ
ሻ
ሻ
ሻ
ሻ
Ɍɨɝɞɚ
ۃ
ܴሺݐ
ஶ
න݀
ଵ
ሻ
ۄ
ൌʹԜߨන݀
ܲ
ݑ
ିଵȀଶାఒ
ݑͳ
ஶ
ߣԜߣ
ሺݑሻ
ൌ
Ԝሺߨߣ
ଶ
ሺߨߣሻ
ߨ
ሺߨߣ
ሻ
ି
Ԝ݁
ଵ
ఛ
మ
Ǥ
ሻ
ሺ
ଵȀସାఒ
ሺ
ሻ
ͻǤʹ
మ
ሻ
௧
ಽ
ሺ
ͻǤ͵
ሻ
ɉɪɢ ݐ
ɦɨɠɧɨ ɩɨɥɨɠɢɬɶ
ൌܮȀݒ߬ଶبͳ ɜɤɥɚɞ ɜ ɢɧɬɟɝɪɚɥ ɞɚɸɬ ɦɚɥɵɟ ߣ߬ଶȀݐ
Ȁ߬ଶ
ଶ
ሺߨߣሻൎͳ ɢ ሺߨߣሻൎߨߣ, ɬɨɝɞɚ ɨɤɨɧɱɚɬɟɥɶɧɨ ɩɨɥɭɱɚɟɦ:
ଷȀଶ
ହȀଶ
ߨ
߬
ۦ
ܴሺݐ
ሻ
ۧ
ൎ
ʹԜሺݐ
ଶ
ଷȀଶ
ሻ
ɉɨɞɫɬɚɜɥɹɟɦ ɷɬɨɬ ɪɟɡɭɥɶɬɚɬ ɜ (9.3), ɢ ɭɱɢɬɵɜɚɹ ݐ
ି௧ಽȀସఛ
Ԝ݁
మ
Ǥ
ൌܮȀݒ
, ɩɨɥɭɱɚɟɦ ɜɵɪɚɠɟ-
, ɩɨɷɬɨɦɭ
ሺ
ͻǤͶ
ɧɢɟ ɞɥɹ ɩɪɨɜɨɞɢɦɨɫɬɢ
ߪሺͲሻൌ
ଶ
ሺ
݁
ߨݒ
߬ଶ
ଷȀଶ
ሻ
ିȀସ௩బఛ
݁
మ
Ǥ
ሺ
ͻǤͷ
Ͷξܮ
Ɍɚɤɢɦ ɨɛɪɚɡɨɦ, ɫɬɚɬɢɱɟɫɤɚɹ ɩɪɨɜɨɞɢɦɨɫɬɶ ɨɞɧɨɦɟɪɧɨɝɨ ɦɟɬɚɥɥɚ ɷɤɫɩɨɧɟɧ-
ɰɢɚɥɶɧɨ ɦɚɥɚ ɞɥɹ ɨɛɪɚɡɰɚ ɤɨɧɟɱɧɨɣ ɞɥɢɧɵ ɩɪɢ ܮبݒ
߬ଶ
.
