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ɐሺߗሻൌ݅
ଶ
ݒ߬
݁
ʹߨ
݀ݒ
ଶ
ଶ
ሾ
ͳݒܴሺെሺݒͳሻǡλ
න
ݒ
ሻ
ሿ
ஶ
න݀
ݏߔሺݒǡݏሻǡሺͳͲǤͷ
ɡɞɟɫɶ ɤɨɧɬɭɪ ɢɧɬɟɝɪɢɪɨɜɚɧɢɹ ܥ ɟɫɬɶ ɥɢɧɢɹ ɩɚɪɚɥɥɟɥɶɧɚɹ ɦɧɢɦɨɣ ɨɫɢ, ɩɟɪɟɫɟɤɚɸɳɚɹ ɞɟɣɫɬɜɢɬɟɥɶɧɭɸ ɨɫɶ ɜ ɬɨɱɤɟ െͳȀʹ.
10.4.2. Ɋɟɲɟɧɢɟ ɭɪɚɜɧɟɧɢɹ ɞɥɹ ࡾሺ࢜ǡ࢙
ሻ
Ɋɟɲɟɧɢɟ ɭɪɚɜɧɟɧɢɟ (10.60) ɞɨɥɠɧɨ ɭɞɨɜɥɟɬɜɨɪɹɬɶ ɧɚɱɚɥɶɧɨɦɭ ɭɫɥɨɜɢɸ
ܴሺݒǡͲሻൌͲ ɢ ɞɨɥɠɧɨ ɛɵɬɶ ɪɟɝɭɥɹɪɧɵɦ ɩɪɢ ݒ՜Ͳ. ɍɪɚɜɧɟɧɢɟ ɢɦɟɟɬ ɫɜɨɛɨɞɧɵɣ
ɱɥɟɧ, ɢ ɱɬɨɛɵ ɢɡɛɚɜɢɬɶɫɹ
ɨɬ ɧɟɝɨ, ɛɭɞɟɦ ɢɫɤɚɬɶ ɪɟɲɟɧɢɟ ɜ ɜɢɞɟ
ሻ
ଶ
ሻ
݅ߚԜ
ሻ
ɢɦɟɟɬ ɜɢɞ
ɍɪɚɜɧɟɧɢɟ ɞɥɹ ݎሺݒ
ሺ
ݒͳ
ͳ
ɂɧɬɟɝɪɢɪɭɟɦ ɩɨ ݒ, ɩɨɥɭɱɚɟɦ
ܥെ
ͳ
ሺ
ݒͳ
ሻ
ɢɥɢ
ܥ
ݒሺݒͳ
ȼɵɛɟɪɟɦ ܥൌͳ, ɬɨɝɞɚ
ɉɨɥɚɝɚɟɦ
ሺݑሻ
ɞɥɹ ݎ
ɩɨɥɭɱɚɟɦ ɭɪɚɜɧɟɧɢɟ
ଵ
݅ߚԜ
ሺ
ܴሺݒǡݏሻൌܴ
݀
ݒሺݒͳሻݎሺݒሻ
݀ݒ
ݒǡݏሻݎሺݒሻǤ
ଵ
݀
݀ݒ
ݒሺݒͳ
݅ߚԜݒሺݒͳሻݎሺݒሻݒሺݒͳ
ͳ
െ
ሻ
ሺ
ݒͳ
ሺ
ݒͳ
ݒሺݒͳ
ͳ
ଶ
ሻ
ݎሺݒሻൌ
ͳ
ሻ
݅ߚԜݎሺݒሻ
ଶ
ሻ
݅ߚԜݎሺݒሻ
ͳ
ሺ
ሻ
ݒͳ
ሺݒሻ
݅ߚԜݎ
ଵ
݀ݎሺݒ
ݎ
݀ݒ
ଵ
݀ݎ
ሺݒሻ
ଵ
݀ݒ
