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Ƚɥɚɜɚ 3
ȽɊɂɇɈȼɋɄɂȿ ɎɍɇɄɐɂɂ ɗɅȿɄɌɊɈɇɈȼ
3.1. Ɉɩɟɪɚɬɨɪɵ ɪɨɠɞɟɧɢɹ ɢ ɭɧɢɱɬɨɠɟɧɢɹ
Ɋɚɫɫɦɨɬɪɢɦ ɫɢɫɬɟɦɭ ɧɟɜɡɚɢɦɨɞɟɣɫɬɜɭɸɳɢɯ ɷɥɟɤɬɪɨɧɨɜ ɜɨ ɜɧɟɲɧɟɦ ɩɨɥɟ.
ɉɭɫɬɶ ɨɞɧɨɱɚɫɬɢɱɧɵɣ ɝɚɦɢɥɶɬɨɧɢɚɧ ɷɥɟɤɬɪɨɧɚ ɢɦɟɟɬ ɜɢɞ
ଶ
ሺxሻ
ɚ ߮
ɜ ɫɨɫɬɨɹɧɢɢ ߮
ɩɨɥɧɵɣ ɧɚɛɨɪ ɫɨɛɫɬɜɟɧɧɵɯ ɫɨɫɬɨɹɧɢɣ ɷɬɨɝɨ ɝɚɦɢɥɶɬɨɧɢɚɧɚ:
݄Ԝ߮
ȼɜɟɞɟɦ ܽො
ሺxሻ
ൌߝ
߮
ା
, ܽො
ɮɟɪɦɢɟɜɫɤɢɟ ɨɩɟɪɚɬɨɪɵ ɪɨɠɞɟɧɢɹ ɢ ɭɧɢɱɬɨɠɟɧɢɹ ɷɥɟɤɬɪɨɧɚ
ሺxሻ
, ɷɬɢ ɨɩɟɪɚɬɨɪɵ ɭɞɨɜɥɟɬɜɨɪɹɸɬ ɫɥɟɞɭɸɳɢɦ ɩɟɪɟɫɬɚɧɨɜɨɱɧɵɦ
ɫɨɨɬɧɨɲɟɧɢɹɦ
ቐ
ɋ ɩɨɦɨɳɶɸ ɨɩɟɪɚɬɨɪɨɜ ܽො
ɩɢɲɟɬɫɹ ɬɚɤ
ሺxሻ
ǡԜԜԜ߮
ܽො
ܽො
ܽො
ା
ܽො
ା
, ܽො
pො
݄ൌ
ʹ݉
ሺxሻ
ା
ܽො
ܽො
ܽොܽො
ା
ܽො
ܽො
ɦɧɨɝɨɱɚɫɬɢɱɧɵɣ ɝɚɦɢɥɶɬɨɧɢɚɧ ɷɥɟɤɬɪɨɧɨɜ ɡɚ-
ሺxሻ
ܷ
כ
ᇱ
ሺ
ሻൌߜሺ
߮
x
ା
ܽොൌߜǡ
ൌͲǡ
ା
ା
ܽො
ൌͲǤ
ǡ
ᇱ
x െx
ሻ
ǡԜԜԜۦ߮
ȁ߮
ۧ
ൌߜ
ሺ͵Ǥ͵ሻ
ሺ
͵Ǥͳ
Ǥሺ͵Ǥʹ
ሻ
ሻ
ᇱ
ܪ
ൌߝ
ା
ܽො
ܽොǡ
ɚ ɨɩɟɪɚɬɨɪ ɱɢɫɥɚ ɱɚɫɬɢɰ
ܰൌܽො
ା
ܽොǤ
ȼ ɞɚɥɶɧɟɣɲɟɦ ɛɭɞɟɬ ɭɞɨɛɧɨ ɨɬɫɱɢɬɵɜɚɬɶ ɷɧɟɪɝɢɸ ɷɥɟɤɬɪɨɧɨɜ ɨɬ ɭɪɨɜɧɹ ɯɢ-
ɦɢɱɟɫɤɨɝɨ ɩɨɬɟɧɰɢɚɥɚ ߤ, ɩɨɷɬɨɦɭ ɜɜɟɞɟɦ «ɩɨɞɩɪɚɜɥɟɧɧɵɣ» ɝɚɦɢɥɶɬɨɧɢɚɧ (ɤɨɬɨ-
ɪɵɣ ɭɠɟ ɛɭɞɟɬ ɛɟɡ ɲɬɪɢɯɚ)
ᇱ
ܪൌܪ
െߤܰൌԜߦܽො
ɋ ɩɨɦɨɳɶɸ ɝɚɦɢɥɶɬɨɧɢɚɧɚ ܪ ɨɩɪɟɞɟɥɢɦ ɨɩɟɪɚɬɨɪɵ ܽො
ା
ܽොǡԜԜԜߦൌሺߝെߤሻȀ
ା
ሺݐሻ
, ܽො
ሺݐሻ
ሺ
͵ǤͶ
ɜ ɝɟɣɡɟɧɛɟɪ-
ɝɨɜɫɤɨɦ ɩɪɟɞɫɬɚɜɥɟɧɢɢ
ൌ݁
௧ு
ା
ሺݐሻ
ܽො
௧ு
ି
ା
ܽො
݁
ǡԜԜԜԜԜܽො
ሺݐሻ
ൌ݁
௧ு
ܽො݁
௧ு
ି
ǡ
ሺ
͵Ǥͷ
21
ሻ
ሻ
ɤɨɬɨɪɵɟ ɭɞɨɜɥɟɬɜɨɪɹɸɬ ɫɥɟɞɭɸɳɢɦ ɭɪɚɜɧɟɧɢɹɦ
ା
݀ܽො
െ݅
ɝɞɟ ൣܣመǡܤ൧ൌܣመܤെܤܣ
ሺݐሻ
ൌൣܪǡܽො
݀ݐ
መ
ɤɨɦɦɭɬɚɬɨɪ ɨɩɟɪɚɬɨɪɨɜ. Ʉɨɦɦɭɬɚɬɨɪɵ ɜ ɩɪɚɜɵɯ ɱɚɫɬɹɯ
ା
ሺݐሻ
൧ǡԜԜԜെ݅
݀ܽො
݀ݐ
ሺݐሻ
ൌൣܪǡܽො
ሺݐሻ
൧ǡ
ሺ
͵Ǥ
ɭɪɚɜɧɟɧɢɣ (3.6) ɥɟɝɤɨ ɜɵɱɢɫɥɹɸɬɫɹ ɫ ɩɨɦɨɳɶɸ ɩɟɪɟɫɬɚɧɨɜɨɱɧɵɯ ɫɨɨɬɧɨɲɟɧɢɣ
(3.3), ɜ ɪɟɡɭɥɶɬɚɬɟ ɩɨɥɭɱɢɦ
ା
െ݅
݀ܽො
݀ݐ
ሺݐሻ
ൌԜߦܽො
ା