ɇɚ ɨɬɫɭɬɫɬɜɢɟ ɫɬɚɬɢɱɟɫɤɨɣ ɩɪɨɜɨɞɢɦɨɫɬɢ ɨɞɧɨɦɟɪɧɨɝɨ ɦɟɬɚɥɥɚ ɜ ɩɨɥɟ ɫɥɭɱɚɣɧɵɯ ɩɪɢɦɟɫɟɣ ɜɩɟɪɜɵɟ ɭɤɚɡɚɥɢ Ɇɨɬɬ ɢ Ɍɨɭɡ [10]. Ɉɧɢ ɪɚɫɫɦɨɬɪɟɥɢ ɫɥɭɱɚɣɧɭɸ ɨɞɧɨɦɟɪɧɭɸ ɦɨɞɟɥɶ Ʉɪɨɧɢɝɚ – ɉɟɧɧɢ ɢ ɩɪɢɜɟɥɢ ɚɪɝɭɦɟɧɬɵ ɜ ɩɨɥɶɡɭ ɬɨɝɨ,
ɱɬɨ ɜ ɨɞɧɨɦɟɪɧɨɣ ɫɢɫɬɟɦɟ ɫ ɩɪɨɢɡɜɨɥɶɧɵɦ ɫɥɭɱɚɣɧɵɦ ɩɨɥɟɦ ɜɫɟ ɷɥɟɤɬɪɨɧɵ
ɞɨɥɠɧɵ ɛɵɬɶ ɥɨɤɚɥɢɡɨɜɚɧɵ. ɉɨɫɤɨɥɶɤɭ ɨɬɥɢɱɧɚɹ ɨɬ ɧɭɥɹ ɫɬɚɬɢɱɟɫɤɚɹ
ɩɪɨɜɨɞɢɦɨɫɬɶ ɦɨɠɟɬ ɛɵɬɶ ɨɛɟɫɩɟɱɟɧɚ ɬɨɥɶɤɨ ɩɟɪɟɧɨɫɨɦ ɡɚɪɹɞɚ ɧɚ ɦɚɤɪɨɫɤɨɩɢɱɟɫɤɢɟ
ɪɚɫɫɬɨɹɧɢɹ, ɬɨ ɜ ɫɥɭɱɚɟ, ɤɨɝɞɚ ɜɫɟ ɷɥɟɤɬɪɨɧɧɵɟ ɫɨɫɬɨɹɧɢɹ ɥɨɤɚɥɢɡɨɜɚɧɵ, ɩɪɨ-
ሺ
ɜɨɞɢɦɨɫɬɶ ɪɚɜɧɚ ɧɭɥɸ. ȼɵɪɚɠɟɧɢɟ ɞɥɹ ɫɬɚɬɢɱɟɫɤɨɣ ɩɪɨɜɨɞɢɦɨɫɬɢ
ͻǤͷ
ሻ
ɩɨɡ-
ɜɨɥɹɟɬ ɡɚɤɥɸɱɢɬɶ, ɱɬɨ ɞɥɹ ɞɥɢɧɵ ɥɨɤɚɥɢɡɚɰɢɢ ɷɥɟɤɬɪɨɧɧɵɯ ɫɨɫɬɨɹɧɢɣ ɢɦɟɟɬ
ɨɰɟɧɤɭ ߣ
ݒ߬ଶ
Ǥ
Ʌɢɬɟɪɚɬɭɪɚ
Ɋɟɡɭɥɶɬɚɬ (9.75) ɦɟɬɨɞɨɦ S-ɦɚɬɪɢɰɵ ɛɵɥ ɩɨɥɭɱɟɧ ɜ ɪɚɛɨɬɟ Ⱥ. Ⱥ. Ⱥɛɪɢɤɨɫɨɜɚ
ɢ ɂ. Ⱥ. Ɋɵɠɤɢɧɚ // ɀɗɌɎ. 1976. Ɍ. 71. ɋ. 12041224. Ɋɚɧɟɟ ɨɧ ɞɪɭɝɢɦ ɦɟ-
ɬɨɞɨɦ ɛɵɥ ɩɨɥɭɱɟɧ ȼ. Ʌ. Ȼɟɪɟɡɢɧɫɤɢɦ // ɀɗɌɎ. 1974. Ɍ. 65. ɋ. 1251.
Ɏɨɪɦɭɥɵ (9.63), (9.64) ɛɵɥɢ ɞɨɤɚɡɚɧɵ ȼ. Ⱥ. Ɏɨɤɨɦ ɜ ɪɚɛɨɬɟ «Ɉ
ɩɪɨɢɡɜɨɥɶɧɨɣ ɮɭɧɤɰɢɢ ɜ ɢɧɬɟɝɪɚɥ ɩɨ ɮɭɧɤɰɢɹɦ Ʌɟɠɚɧɞɪɚ ɫ ɤɨɦɩɥɟɤɫɧɵɦ ɡɧɚɱɤɨɦ» // ȾȺɇ ɋɋɋɊ. 1943. Ɍ. 39. ɋ. 279283.