ሻ
ሺݒሻ
ሻ
݀ݎሺݒ
݀ݒ
ൌͲǤ
ǡ
ൌͲǡ
݀
ሻ
ݎሺݒሻൌͲǤሺͳͲǤ
݀ݒ
݀
ݎሺݒሻൌͲǡ
݀ݒ
ሻ
ൌͲǤ
ሺ
ͳͲǤ
ሺ
ͳͲǤͺ
ሺ
ͳͲǤͻ
ሺ
ͳͲǤͲ
ሺ
ͳͲǤͳ
ሺ
ͳͲǤʹ
ሻ
ሻ
ሻ
ሻ
ሻ
ሻ
ሻ
ɪɟɲɟɧɢɟ ɤɨɬɨɪɨɝɨ ɧɚɯɨɞɢɬɫɹ ɫɬɚɧɞɚɪɬɧɵɦ ɨɛɪɚɡɨɦ. Ɋɟɝɭɥɹɪɧɨɟ ɪɟɲɟɧɢɟ ɩɪɢ
ݒ՜Ͳ ɦɨɠɧɨ ɜɵɛɪɚɬɶ ɜ ɜɢɞɟ
ஶ
ఉ௩
భ
݁
݀ݒ
ݎ
ଵ
ሺݒሻ
ൌ݅ߚԜන
ݒଵݒͳ
ଵ
Ǥ
ሺ
ͳͲǤ͵
111
ሻ
ሺ
ሻ
ݒǡݏ
Ɏɭɧɤɰɢɹ ܴ
μܴ
ሺ
ଵ
μݏ
ɢɡ (10.66) ɭɞɨɜɥɟɬɜɨɪɹɟɬ ɨɞɧɨɪɨɞɧɨɦɭ ɭɪɚɜɧɟɧɢɸ
ଵ
ݒǡݏ
ሻ
ൌ݅ߚ
μ
ݒሺݒͳሻܴ
μݒ
ሺ
ݒǡݏሻ
ଵ
μ
ݒሺݒͳ
μݒ
μ
ሻ
μݒ
ɢ ɧɚɱɚɥɶɧɨɦɭ ɭɫɥɨɜɢɸ, ɜɵɬɟɤɚɸɳɟɦɭ ɢɡ ɭɫɥɨɜɢɹ ܴሺݒǡͲሻൌͲ
ሺ
ݒǡݏሻǡሺͳͲǤͶ
ܴ
ଵ
ሻ
ሺ
ݒǡͲሻൌെݎሺݒሻǤ
ܴ
ଵ
ሺ
ͳͲǤͷ
Ɋɟɲɢɦ ɫɧɚɱɚɥɚ ɭɪɚɜɧɟɧɢɟ (10.74) ɩɪɢ ߚൌͲ
ሺ
ሻ
μܴ
ݒǡݏ
ଵ
μݏ
ൌ
μ
ݒሺݒͳ
μݒ
μ
ሺ
ሻ
μݒ
ݒǡ݈ሻǡ
ܴ
ଵ
ሺ
ͳͲǤ
ɫɞɟɥɚɟɦ ɡɚɦɟɧɭ ɩɟɪɟɦɟɧɧɨɣ ݒൌሺݑെͳሻȀʹ, ɬɨɝɞɚ
μܴ
ሺ
ሻ
ݑǡݏ
ଵ
μݏ
ൌʹݑ
μ
μݑ
ሺ
ݑǡ݈ሻሺݑ
ܴ
ଵ
ଶ
െͳ
ଶ
μ
μݑ
ሺ
ݒǡ݈ሻǡ
ܴ
ଵ
ଶ
ሻ
ሺ
ͳͲǤ
ɷɬɨ ɜ ɬɨɱɧɨɫɬɢ ɭɪɚɜɧɟɧɢɟ (9.56) ɞɥɹ ɮɭɧɤɰɢɢ ɪɚɫɩɪɟɞɟɥɟɧɢɹ ɫɥɭɱɚɣɧɨɝɨ ɛɥɭɠɞɚɧɢɹ ɧɚ ɩɥɨɫɤɨɫɬɢ Ʌɨɛɚɱɟɜɫɤɨɝɨ. Ɏɭɧɤɰɢɹ Ƚɪɢɧɚ ɷɬɨɝɨ ɭɪɚɜɧɟɧɢɹ ɢɦɟɟɬ ɜɢɞ