ሺݐሻ
ǡԜԜԜെ݅
݀ܽො
݀ݐ
ሺݐሻ
ൌെԜߦܽො
ሺݐሻ
ǡ
ሺ
͵Ǥ
ɱɬɨ ɩɨɡɜɨɥɹɟɬ ɨɩɪɟɞɟɥɢɬɶ ɹɜɧɭɸ ɡɚɜɢɫɢɦɨɫɬɶ ɨɬ ɜɪɟɦɟɧɢ ɝɟɣɡɟɧɛɟɪɝɨɜɫɤɢɯ ɨɩɟɪɚɬɨɪɨɜ
ା
ሺݐሻ
ܽො
ൌ݁
௧క
ା
ೖ
ܽො
ǡԜԜԜܽො
ሺݐሻ
ൌ݁
ି௧క
ೖ
ܽොǤ
ሺ
͵Ǥͺ
ȼɜɟɞɟɦ ɬɟɩɟɪɶ ɨɩɟɪɚɬɨɪɵ ɪɨɠɞɟɧɢɹ ɢ ɭɧɢɱɬɨɠɟɧɢɹ ɷɥɟɤɬɪɨɧɚ ɜ ɬɨɱɤɟ x:
ሺxሻ
߰
ൌ߮
ା
ሺxሻ
ൌ߮
߰
ሺxሻ
Ԝܽො
ǡ
ା
כ
ሺxሻ
Ԝܽො
ǡ
ሺ
ሺ
͵ǤͳͲ
͵Ǥͻ
ɥɟɝɤɨ ɩɪɨɜɟɪɢɬɶ, ɱɬɨ ɷɬɢ ɨɩɟɪɚɬɨɪɵ ɭɞɨɜɥɟɬɜɨɪɹɸɬ ɫɥɟɞɭɸɳɢɦ ɚɧɬɢɤɨɦɦɭɬɚɰɢɨɧɧɵɦ ɫɨɨɬɧɨɲɟɧɢɹɦ
ሻ
ሻ
ሻ
ሻ
ሻ
ା
ሺxሻ
ሺxሻ
ା
ሺxሻ
ሺ
߰
x
ሺ
߰
x
ଵ
ା
ሺ
߰
߰
ቐ
߰
߰
ଵ
ሻ
x
ሻ
߰
߰
ሻ
ଵ
ሺ
x
߰
ା
ሺ
ሻ
ሺxሻ
x
߰
ଵ
ሻ
ሺxሻ
߰
ଵ
ା
ା
ሺ
ሻ
ሺxሻ
x
߰
ଵ
Ƚɚɦɢɥɶɬɨɧɢɚɧ (3.4), ɡɚɩɢɫɚɧɧɵɣ ɱɟɪɟɡ ɨɩɟɪɚɬɨɪɵ ߰
ଶ
pො
ା
ܪൌනܸ݀߰
ሺxሻ
ቈ
ʹ݉
ܷ
ሺxሻ
ȼ ɝɟɣɡɟɧɛɟɪɝɨɜɫɤɨɦ ɩɪɟɞɫɬɚɜɥɟɧɢɢ ɨɩɟɪɚɬɨɪɵ ߰
ɥɹɸɬɫɹ ɫ ɩɨɦɨɳɶɸ (3.8) ɢ ɢɦɟɸɬ ɜɢɞ
ሺ
߰
xǡݐሻൌ߮
ା
ሺ
߰
xǡݐሻൌ߮
ሺxሻ
Ԝܽො
כ
ሺxሻ
22
ൌߜሺx െx
ൌͲǡ
ൌͲǡ
ሺxሻ
ሺxሻ
െߤ߰
ሺ
xǡݐሻ ɢ ߰
ି௧క
ೖ
Ԝ݁
Ԝܽො
ǡ
ା
௧క
ೖ
݁
Ǥ
ɢ ߰
Ǥ
ሻ
ǡ
ଵ
ା
ሺxሻ
ା
ሺ
xǡݐ
, ɢɦɟɟɬ ɜɢɞ
ሺ
͵Ǥͳʹ
ሻ
ɥɟɝɤɨ ɜɵɱɢɫ-
ሺ
͵Ǥͳ͵
ሺ
͵ǤͳͶ
ሻ
ሻ
ሻ
3.2. Ɂɚɩɚɡɞɵɜɚɸɳɚɹ ɢ ɨɩɟɪɟɠɚɸɳɚɹ ɮɭɧɤɰɢɢ Ƚɪɢɧɚ
Ɋɚɫɫɦɨɬɪɢɦ ɬɟɩɟɪɶ ɷɥɟɤɬɪɨɧɧɵɣ ɝɚɡ ɩɪɢ ɤɨɧɟɱɧɨɣ ɬɟɦɩɟɪɚɬɭɪɟ ܶ. ȼ ɷɬɨɦ ɫɥɭ-
ɱɚɟ ɞɥɹ ɩɨɥɭɱɟɧɢɹ ɮɢɡɢɱɟɫɤɢɯ ɜɟɥɢɱɢɧ ɧɟɨɛɯɨɞɢɦɨ ɩɪɨɜɨɞɢɬɶ ɭɫɪɟɞɧɟɧɢɟ ɩɨ
ɝɢɛɛɫɨɜɫɤɨɦɭ ɚɧɫɚɦɛɥɸ ɩɨ ɮɨɪɦɭɥɟ (2.3). Ɇɵ ɷɬɨ ɛɭɞɟɦ ɞɟɥɚɬɶ ɫ ɩɨɦɨɳɶɸ ɝɚɦɢɥɶɬɨɧɢɚɧɚ (3.4), ɱɬɨ ɛɭɞɟɬ ɨɡɧɚɱɚɬɶ, ɱɬɨ ɦɵ ɪɚɫɫɦɚɬɪɢɜɚɟɦ ɫɨɫɬɨɹɧɢɹ ɧɟ ɩɪɢ
ɡɚɞɚɧɧɨɦ ɱɢɫɥɟ ɱɚɫɬɢɰ ܰ, ɚ ɩɪɢ ɡɚɞɚɧɧɨɦ ɯɢɦɢɱɟɫɤɨɦ
ɯɨɞ ɨɤɚɡɵɜɚɟɬɫɹ ɛɨɥɟɟ ɭɞɨɛɟɧ ɜ ɬɟɨɪɢɢ ɝɪɢɧɨɜɫɤɢɯ ɮɭɧɤɰɢɣ. Ɍɨɝɞɚ ɢɦɟɟɦ ɫɥɟɞɭɸɳɢɟ ɫɪɟɞɧɢɟ
ା
ۦ
ܽො
ɝɞɟ ݊
ɮɟɪɦɢɟɜɫɤɨɟ ɱɢɫɥɨ ɡɚɩɨɥɧɟɧɢɹ
ܽො
்ۧൌ݊
݊
ൌ݊ி
ሺ
ǡԜԜԜۦܽොܽො
ሻ
ߦ
ൌ
ା
ۧ
்
ఉక
݁
ൌͳെ݊ǡ
ͳ
ೖ
ɚ ߚൌȀܶ ɤɚɤ ɢ ɜ ɩɪɟɞɵɞɭɳɟɣ ɝɥɚɜɟ.