ɋɢɫɬɟɦɚɬɢɱɟɫɤɨɟ ɢɡɥɨɠɟɧɢɟ ɬɟɨɪɢɢ ɩɪɟɨɛɪɚɡɨɜɚɧɢɹ Ɇɟɥɟɪɚ – Ɏɨɤɚ ɦɨɠɧɨ
ɧɚɣɬɢ ɜ ɤɧɢɝɟ Ⱦɢɬɤɢɧ ȼ. Ⱥ., ɉɪɭɞɧɢɤɨɜ Ⱥ. Ɍ. «ɂɧɬɟɝɪɚɥɶɧɵɟ ɩɪɟɨɛɪɚɡɨɜɚɧɢɹ ɢ
ɨɩɟɪɚɰɢɨɧɧɨɟ ɢɫɱɟɫɥɟɧɢɟ» // Ɇ: Ɏɢɡɦɚɬɝɢɡ. 1961.
ɪɚɡɥɨɠɟɧɢɢ
ሻ
ሻ
99
Ʉɨɧɬɪɨɥɶɧɵɟ ɜɨɩɪɨɫɵ
1. Ʉɚɤɢɦ ɨɛɪɚɡɨɦ ɷɥɟɦɟɧɬɵ ܵ
ɡɨɜɚɧɢɟɦ, ɩɟɪɟɜɨɞɹɳɢɦ ɟɞɢɧɢɱɧɵɣ ɤɪɭɝ ɜ ɫɟɛɹ?
2. ɉɨɤɚɡɚɬɶ, ɱɬɨ ɞɪɨɛɧɨ-ɥɢɧɟɣɧɨɟ ɩɪɟɨɛɪɚɡɨɜɚɧɢɟ, ɩɨɪɨɠɞɚɟɦɨɟ ܵ
ɹɜɥɹɟɬɫɹ ɞɜɢɠɟɧɢɟɦ ɦɟɬɪɢɤɢ ɩɥɨɫɤɨɫɬɢ Ʌɨɛɚɱɟɜɫɤɨɝɨ.
3. Ʉɚɤ ɫɜɹɡɚɧɚ ɤɨɧɟɱɧɚɹ ɬɨɱɤɚ ɫɥɭɱɚɣɧɨɝɨ ɛɥɭɠɞɚɧɢɹ ɫ ɩɪɨɜɨɞɢɦɨɫɬɶɸ ɨɞɧɨ-
ɦɟɪɧɨɣ ɫɢɫɬɟɦɵ ɷɥɟɤɬɪɨɧɨɜ?
4. ɉɨɱɟɦɭ ɮɭɧɤɰɢɹ ɪɚɫɩɪɟɞɟɥɟɧɢɹ ɤɨɧɟɱɧɨɣ
ɭɞɨɜɥɟɬɜɨɪɹɟɬ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɨɦɭ ɭɪɚɜɧɟɧɢɸ, ɚɧɚɥɨɝɢɱɧɨɦɭ ɭɪɚɜɧɟɧɢɸ ɞɢɮɮɭɡɢɢ?
5. Ɂɚɩɢɫɚɬɶ ɨɩɟɪɚɬɨɪ Ʌɚɩɥɚɫɚ – Ȼɟɥɶɬɪɚɦɢ ɧɚ ɩɥɨɫɤɨɫɬɢ Ʌɨɛɚɱɟɜɫɤɨɝɨ ɜ ɩɨ-
ɥɹɪɧɵɯ ɢ ɞɟɤɚɪɬɨɜɵɯ ɤɨɨɪɞɢɧɚɬɚɯ.
መ
-ɦɚɬɪɢɰɵ ɫɜɹɡɚɧɵ ɫ ɞɪɨɛɧɨ-ɥɢɧɟɣɧɵɦ ɩɪɟɨɛɪɚ-
መ
-ɦɚɬɪɢɰɟɣ,
ɬɨɱɤɢ ɫɥɭɱɚɣɧɨɝɨ ɛɥɭɠɞɚɧɢɹ
100
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