ஶ
௦
ܩ
ି
ሻ
ൌ݁
ସ
න݀
ߣԜߣԜሺߨߣሻԜ݁
ሺ
ݑǡݑ
௦
ଵ
ିఒమԜ௦
Ԝܲ
ିଵȀଶାఒ
ሺݑሻ
Ԝܲ
ିଵȀଶାఒ
ሺ
ሻǡሺ
ݑ
ͳͲǤͺ
ଵ
ɬɨɝɞɚ (ɩɪɢ ߚൌͲ)
ஶ
ൌെ݁
௦
ି
ସ
ሺ
ݑǡݏሻൌെන݀
ܴ
ଵ
ஶ
න݀
ߣԜߣԜሺߨߣሻԜ݁
ଵ
ିఒమԜ௦
ݑଵܩ
ܲ
ሺ
ݑǡݑ
ିଵȀଶାఒ
ଵ
ሺݑሻ
ሻ
ሾ
ݎ
ሺ
ݑ
െͳሻȀʹሿൌ
ଵ
ஶ
Ԝන ݀
ଵ
ሺ
ሻ
ʹܲ
ିଵȀଶାఒ
ݑ
ଵ
ሺ
ݑ
ଵ
ǡሺͳͲǤͻ
ሻ
ݑ
ͳ
ଵ
ɩɪɢɧɢɦɚɹ ɜɨ ɜɧɢɦɚɧɢɟ (9.72), ɩɨɥɭɱɢɦ
ஶ
௦
ି
ܴ
ሺ
ݑǡݏሻൌെɎ݁
ଵ
ସ
න݀
ߣԜߣԜ
Ɍɚɤɢɦ ɨɛɪɚɡɨɦ, ɦɵ ɩɨɥɭɱɢɥɢ, ɱɬɨ ɩɪɢ ݏ՜λ ɮɭɧɤɰɢɹ ܴ
ሺߨߣ
ଶ
ሺߨߣሻ
ሻ
Ԝ݁
ିఒమ௦
ሺݑሻ
ܲ
ଵ
ି
ାఒ
ଶ
ሺ
ݑǡݏ
ଵ
ሺ
Ǥ
ሻ
ͳͲǤͺͲ
ɷɤɫɩɨɧɟɧɰɢ-
ɚɥɶɧɨ ɫɬɪɟɦɢɬɫɹ ɤ ɧɭɥɸ. ɗɬɨɬ ɜɵɜɨɞ ɨɫɬɚɟɬɫɹ ɜ ɫɢɥɟ ɢ ɩɪɢ ߚ്Ͳ, ɜ ɱɟɦ ɦɨɠɧɨ
ɭɛɟɞɢɬɶɫɹ, ɫɱɢɬɚɹ ɫɥɚɝɚɟɦɵɟ ɫ ߚ ɜ ɭɪɚɜɧɟɧɢɢ (10.60) ɢ ɜ ɧɚɱɚɥɶɧɨɦ ɭɫɥɨɜɢɢ
(10.71), (10.73) ɜɨɡɦɭɳɟɧɢɟɦ ɢ ɜɵɱɢɫɥɹɹ ɩɨɩɪɚɜɤɢ ɩɨ ɷɬɨɦɭ ɜɨɡɦɭɳɟɧɢɸ. Ɍɚɤɢɦ
ɨɛɪɚɡɨɦ, ɦɵ ɩɨɥɭɱɚɟɦ
ሺ
ݒͳ
ͳ
ɝɞɟ ݎ
ܴሺݒǡλሻൌݎሺݒሻൌ
ሺݒሻ
ɞɚɟɬɫɹ (10.73). ɉɪɢɧɢɦɚɹ ɜɨ ɜɧɢɦɚɧɢɟ ɮɨɪɦɭɥɭ ɞɥɹ ɩɪɨɜɨɞɢɦɨɫɬɢ