ோ
Ɋɚɫɫɦɨɬɪɢɦ ɬɟɩɟɪɶ ɡɚɩɚɡɞɵɜɚɸɳɭɸ ܩ
ሺ
ܩ
ݐ
ଵെݐଶǡxଵǡxଶ
ሻ
ɮɭɧɤɰɢɢ Ƚɪɢɧɚ ɷɥɟɤɬɪɨɧɨɜ ɩɪɢ ɤɨɧɟɱɧɨɣ ɬɟɦɩɟɪɚɬɭɪɟ. Ɉɧɢ
ሺ
ݐ
ɨɩɪɟɞɟɥɹɸɬɫɹ ɫɨɨɬɧɨɲɟɧɢɹɦɢ:
ோ
ሺ
ܩ
ݐ
ଵെݐଶǡxଵǡxଶ
ଵǡݐଵ
ା
ሻ
ሺ
߰
x
ଶǡݐଶ
ሺ
െ݅ൻ߰
ൌ
ቊ
x
Ͳǡ ݐ
ሻ
߰
ା
ሻ
ሺ
ሻ
x
߰
ଶǡݐଶ
ሺ
ܩ
ݐ
ଵെݐଶǡxଵǡxଶ
ሻ
Ͳǡ ݐ
ൌ
ቊ
݅ൻ߰
ଵǡݐଵ
ା
ሻ
ሺ
߰
x
ଶǡݐଶ
ሺ
x
ሻ
߰
ା
ሺ
ሻ
x
߰
ଶǡݐଶ
Ɋɚɫɫɦɨɬɪɢɦ ɬɚɤɠɟ ɢɯ ɮɭɪɶɟ-ɤɨɦɩɨɧɟɧɬɵ ɩɨ ɜɪɟɦɟɧɢ
ோ
ሺ
ܩ
߱ǡx
ሺ
߱ǡx
ܩ
ଵǡxଶ
ଵǡxଶ
ሻ
ൌെ݅න݁
ሻ
ஶ
ൌ݅න ݁
ିஶ
ఠ௧
ఠ௧
Ԝൻ߰
Ԝൻ߰
x
ଵǡݐଵ
ଵǡݐଵ
ሻ
ሺ
߰
ା
ሻ
߰
ሻ
x
߰
ଶǡݐଶ
ሺ
ሻ
x
߰
ଶǡݐଶ
ሺ
ሺ
x
ା
ɉɨɞɫɬɚɜɥɹɟɦ (3.13) ɢ (3.14), ɢ ɢɫɩɨɥɶɡɭɹ (3.3), ɩɨɥɭɱɢɦ
ሺ
߮
x
߱െߦ
ሺ
߮
߱െߦ
ଵ
x
ଵ
ோ
ሺ
ܩ
߱ǡx
ሺ
߱ǡx
ܩ
ଵǡxଶ
ଵǡxଶ
ሻ
ൌ
ሻ
ൌ
23
ɩɨɬɟɧɰɢɚɥɟ ߤ. Ɍɚɤɨɣ ɩɨɞ-
ǡ
ͳ
ሻ
ଵെݐଶǡxଵǡxଶ
ɢ ɨɩɟɪɟɠɚɸɳɭɸ
ൌ
ሺ
ሻ
x
ൿ
ǡݐଵݐଶǡ
ଵǡݐଵ
்
ଵ൏ݐଶ
ൌ
ሻ
ൿ
ǡݐଵ൏ݐଶǤ
்
ሻ
߰
ଶǡݐଶ
ሻ
߰
ሻ
ǡ
ሻ
Ǥ
ଵݐଶ
ሺ
x
ሺ
x
ଵǡݐଵ
ଵǡݐଵ
ሻ
ሻ
ሺ
߮
߮
x
ଵǡݐଵ
ା
ା
ሺ
כ
ሺ
݅ߜ
כ
ሺ
െ݅ߜ
ሺ
x
x
x
x
ଶǡݐଶ
ଶ
ଶ
ǡ
ǡ
ሻ
ൿ
݀ݐǡሺ͵Ǥͳͻ
்
ሻ
ൿ
݀ݐǤሺ͵ǤʹͲ
்
ሺ
͵Ǥͳͷ
ሺ
͵Ǥͳ
ሺ
͵Ǥͳ
ሺ
͵Ǥͳͺ
ሺ
͵Ǥʹͳ
ሺ
͵Ǥʹʹ
ሻ
ሻ
ሻ
ሻ
ሻ
ሻ
ሻ
ሻ
ȼɵɩɢɲɟɦ ɩɨɥɟɡɧɨɟ ɜ ɞɚɥɶɧɟɣɲɟɦ ɭɪɚɜɧɟɧɢɟ, ɤɨɬɨɪɨɦɭ ɭɞɨɜɥɟɬɜɨɪɹɸɬ ɨɛɟ
ȁ
ɷɬɢ ɮɭɧɤɰɢɢ. ɗɬɨ ɭɪɚɜɧɟɧɢɟ ɥɟɝɤɨ ɩɨɥɭɱɚɟɬɫɹ ɢɡ ɫɨɨɬɧɨɲɟɧɢɣ (3.2), ɞɟɣɫɬɜɢ-
ɬɟɥɶɧɨ, ɩɪɢɦɟɧɢɜ ɨɩɟɪɚɬɨɪ ߱െ݄ȀߤȀ ɤ ɥɸɛɨɣ ɢɡ ɮɭɧɤɰɢɣ (3.21) ɢ (3.22),
ɩɨɥɭɱɢɦ
ߤ
ቆ߱െ
݄
ቇܩሺ߱ǡxଵǡx
ሻൌߜሺ
ଶ
x
ଵെxଶ
ሻ
ǡ
ሺ
͵Ǥʹ͵
ɢɥɢ ɫ ɭɱɟɬɨɦ (3.1)
ቆ߱െ
ଶ
ෝ
ଵ
ʹ݉
െ
ܷሺ࢞
ሻ
ߤ
ቇܩሺ߱ǡxଵǡx
ሻൌߜሺ
ଶ
x
ଵെxଶ
ሻ
Ǥ
ሺ
͵ǤʹͶ
ɗɬɨ ɢ ɟɫɬɶ ɭɪɚɜɧɟɧɢɟ, ɤɨɬɨɪɨɦɭ ɭɞɨɜɥɟɬɜɨɪɹɸɬ ɨɛɟ ɮɭɧɤɰɢɢ (3.21) ɢ (3.22),
ɮɚɤɬɢɱɟɫɤɢ ɨɧɢ ɹɜɥɹɸɬɫɹ ɪɟɲɟɧɢɹɦɢ ɷɬɨɝɨ ɭɪɚɜɧɟɧɢɹ ɫ ɪɚɡɥɢɱɧɵɦɢ ɧɚɱɚɥɶɧɵɦɢ ɭɫɥɨɜɢɹɦɢ.
ɂɫɩɨɥɶɡɭɹ ɷɬɨ ɭɪɚɜɧɟɧɢɟ, ɥɟɝɤɨ ɩɨɥɭɱɢɬɶ ɮɭɧɤɰɢɢ Ƚɪɢɧɚ ɞɥɹ ɫɜɨɛɨɞɧɵɯ
ɷɥɟɤɬɪɨɧɨɜ ܩ
ሺ
߱ǡx
ሻ
. ȼ ɷɬɨɦ ɫɥɭɱɚɟ ܷ
ଵǡxଶ
ଶ
ෝ
ቆ߱െ
ଵ
ʹ݉
ሺxሻ
ൌͲ ɢ ɭɪɚɜɧɟɧɢɟ ɩɪɢɧɢɦɚɟɬ ɜɢɞ
ߤ
ሺ
ቇܩ
߱ǡx
ଵǡxଶ
ሻൌߜሺ
x
ଵെxଶ
ሻ
Ǥ
ሺ
͵Ǥʹͷ
ሻ
ሻ
ሻ
Ɉɱɟɜɢɞɧɨ, ɱɬɨ ɮɭɧɤɰɢɹ ܩ
x
, ɢ ɩɨɷɬɨɦɭ ɭɞɨɛɧɨ ɩɟɪɟɣɬɢ ɤ ɬɚɤ ɧɚɡɵɜɚɟɦɨɦɭ ɢɦɩɭɥɶɫɧɨɦɭ ɩɪɟɞɫɬɚɜɥɟ-
ଵെxଶ
ሺ
߱ǡx
ሻ
ɡɚɜɢɫɢɬ ɬɨɥɶɤɨ ɨɬ ɪɚɡɧɨɫɬɢ ɤɨɨɪɞɢɧɚɬ
ଵǡxଶ
ɧɢɸ
ௗ
݀
ܩ
ɡɞɟɫɶ ɧɚ ɫɚɦɨɦ ɞɟɥɟ x
ሺ
߱ǡx
ଵെxଶ
ଵെxଶ
ሻ
ൌන
ሺʹߨሻ
p
݁
ௗ
– ɜɨɥɧɨɜɨɣ ɜɟɤɬɨɪ, ɚ ɧɟ ɢɦɩɭɥɶɫ, ɧɨ ɦɵ ɛɭɞɟɦ ɢɝɧɨ-
ሻ
ሺ
p
x
భିxమ
ሺ
߱ǡpሻǡ
ܩ
ሺ
͵Ǥʹ
ɪɢɪɨɜɚɬɶ ɷɬɨ ɨɬɥɢɱɢɟ (ɨɧɨ ɩɨɥɧɨɫɬɶɸ ɢɫɱɟɡɚɟɬ, ɟɫɥɢ ɜɵɛɪɚɬɶ ɫɢɫɬɟɦɭ ɟɞɢɧɢɰ,
ሺ
ሻ
߱ǡp
ɝɞɟ ൌͳ), ɞɥɹ ɮɭɧɤɰɢɢ ܩ
ɢɦɩɭɥɶɫɧɨ-ɱɚɫɬɨɬɧɨɦ ɩɪɟɞɫɬɚɜɥɟɧɢɢ ɢɦɟɟɦ
ɭɪɚɜɧɟɧɢɟ
൫߱െߦ
൯ܩ
p
ሺ
߱ǡpሻൌͳǡ
ሺ
͵Ǥʹ
ɝɞɟ, ɤɚɤ ɢ ɜɵɲɟ ɨɛɨɡɧɚɱɟɧɨ
ଶ
ͳ
ȁ
p
ൌ
ߦ
p
ቆ
ʹ݉
െߤቇǤ
ሺ
͵Ǥʹͺ
Ɋɟɲɟɧɢɹɦɢ ɭɪɚɜɧɟɧɢɹ (3.27) ɨɱɟɜɢɞɧɨ ɟɫɬɶ
߱െߦ
߱െߦ
ͳ
p
ͳ
p
݅ߜ
െ݅ߜ
ǡ
Ǥ
ሺ
͵Ǥʹͻ
ሺ
͵Ǥ͵Ͳ
ோ
ܩ
ܩ
ሺ
߱ǡpሻൌ
ሺ
߱ǡpሻൌ
24
ሻ
ሻ
ሻ
ሻ
ሻ
ɍɪɚɜɧɟɧɢɟ (3.24) ɦɨɠɧɨ ɩɟɪɟɩɢɫɚɬɶ ɜ ɜɢɞɟ ɢɧɬɟɝɪɚɥɶɧɨɝɨ ɭɪɚɜɧɟɧɢɹ
ܩሺ߱ǡx
ଵǡxଶ
ሻ
ൌܩ
ሺ
߱ǡx
ଵെxଶ
ሻ
݀
ௗ
ሺ
߱ǡx
yܩ
ଵ
െy
ሻ
ܷ
ܩሺ߱ǡyǡx
ሻǡሺ
ଶ
͵Ǥ͵ͳ
ሺyሻ
ɞɟɣɫɬɜɢɬɟɥɶɧɨ, ɩɨɞɟɣɫɬɜɨɜɚɜ ɨɩɟɪɚɬɨɪɨɦ
ଶ
pො
ߤ
y
ܩ൫߱ǡxଵെyଵ൯
ଵ
൯
ܷ൫y
ଵ
൯
ܷ൫y
ଵ
ܩ൫߱ǡyଵെxଶ൯
ܩ൫߱ǡyଵെyଶ൯ൈ
ሺ
͵Ǥ͵ͳ
ሻ
ɧɚ
ɭɪɚɜɧɟɧɢɹ
ܩሺ߱ǡx
ଵǡxଶ
߱െ
ʹ݉
ɢ ɜɨɫɩɨɥɶɡɨɜɚɜɲɢɫɶ (3.25), ɩɨɥɭɱɢɦ (3.26). Ɇɨɠɧɨ ɜɵɩɢɫɚɬɶ ɪɟɲɟɧɢɟ
ሺ
ሻ
͵Ǥ͵ͳ
ɜ ɜɢɞɟ ɮɨɪɦɚɥɶɧɨɝɨ ɪɹɞɚ, ɢɬɟɪɢɪɭɹ ɟɝɨ ɦɧɨɝɨɤɪɚɬɧɨ
ሻ
ൌܩ
න݀
ሺ
߱ǡx
ௗ
y
ଵ
ሻ
ଵെxଶ
ௗ
݀
y
ܩ൫߱ǡxଵെyଵ൯
ଶ
න݀
ௗ
ܷ൫yଶ൯
ൈ
ܩ൫߱ǡyଶെxଶ൯ڮǡ
ɷɬɨ ɪɚɡɥɨɠɟɧɢɟ ɜ ɪɹɞ ɩɨ ɩɨɬɟɧɰɢɚɥɭ, ɩɪɢɦɟɧɟɧɧɨɟ ɤ ɮɭɧɤɰɢɹɦ Ƚɪɢɧɚ ܩ
ோ
ሺ
͵Ǥ͵ʹ
ሺ
͵Ǥ͵͵
ɢ ܩ
ɦɵ ɢɫɩɨɥɶɡɭɟɦ ɜ ɝɥɚɜɟ 5 ɞɥɹ ɩɪɨɰɟɞɭɪɵ ɭɫɪɟɞɧɟɧɢɹ ɪɚɡɥɢɱɧɵɯ ɮɢɡɢɱɟɫɤɢɯ ɜɟɥɢɱɢɧ ɩɨ ɫɥɭɱɚɣɧɨɦɭ ɪɚɫɩɨɥɨɠɟɧɢɸ ɩɪɢɦɟɫɟɣ.