ଵ
(10.65), ɦɵ ɜɢɞɢɦ, ɱɬɨ ɜ ɧɟɟ ɜɯɨɞɢɬ ܴሾെሺݒͳሻǡλ
ሺݒሻ
ሻ
ݎ
ǡ
ଵ
ሿ
, ɞɥɹ ɤɨɬɨɪɨɣ ɩɨɥɭɱɚɟɦ
ሺ
ͳͲǤͺͳ
ሻ
ሻ
ሻ
ሻ
ሻ
ሻ
ሻ
112
ܴሾെሺݒͳሻǡλሿൌݎ൫െሺݒͳሻ൯ൌെ
ɩɨɞɫɬɚɜɥɹɹ ɷɬɨ ɜ (10.65), ɩɨɥɭɱɢɦ
ଶ
݁
ݒ߬
ߪሺߗሻൌ݅
ʹߨ
ଶ
ଶ
නԜ݀ݒԜݎ
Ⱦɚɥɟɟ ɩɨɞɫɬɚɜɥɹɟɦ (10.73) ɜ ɷɬɭ ɮɨɪɦɭɥɭ
ଶ
ݒ߬ଶߚ
ߪሺߗሻൌെ
݁
ʹߨ
ଶ
නԜ݀ݒԜන
ሾ
െሺݒͳ
ଵ
ஶ
݁
ሻ
ఉ௩
భ
݀ݒ
ݒଵെݒ
ͳ
ݎ
ݒ
ஶ
ሿ
න݀
ଵ
න݀
ሾ
െሺݒͳ
ଵ
ݏԜߔሺݒǡݏሻǤ
ஶ
ݏԜߔሺݒǡݏሻǤ
ሻ
ሺ
ሿ
ǡ
ሺ
ሺ
ͳͲǤͺʹ
ͳͲǤͺ͵
ͳͲǤͺͶ
ሻ
ሻ
ሻ
Ɏɭɧɤɰɢɹ ߔሺݒǡݏ
ሻ
ܥሺݔǡݏ
ɚɧɚɥɢɬɢɱɧɚ ɜ ɤɪɭɝɟ
ɫ ɩɨɦɨɳɶɸ ɜɵɱɟɬɚ ɜ ɩɨɥɸɫɟ ݒൌݒ
ɤɚɤ ɦɵ ɜɢɞɢɦ ɜ ɷɬɭ ɮɨɪɦɭɥɭ ɜɯɨɞɢɬ ɬɨɥɶɤɨ ɢɧɬɟɝɪɚɥ ɩɨ ݏ ɨɬ ߔሺݒǡݏ
ሻ
ɚɧɚɥɢɬɢɱɧɚ ɜ ɩɪɚɜɨɣ ɩɨɥɭɩɥɨɫɤɨɫɬɢ, ɬɚɤ ɤɚɤ ɮɭɧɤɰɢɹ
ȁݔȁ
൏ͳ, ɩɨɷɬɨɦɭ ɢɧɬɟɝɪɚɥ ɩɨ ɤɨɧɬɭɪɭ ܥ ɜɵɱɢɫɥɹɟɬɫɹ
ɜ ɢɬɨɝɟ (ɩɨɫɥɟ ɡɚɦɟɧɵ ݒଵ՜ݒ)
ଵ
ߪሺߗሻൌെ݅
݁
ଶ
ݒ߬ଶߚ
ߨ
ஶ
න݀
ݒ݁
ఉ௩
ஶ
න݀
ݏߔሺݒǡݏሻǡ
ሺ
ሻ
, ɩɨɷɬɨɦɭ
ͳͲǤͺͷ
ɨɛɨɡɧɚɱɢɦ
ஶ
݂ሺݒሻൌන݀
ݏԜߔሺݒǡݏሻǡ
ሺ
ͳͲǤͺ
ɬɨɝɞɚ
ஶ
න݀
ݒ݁
ఉ௩
Ԝ݂ሺݒሻǡ