3.3. Ɇɚɰɭɛɚɪɨɜɫɤɚɹ ɮɭɧɤɰɢɹ Ƚɪɢɧɚ
ȼɜɟɞɟɦ ɬɟɩɟɪɶ ɦɚɰɭɛɚɪɨɜɫɤɭɸ ɮɭɧɤɰɢɸ Ƚɪɢɧɚ ɷɥɟɤɬɪɨɧɨɜ. Ⱦɥɹ ɷɬɨɝɨ ɩɟɪɟɣ-
ɞɟɦ ɤ ɦɚɰɭɛɚɪɨɜɫɤɨɦɭ ɜɪɟɦɟɧɢ ߬ ɢ ɪɚɫɫɦɨɬɪɢɦ ɨɩɟɪɚɬɨɪɵ ɜ ɦɚɰɭɛɚɪɨɜɫɤɨɦ ɝɟɣ-
ɡɟɧɛɟɪɝɨɜɫɤɨɦ ɩɪɟɞɫɬɚɜɥɟɧɢɢ
߰
߰
ெ
ெ
ሺ
xǡ߬ሻൌ݁
ሺ
xǡ߬ሻൌ݁
ఛு
ఛு
߰
ା
߰
ሺxሻ
ሺxሻ
ఛு
ି
݁
ǡ
ఛு
ି
݁
ǡԜԜԜԜԜ
ሺ
͵Ǥ͵Ͷ
ሺ
͵Ǥ͵ͷ
ሻ
ሻ
ሻ
,
ሻ
ሻ
ɬɨɝɞɚ ɩɨ ɨɩɪɟɞɟɥɟɧɢɸ
ெ
ܩ
ሺ
߬
ଵǡ߬ଶǡxଵǡxଶ
ሻ
ൌെർT
ெ
ሺ
x
߰
ఛ
ଵǡ߬ଵ
ெ
ሻ
ሺ
߰
x
ଶǡ߬ଶ
Ʌɟɝɤɨ ɜɢɞɟɬɶ, ɱɬɨ ɭɠɟ ɩɨ ɫɚɦɨɦɭ ɨɩɪɟɞɟɥɟɧɢɸ (3.36) ɮɭɧɤɰɢɹ ܩ
ɪɚɡɧɨɫɬɢ ߬ൌ߬
ൌ
. ɉɭɫɬɶ, ɧɚɩɪɢɦɟɪ, ߬ଵ൏߬
ଵെ߬ଶ
ͳ
ெ
ܩ
ൌ
Tr ቆ݁
ܼ
ͳ
ିఉுȀ݁ఛమுȀ
Tr൫݁
ܼ
ିఉுȀ
ା
߰
߰
ሺ
ሻ
x
ଶ
, ɬɨɝɞɚ ɢɦɟɟɦ
ଶ
ெ
ሺ
ሻ
x
ଶǡ߬ଶ
ିఛమுȀ݁ఛభுȀ
݁
ெ
ሺ
߰
x
ଵǡ߬ଵ
ሻ
ሺ
߰
x
ଵ
25
ሻ
ሻ
ቇൌ
ିఛభுȀ
݁
ሺ
Ǥ
்
ெ
ɡɚɜɢɫɢɬ ɨɬ
͵Ǥ͵
ሻ
൯ൌ
Tr ൭݁
ఉு
ି
݁
ͳ
ൌ
ܼ
ሺ
ఛ
మିఛభ
ሻ
ு
߰
ି
ା
ሺ
ሻ
x
݁
ଶ
ሺ
ఛ
మିఛభ
ሻ
ு
ሻ
ሺ
൱ǡ
߰
x
ଵ
ሺ
͵Ǥ͵
ɨɬɤɭɞɚ ɨɱɟɜɢɞɧɨ ɫɞɟɥɚɧɧɨɟ ɭɬɜɟɪɠɞɟɧɢɟ (ɦɵ ɩɪɨɢɡɜɟɥɢ ɩɨɞ ɡɧɚɤɨɦ Tr ɰɢɤɥɢɱɟɫɤɭɸ ɩɟɪɟɫɬɚɧɨɜɤɭ ɦɧɨɠɢɬɟɥɟɣ). ɋ ɭɱɟɬɨɦ ɷɬɨɝɨ ɨɛɫɬɨɹɬɟɥɶɫɬɜɚ, ɦɵ ɢɡɦɟɧɢɦ
ɚɪɝɭɦɟɧɬɵ ɮɭɧɤɰɢɢ ܩ
ெ
ɫɥɟɞɭɸɳɢɦ ɨɛɪɚɡɨɦ:
ெ
ሺ
߬ǡx
ܩ
ଵǡxଶ
ሻ
ൌെർT
x
ǡ߬ሻ߰
ଵ
ெ
ሺ
x
ଶ
ǡͲሻ்ǡ
ሺ
͵Ǥ͵ͺ
ெ
ሺ
߰
ఛ
ɩɪɢ ɷɬɨɦ ɩɟɪɟɦɟɧɧɚɹ ߬ ɛɭɞɟɬ ɮɚɤɬɢɱɟɫɤɢ ɩɪɨɛɟɝɚɬɶ ɡɧɚɱɟɧɢɹ ɜ ɤɨɧɟɱɧɨɦ ɢɧɬɟɪ-
ɜɚɥɟ
ሺ
͵Ǥ͵ͻ
ɉɪɢ ɷɬɨɦ ɡɧɚɱɟɧɢɹ ɮɭɧɤɰɢɢ ܩ
െߚ߬ߚǤ
ெ
ሺ߬ሻ
ɩɪɢ ߬൏Ͳ ɢ ߬Ͳ ɫɜɹɡɚɧɵ ɞɪɭɝ ɫ ɞɪɭɝɨɦ
ɩɪɨɫɬɵɦ ɫɨɨɬɧɨɲɟɧɢɟɦ. Ⱦɟɣɫɬɜɢɬɟɥɶɧɨ, ɩɪɢ ߬൏Ͳ ɢɦɟɟɦ
ሻ
ሻ
ሻ
ܩ
ெ
ܩ
ெ
ሺ
߬ߚሻൌെ
ሺ߬ሻ
ఉு
ͳ
ൌ
Tr ൭݁
ܼ
ͳ
Tr ൭݁
ܼ
ఛு
ି
ି
ା
݁
ሺ
݁
ఛାఉሻு
߰
߰
ఉு
ି
ఛு
ሺ
ሻ
x
݁
ଶ
ሻ
ሺ
݁
x
ଵ
ሺ
ି
ሺ
߰
x
ఛାఉሻு
ሻ
൱ǡ
ଵ
ା
ሺ
ሻ
x
߰
൱ǡ
ଶ
ሺ
͵ǤͶͲ
ሺ
͵ǤͶͳ
ɜɨ ɜɬɨɪɨɣ ɮɨɪɦɭɥɟ ɰɢɤɥɢɱɟɫɤɢ ɩɟɪɟɫɬɚɜɥɹɹ ɨɩɟɪɚɬɨɪɵ, ɩɨɥɭɱɢɦ
ெ
ሺ
ܩ
߬ߚሻൌെ
ͳ
ܼ
Tr ൭݁
ఉு
ఛு
ି
ି
ା
݁
߰
ఛு
ሺ
ሻ
x
݁
ଶ
ሻ
ሺ
൱ǡ
߰
x
ଵ
ሺ
͵ǤͶʹ
ɬɚɤɢɦ ɨɛɪɚɡɨɦ,
ெ
ܩ
Ɋɚɡɥɨɠɢɦ ɬɟɩɟɪɶ ɮɭɧɤɰɢɸ ܩ
ெ
ሺ
߬ǡx
ܩ
ଵǡxଶ
ሺ߬ሻ
ெ
ሻ
ሺ߬ሻ
ൌ
ெ
ൌെܩ
ሺ
߬ߚሻǤ
ɜ ɪɹɞ Ɏɭɪɶɟ ɩɨ ߬ ɧɚ ɢɧɬɟɪɜɚɥɟ (3.39)
ஶ
ͳ
ߚ
݁
௦ୀିஶ
ିೞఛ
ሺ
ሻሺ
x
݃
௦
ଵǡxଶ
ሺ
͵ǤͶ͵
͵ǤͶͶ
ɩɪɢ ɷɬɨɦ ɥɟɝɤɨ ɩɪɨɜɟɪɢɬɶ, ɱɬɨ ɞɥɹ ɜɵɩɨɥɧɟɧɢɹ ɫɜɨɣɫɬɜɚ (3.43) ɧɟɨɛɯɨɞɢɦɨ, ɱɬɨɛɵ
ൌሺʹݏͳሻߨȀߚൌሺʹݏͳሻߨܶȀǤ
ߞ
௦
ሺ
͵ǤͶͷ
Ɉɛɪɚɬɧɨɟ ɤ (3.44) ɩɪɟɨɛɪɚɡɨɜɚɧɢɟ ɢɦɟɟɬ ɜɢɞ
ఉ
ೞఛ
ሺ
݃
ሻ
x
ଵǡxଶ
ൌන݁
௦
ெ
ሺ
ܩ
߬ǡx
ଵǡxଶ
ሻ
Ǥ
ሺ
͵ǤͶ
ሻ
ሻ
ሻ
ሻ
ሻ
ሻ
ሻ
26
Ɉɤɚɡɵɜɚɟɬɫɹ ɞɥɹ ɞɚɥɶɧɟɣɲɟɝɨ ɭɞɨɛɧɨ ɨɩɪɟɞɟɥɢɬɶ ɬɚɤɭɸ ɮɭɧɤɰɢɸ ɤɨɦɩɥɟɤɫ-
ɧɨɝɨ ɩɟɪɟɦɟɧɧɨɝɨ ܩ
ெ
ሺ
ݖǡx
ሻ
, ɱɬɨ ɞɥɹ ɜɫɟɯ ɰɟɥɵɯ ݏ
ଵǡxଶ
ெ
ሺ
ܩ
݅ߞ
௦ǡxଵǡxଶ
ሻ
ൌ݃
ሺ
ሻ
x
ଵǡxଶ
ǡ
௦
ɬɨ ɟɫɬɶ
ெ
ሺ
݅ߞ
ܩ
௦ǡxଵǡxଶ
ሻ
ൌන݁
ఉ
ೞఛ
ெ
ሺ
߬ǡx
ܩ
ଵǡxଶ
ሻ
Ǥ
ȼɨɫɩɨɥɶɡɭɟɦɫɹ ɬɟɩɟɪɶ (3.9) ɢ (3.10) ɢ ɬɟɦ, ɱɬɨ
ܽො
ൌ݁
ఛு
ା
ሺ߬ሻ
ఛு
ି
ା
ܽො
݁
ା
ఛక
ൌܽො
ೖ
ǡԜԜԜԜԜܽො
ሺ߬ሻ
݁
ൌ݁
ఛு
ܽො݁
ି
ఛு
ൌܽො݁
ିఛక
ೖ
ǡሺ͵ǤͶͻ
ɬɨɝɞɚ ɨɩɟɪɚɬɨɪɵ (3.35) ɦɨɠɧɨ ɩɪɟɞɫɬɚɜɢɬɶ ɜ ɜɢɞɟ
ெ
ሺ
xǡ߬ሻൌ߮
߰
ெ
ሺ
xǡ߬ሻൌ߮
߰
ɗɬɨ ɩɨɡɜɨɥɢɬ ɧɚɦ ɩɪɟɞɫɬɚɜɢɬɶ ɮɭɧɤɰɢɸ ܩ
ା
כ
ሺxሻ
ሺxሻ
Ԝܽො
Ԝܽො
ெ
ఛక
݁
ିఛక
݁
ሺ
ߞ
௦ǡxଵǡxଶ
ೖ
ǡ
ೖ
Ǥ
ሻ
ɜ ɮɨɪɦɟ ɚɧɚɥɨɝɢɱɧɨɣ
(3.22), (3.23), ɞɟɣɫɬɜɢɬɟɥɶɧɨ
ெ
ሺ
ܩ
߬ǡx
ଵǡxଶ
ሻ
ൌെߠሺ߬ሻർ߰
ெ
ሺ
x
ǡ߬ሻ߰
ଵ
ெ
ሺ
x
ǡͲሻ்ߠሺെ߬ሻർ߰
ଶ
ெ
ሺ
x
ଶ
ǡͲሻ߰
ெ
ሺ
x
ଵ
ሺ
͵ǤͶ
ሺ
͵ǤͶͺ
ሺ
͵ǤͷͲ
ሺ
͵Ǥͷͳ
ǡ߬ሻ்ൌ
ሻ
ሻ
ሻ
ሻ
ሻ
כ
ሺ
ሻ
ሺ
ൌ߮
x
ଵ
ሻ
߮
x
ଶ
Ⱦɥɹ ɫɪɟɞɧɢɯ ɨɬ ɨɩɟɪɚɬɨɪɨɜ ܽො
ା
ۧ
ۦ
ܽො
ܽො
்ൌߜ
ା
ۦ
ܽො
ܽො
்ۧൌߜԜ݊
ିఛక
ೖ
ሺെߠሺ߬ሻ
݁
ା
, ܽො
ɢɦɟɟɦ
ሺ
ͳെ݊
ା
ۧ
ۦ
ܽො
ܽො
ߠሺെ߬
்
ߜ
ሻ
ൌ
ൌ
ିఉక
ೖ
݁
ͳ
ߜ
ೖ
ͳ
ǡ
ఉక
݁
ା
ሻ
ۦ
ܽො
ǡ
ሻǤሺ
ܽො
்ۧ
͵Ǥͷʹ
ሺ
͵Ǥͷ͵
ሺ
͵ǤͷͶ
ɡɞɟɫɶ ɦɵ ɢɫɩɨɥɶɡɨɜɚɥɢ ɜɵɪɚɠɟɧɢɟ ɞɥɹ ɫɪɟɞɧɟɝɨ ɱɢɫɥɚ ɡɚɩɨɥɧɟɧɢɹ ɮɟɪɦɢɨɧɨɜ:
݊
ൌ݊ிሺߦ
ሻൌ
ͳ
ೖ
ͳ
ǡ
ఉక
݁
ሺ
͵Ǥͷͷ
ɬɚɤɢɦ ɨɛɪɚɡɨɦ,
ெ
ሺ
߬ǡx
ܩ
ଵǡxଶ
ሻ
ൌ߮
כ
ିఛక
ሺ
ሻ
ሺxሻ
x
߮
ଵ
ೖ
݁
ሾ
െߠሺ߬ሻሺͳെ݊
ሻߠሺെ߬ሻ
݊
ሺ
ሿ
Ǥ
͵Ǥͷ
27
ሻ
ሻ
ሻ
ሻ
ሻ
Ⱦɚɥɟɟ ɜɵɱɢɫɥɹɟɦ ɮɭɪɶɟ-ɨɛɪɚɡ
ெ
ሺ
݅ߞ
ܩ
௦ǡxଵǡxଶ
ൌെ߮
ሺ
ሻ
ሻ
x
߮
ଵ
ൌെ߮
െ݁
כ
ሺ
ሻ
x
ଶ
݅ߞ
ሺ
x
ିఉక
ೖ
௦െߦ
ሻ
ଵ
െͳ
כ
߮
ఉ
ఛ
ିఛక
න݁
ͳ
ೖ
ͳ
ೞ
݁
ൌ
ሺ
ሻ
x
ଶ
Ԝ
ିఉక
݁
ͳ
ೖ
Ԝ
ିఉక
݁
ሺ
߮
x
݅ߞ
ೖ
ͳ
כ
ሻ
߮
ଵ
௦െߦ
ൌ
ሺ
ሻ
x
ଶ
Ɍɚɤɢɦ ɨɛɪɚɡɨɦ,
כ
ሺ
ሻ
ሺ
߮
x
݅ߞ
ଵ
௦െߦ
ெ
ሺ
݅ߞ
ܩ
௦ǡxଵǡxଶ
ሻ
ൌ
ሻ
߮
x
ଶ
Ǥ
ɋɪɚɜɧɢɜɚɹ ɷɬɨ ɜɵɪɚɠɟɧɢɟ ɫ (3.22), (3.23), ɩɨɥɭɱɚɟɦ
ெ
ሺ
݅ߞ
ܩ
௦ǡxଵǡxଶ
ெ
ሺ
ܩ
݅ߞ
௦ǡxଵǡxଶ
ሻ
ሻ
ൌܩ
ൌܩ
ோ
ሺ
݅ߞ
ሺ
݅ߞ
௦ǡxଵǡxଶ
௦ǡxଵǡxଶ
ሻ
ԜԜԜɩɪɢԜԜߞ
ሻ
ԜԜԜɩɪɢԜԜߞ
௦
௦
Ͳǡ
൏ͲǤ
ɗɬɢ ɪɚɜɟɧɫɬɜɚ ɨɡɧɚɱɚɸɬ, ɱɬɨ ɚɧɚɥɢɬɢɱɟɫɤɚɹ ɜ ɜɟɪɯɧɟɣ ɩɨɥɭɩɥɨɫɤɨɫɬɢ ߱
ɮɭɧɤɰɢɹ ܩ
ሺ߱ሻ
ɞɥɹ ܩ
ோ
ሺ߱ሻ
ɜ ɞɢɫɤɪɟɬɧɵɯ ɬɨɱɤɚɯ ɦɧɢɦɨɣ ɨɫɢ ݅ߞ
ɜ ɧɢɠɧɟɣ ɩɨɥɭɩɥɨɫɤɨɫɬɢ.