ሺ
ͳͲǤͺ
ߪሺߗሻൌെ݅
ଶ
݁
ݒ߬ଶߚ
ߨ
ɩɪɟɨɛɪɚɡɭɟɦ ɬɟɩɟɪɶ ɷɬɨɬ ɢɧɬɟɝɪɚɥ ɩɨ ɱɚɫɬɹɦ, ɢɫɩɨɥɶɡɭɹ ɬɨɬ ɮɚɤɬ, ɱɬɨ ݂ሺλሻൌͲ,
ɬɨɝɞɚ
ߪሺߗሻൌെ
ଶ
݁
ݒ߬
ߨ
ଶ
න݀
ஶ
ݒԜ൫ͳെ݁
ఉ௩
൯Ԝ
݂݀ሺݒ
݀ݒ
ሻ
ǡ
ሺ
ͳͲǤͺͺ
ɩɟɪɟɣɞɟɦ ɤ ɩɟɪɟɦɟɧɧɨɣ ߟൌߚݒ ɢ ɩɪɟɞɫɬɚɜɢɦ ɩɪɨɢɡɜɨɞɧɭɸ ݂݀Ȁ݀ߟ ɜ ɜɢɞɟ
݂݀
݀ߟ
ൌߚܨ
ଵ
ሺߟሻ
ߚ
ଶ
ሺߟሻ
ǡ
ܨ
ଶ
ሺ
ͳͲǤͺͻ
ሻ
ሻ
ሻ
ሻ
ሻ
ɬɨɝɞɚ ɩɨɥɭɱɢɦ
ߪሺߗሻൌെ
ݒ߬
ߨ
ଶ
න݀
ஶ
ߟԜ൫ͳെ݁ఎ൯Ԝ൫ߚܨ
ଵ
ሺߟሻ
ߚ
ଶ
ሺߟሻ
൯Ǥ
ܨ
ଶ
ሺ
ͳͲǤͻͲ
ሻ
ଶ
݁
113
10.4.3. Ɋɟɲɟɧɢɟ ɭɪɚɜɧɟɧɢɹ ɞɥɹ ሺܞǡܛ
ሻ
.
ȼɵɱɢɫɥɟɧɢɟ ɩɪɨɜɨɞɢɦɨɫɬɢ
Ɂɚɣɦɟɦɫɹ ɪɟɲɟɧɢɟɦ ɭɪɚɜɧɟɧɢɹ (10.61) ɧɚ ɮɭɧɤɰɢɸ ߔሺݒǡݏ
ሻ
, ɩɪɢ ɷɬɨɦ ɦɵ ɛɭ-
ɞɟɦ ɪɚɫɫɦɚɬɪɢɜɚɬɶ ɫɥɭɱɚɣ ɦɚɥɵɯ ɱɚɫɬɨɬ ߚاͳ. Ƚɪɚɧɢɱɧɨɟ ɭɫɥɨɜɢɟ ɞɥɹ ɪɟɲɟɧɢɹ
ɩɨɥɭɱɚɟɬɫɹ ɢɡ (10.57) ɢ ɢɦɟɟɬ ɜɢɞ
ஶ
ఉ௩
భ
݁
݀ݒ
ߔሺݒǡͲሻൌെݎ
ଵ
ሺݒሻ
ൌെ݅ߚԜන
ݒଵݒͳ
ଵ
Ǥ
ሺ
ͳͲǤͻͳ
ɉɪɨɢɡɜɟɞɟɦ ɧɚɞ ɭɪɚɜɧɟɧɢɟɦ (10.61) ɩɪɟɨɛɪɚɡɨɜɚɧɢɹ Ʌɚɩɥɚɫɚ ɨɬɧɨɫɢɬɟɥɶɧɨ
ɩɟɪɟɦɟɧɧɨɣ ݏ, ɞɥɹ ɷɬɨɝɨ ɜɜɟɞɟɦ ɮɭɧɤɰɢɸ
ஶ
݀
݀ݒ
ିச௦
Ԝߔሺݒǡݏሻ݀ݏǡ
μ
ሻ