ɫɨɜɩɚɞɚɟɬ ɫ ܩ
௦
ெ
, ɚɧɚɥɨɝɢɱɧɨ
ɉɨɤɚɠɟɦ ɜ ɤɚɱɟɫɬɜɟ ɢɥɥɸɫɬɪɚɰɢɢ ɤɚɤ ɫɭɦɦɢɪɨɜɚɧɢɟ ɩɨ ɱɚɫɬɨɬɚɦ ߞ
ɰɢɢ (3.58) ɩɪɢɜɨɞɹɬ ɤ ɮɨɪɦɭɥɟ (3.56). Ɏɚɤɬɢɱɟɫɤɢ ɧɭɠɧɨ ɩɨɤɚɡɚɬɶ
ஶ
ିೞఛ
݁
݅ߞ௦െߦ
ൌ݁
ିఛక
ೖ
ሾ
െߠሺ߬ሻሺͳെ݊
ሻߠሺെ߬ሻ
ൌߚߦȀߨ ɢ ݔൌߨ߬Ȁߚ, ɚ ɢɫɤɨɦɭɸ ɫɭɦɦɭ
ሿ
Ǥ
݊
ͳ
ߚ
௦ୀିஶ
Ɉɛɨɡɧɚɱɢɦ ɞɥɹ ɭɩɪɨɳɟɧɢɹ ɮɨɪɦɭɥ ܽ
ܵ, ɬɨɝɞɚ
ܵൌ
ൌെ
ͳ
ߨ
ஶ
௦ୀିஶ
ܽ
ߨ
ିሺଶ௦ାଵሻ௫
݁
݅ሺʹݏͳሻെܽ
ஶ
௦ୀିஶ
ሾ
ሺ
ʹݏͳ
ሺ
ʹݏͳሻݔ
ଶ
ሻ
ܽ
ஶ
௦ୀିஶ
ͳ
ߨ
݁
ஶ
௦ୀିஶ
ିሺଶ௦ାଵሻ௫
ͳ
ൌ
ߨ
ሿ
െ
ଶ
ሺെ݅ሺ
ሺ
ʹݏͳ
ሺ
ሾ
ʹݏͳሻݔ
ሺ
ʹݏͳ
ʹݏͳሻെܽ
ଶ
ଶ
ሻ
ܽ
ሺ
ሿ
ʹݏͳ
ଶ
ଶ
ሻ
ܽ
ሻ
ൌ
ሻ
Ǥ
Ɇɵ ɩɨɥɭɱɢɥɢ ɬɚɛɥɢɱɧɵɟ ɫɭɦɦɵ ([13], ɫɬɪ. 733)
ȁ
ቀ
ቂܽ
ߨ
ቀ
ʹ
ߨܽ
െȁݔȁቁቃ
ቁ
ʹ
ߨ
ቀ
ʹ
ߨܽ
ቀ
ǡ
െȁݔȁቁቃ
ቁ
ʹ
ǡ
ஶ
௦ୀିஶ
ஶ
௦ୀିஶ
ሾ
ሺ
ʹݏͳ
ሺ
ሾ
ʹݏͳሻݔ
ሺ
ʹݏͳ
ሺ
ʹݏͳሻݔ
ଶ
ሻ
ܽ
ሺ
ሿ
ଶ
ሻ
ܽ
ሿ
ଶ
ʹݏͳ
ଶ
ቂܽ
ߨ
ൌ
ʹܽ
ሻ
ߨݔ
ൌ
ʹȁݔ
Ǥሺ͵Ǥͷ
ሺ
͵Ǥͷͺ
ሺ
͵Ǥͷͻ
ሺ
͵ǤͲ
ɮɭɧɤ-
௦
ሺ
͵Ǥͳ
ሺ
͵Ǥʹ
ሺ
͵Ǥ͵
ሺ
͵ǤͶ
ሻ
ሻ
ሻ
ሻ
ሻ
ሻ
ሻ
ሻ
28
ɜɬɨɪɚɹ ɫɭɦɦɚ ɩɨɥɭɱɚɟɬɫɹ ɢɡ ɩɟɪɜɨɣ ɞɢɮɮɟɪɟɧɰɢɪɨɜɚɧɢɟɦ ɩɨ ݔ, ɬɨɝɞɚ
ܵൌെ
ʹቀ
ͳ
ߨܽ
ቁ
ʹ
൬ቂܽቀ
ߨ
െȁݔȁቁቃ
ʹ
ݔ
ȁݔ
ቂܽቀ
ȁ
ߨ
െȁݔȁቁቃ൰Ǥ
ʹ
Ⱦɚɥɟɟ, ɢɦɟɟɦ ɩɪɢ ݔͲ
గ
ି௫
ቀ
ܵൌെ
݁
ʹቀ
ቁ
ଶ
ൌെ
ߨܽ
ቁ
ʹ
ିఛక
݁
ͳ݁
ೖ
ିఉక
ൌെ݁
ೖ
ିఛక
ೖ
ሺ
ͳെ݊
ሻ
ǡ
ɩɪɢ ݔ൏Ͳ
గ
ି
ି௫
ቀ
ܵൌ
݁
ʹቀ
ଶ
ߨܽ
ቁ
ʹ
ିఛక
ೖ
݁
ൌ
ቁ
ͳ݁
ఉక
ೖ
ൌ݁
ିఛక
ೖ
݊ǡ
ɬɚɤɢɦ ɨɛɪɚɡɨɦ, ɮɨɪɦɭɥɚ (3.61) ɞɨɤɚɡɚɧɚ.
3.4. ɉɪɢɱɢɧɧɚɹ ɮɭɧɤɰɢɹ Ƚɪɢɧɚ
ȼ ɡɚɜɟɪɲɟɧɢɟ ɷɬɨɣ ɝɥɚɜɵ ɪɚɫɫɦɨɬɪɢɦ ɩɪɢɱɢɧɧɭɸ ɮɭɧɤɰɢɸ Ƚɪɢɧɚ. Ɉɧɚ ɨɩɪɟɞɟɥɹɟɬɫɹ ɫɥɟɞɭɸɳɢɦ ɨɛɪɚɡɨɦ:
ሺ
ܩ
ݐǡx
ൌ൝
ଵǡxଶ
െ݅ൻ߰
݅ൻ߰
ሻ
ൌെ݅ൻT߰
ሺ
x
ା
ሺ
ǡݐሻ߰
ଵ
x
ǡͲሻ߰
ଶ
ሺ
ା
ሺ
x
ሺ
ା
x
ଶ
x
ଵ
ሺ
x
ǡݐሻ߰
ଵ
ǡͲሻൿ்ൌ
ଶ
ǡͲሻൿ்ǡݐͲǡ
ǡݐሻൿ்ǡݐ൏ͲǤ
ሺ
͵Ǥͷ
ሺ
͵Ǥ
ሺ
͵Ǥ
ሺ
͵Ǥͺ
ሻ
ሻ
ሻ
ሻ
Ɍɚɤ ɠɟ, ɤɚɤ ɢ ɦɚɰɭɛɚɪɨɜɫɤɚɹ ɮɭɧɤɰɢɹ Ƚɪɢɧɚ, ɷɬɚ ɮɭɧɤɰɢɹ ɜɵɪɚɠɚɟɬɫɹ ɱɟɪɟɡ
ɮɟɪɦɢɟɜɫɤɢɟ ɱɢɫɥɚ ɡɚɩɨɥɧɟɧɢɹ. ɑɬɨɛɵ ɜ ɷɬɨɦ ɭɛɟɞɢɬɶɫɹ, ɩɨɞɫɬɚɜɢɦ ɜ (3.68) ɜɵ-
ା
ɪɚɠɟɧɢɹ (3.9) ɢ (3.10) ɞɥɹ ɨɩɟɪɚɬɨɪɨɜ ߰ ɢ ߰
˔
ሺ
ܩ
ݐǡx
ଵǡxଶ
ሻ
ൌ߮
כ
ሺ
ሻ
ሺ
x
ଵ
ሻ
߮
x
݁
ଶ
ି௧క
ೖ
ሾ
െ݅ߠሺݐሻሺͳെ݊
, ɩɨɥɭɱɢɦ
ሻ
݅ߠሺെݐሻ݊
ሿ
ǡሺ͵Ǥͻ
ɩɪɨɢɡɜɟɞɹ ɮɭɪɶɟ-ɩɪɟɨɛɪɚɡɨɜɚɧɢɟ
˔
ሺ
ܩ
߱ǡx
ଵǡxଶ
ሻ
ൌ߮
ሺ
x
ଵ
ɉɨɥɨɠɢɦ ɜ ɮɨɪɦɭɥɟ (3.69) x
˔
ሺ
ݐൌെͲǡxǡxሻൌȁ߮
െ݅ܩ
כ
ሻ
߮
ଵൌxଶ
ሺ
ሺ
ሻ
x
ቈ
ଶ
߱െߦ
ൌx ɢ ݐ՜െͲ, ɬɨɝɞɚ ɩɨɥɭɱɢɦ
ͳെ݊
ሻ
݅ߜ
ሺxሻ
݊
߱െߦെ݅ߜ
ଶ
ȁ
݊ǡ
Ǥ
ሺ
͵ǤͲ
ሺ
͵Ǥͳ
29
ሻ
ሻ
ሻ
ɜɵɪɚɠɟɧɢɟ ɜ ɩɪɚɜɨɣ ɱɚɫɬɢ ɟɫɬɶ ɤɨɧɰɟɧɬɪɚɰɢɹ ɷɥɟɤɬɪɨɧɨɜ ɜ ɬɨɱɤɟ ɩɪɨɫɬɪɚɧɫɬɜɚ x:
݊
ሺxሻ
ൌȁ߮
ሺxሻ
ȁ
݊ǡ
ሺ
͵Ǥʹ
ଶ
ɞɟɣɫɬɜɢɬɟɥɶɧɨ, ɩɪɨɢɧɬɟɝɪɢɪɭɟɦ ɷɬɨ ɜɵɪɚɠɟɧɢɟ ɩɨ ɜɫɟɦɭ ɩɪɨɫɬɪɚɧɫɬɜɭ
ሺxሻ
න݊
ܸ݀ൌ൬නȁ߮
ሺxሻ
ȁ
ܸ݀൰݊ൌ݊
ǡ
ሺ
͵Ǥ͵
ଶ
ɩɨɫɥɟɞɧɹɹ ɫɭɦɦɚ ɟɫɬɶ ɩɨɥɧɨɟ ɱɢɫɥɨ ɱɚɫɬɢɰ ɜ ɫɢɫɬɟɦɟ ܰ. Ɍɚɤɢɦ ɨɛɪɚɡɨɦ, ɩɨɥɭ-
ɱɚɟɦ ɜɚɠɧɨɟ ɫɨɨɬɧɨɲɟɧɢɟ
˔
Ɋɚɫɫɦɨɬɪɢɦ ɬɟɩɟɪɶ ܩ
െ݅ܩ
˔
ሺ
߱ǡxǡx
ሺ
ݐൌെͲǡxǡxሻൌ݊
ሻ
ɢ ɜɵɞɟɥɢɦ ɭ ɷɬɨɣ ɜɟɥɢɱɢɧɵ ɦɧɢɦɭɸ ɱɚɫɬɶ, ɢɫ-
ሺxሻ
Ǥ
ሺ
͵ǤͶ
ɩɨɥɶɡɭɹ ɮɨɪɦɭɥɭ
߱െߦ
ͳ
േ݅Ͳ
ൌ
࣪
߱െߦ
ט݅ߨߜሺ߱െߦ
ሻ
ǡ
ሺ
͵Ǥͷ
ɩɨɥɭɱɢɦ
ሻ
ሻ
ሻ
ሻ
˔
ሺ
ImԜܩ
߱ǡxǡxሻൌെߨȁ߮
ሺxሻ
ଶ
ሺ
ȁ
ͳെʹԜ݊
ሻߜሺ
߱െߦ
ሻ
ǡ
ɭɱɬɟɦ ɨɱɟɜɢɞɧɨɟ ɫɨɨɬɧɨɲɟɧɢɟ
ఉక
ͳെʹԜ݊
ൌͳെ
ʹ
ఉక
ೖ
݁
ͳ
ɬɨɝɞɚ (ɢɫɩɨɥɶɡɭɟɦ ɫɜɨɣɫɬɜɨ ߜ-ɮɭɧɤɰɢɢ ܨሺߦ
˔
ሺ
ImԜܩ
߱ǡxǡxሻൌെth ൬
ߚ߱
൰Ԝߨȁ߮
ʹ
ೖ
݁
ൌ
െͳ
ఉక
ೖ
݁
ͳ
ሻߜሺ
߱െߦ
ሺxሻ
ߚߦ
ൌth ൬
ሻൌܨሺ߱ሻߜሺ
ଶ
ȁ
ߜሺ߱െߦ
൰ǡ
ʹ
߱െߦ
ሻ
Ǥ
Ɍɟɩɟɪɶ ɨɛɪɚɬɢɦɫɹ ɤ ɮɨɪɦɭɥɚɦ (3.30) ɢ (3.31) ɢ ɪɚɫɫɦɨɬɪɢɦ ɪɚɡɧɨɫɬɶ
ோ
ሺ
ܩ
߱ǡxǡxሻ ɢ ܩ
ܩ
ோ
ሺ
ሺ
߱ǡxǡx
߱ǡxǡxሻെܩ
ሻ
ሺ
߱ǡxǡxሻൌȁ߮
ൌെʹ݅ߨȁ߮
ሺxሻ
ሺxሻ
ଶ
ȁ
߱െߦ
ଶ
ȁ
ߜሺ߱െߦ
ͳ
݅ߜ
ሻ
ǡ
െ
߱െߦ
ͳ
െ݅ߜ
൨ൌ
ɢɫɤɥɸɱɚɹ ɢɡ (3.78) ɢ (3.79) ɨɛɳɭɸ ɱɚɫɬɶ, ɩɨɥɭɱɢɦ
th ൬
ߚ߱
ோ
ሺ
߱ǡxǡxሻെܩ
൰ሾܩ
ʹ
ሺ
߱ǡxǡx
ሻ
ሿ
ൌʹԜ݅ԜImԜܩ
˔
ሺ
߱ǡxǡxሻǤ
30
ሻ
ሺ
ሺ
)
ሺ
ሺ
ሺ
͵Ǥ
͵Ǥ
͵Ǥͺ
͵Ǥͻ
͵ǤͺͲ
ሻ
ሻ
ሻ
ሻ
ሻ
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