μݒ
ݒሺݒͳ
෩
ߔ
ሻ
ሺ
݂݀
ሻ
݀ݒ
μ
ඥ
μݒ
ݒǡߢሻǤ
෩
ିச௦
ݒሺݒͳ
ሺ
ݒǡߢൌͲ
ൌ݅ߚԜන
, ɩɨɥɭɱɢɦ
෩
ሺ
ሻ
ݒǡߢሻ
Ԝߔ
ሻ
, ɤɨɬɨɪɚɹ ɫɨɝɥɚɫɧɨ
ஶ
ఉ௩
భ
݁
݀ݒ
ଵ
ݒଵݒͳ
ǤሺͳͲǤͻͶ
ሺ
ͳͲǤͻʹ
ሺ
ͳͲǤͻ͵
෩
ሺ
ߔ
ݒǡߢሻൌන݁
ɢ ɩɪɨɢɧɬɟɝɪɢɪɭɟɦ (10.61), ɭɦɧɨɠɢɜ ɩɪɟɞɜɚɪɢɬɟɥɶɧɨ ɧɚ ݁
෩
ሺ
െߔሺݒǡͲሻݏԜߔ
ݒǡߢሻൌ݅ߚඥݒሺݒͳ
μ
ݒሺݒͳ
μݒ
Ɏɚɤɬɢɱɟɫɤɢ ɧɚɫ ɢɧɬɟɪɟɫɭɟɬ ɬɨɥɶɤɨ ɮɭɧɤɰɢɹ ߔ
(10.92) ɢ (10.86), ɪɚɜɧɚ ݂ሺݒ
݅ߚඥݒሺݒͳ
݀
ሻ
ቀඥݒሺݒͳሻ݂ቁ
݀ݒ
ሻ
, ɩɨɷɬɨɦɭ
Ɇɵ ɪɚɫɫɦɨɬɪɢɦ ɞɜɟ ɨɛɥɚɫɬɢ ɢɡɦɟɧɟɧɢɹ ɩɟɪɟɦɟɧɧɨɣ: ݒ൏ܸ ɢ ݒܸ, ɝɞɟ
ͳاܸاͳȀߚ. ȼ ɩɟɪɜɨɣ ɨɛɥɚɫɬɢ ɦɵ ɦɨɠɟɦ ɩɪɟɧɟɛɪɟɱɶ ɜ ɭɪɚɜɧɟɧɢɢ (10.94) ɩɟɪ-
ɜɵɦ ɱɥɟɧɨɦ, ɢ ɭɪɚɜɧɟɧɢɟ ɩɪɢɦɟɬ ɜɢɞ
ஶ
ఉ௩
భ
݁
݀
ݒሺݒͳ
݀ݒ
ሻ
݂݀
݀ݒ
ൌ݅ߚԜන
݀ݒ
ଵ
ݒଵݒͳ
Ǥ
ሺ
ͳͲǤͻͷ
ɂɧɬɟɝɪɚɥ ɜ ɩɪɚɜɨɣ ɱɚɫɬɢ ɷɬɨɝɨ ɭɪɚɜɧɟɧɢɹ ɩɪɟɨɛɪɚɡɨɜɚɧɢɟɦ ɩɨ ɱɚɫɬɹɦ ɢ ɧɟɤɨ-
ɬɨɪɵɦɢ ɦɚɧɢɩɭɥɹɰɢɹɦɢ ɫ ɞɪɨɛɹɦɢ ɩɪɟɞɫɬɚɜɢɦ ɜ ɜɢɞɟ
െ
ͳ
ݒͳ
ஶ
݀
ԜݒԜන
݀ݒ
ሺ
ఉ௩
భ
݁
݀ݒ
ݒ
ͳሻሺݒଵݒͳ
ଵ
ଵ
ǡ
ሻ
ሺ
ͳͲǤͻ
ɬɨɝɞɚ ɞɥɹ ɪɟɲɟɧɢɹ ɩɨɥɭɱɚɟɦ ɜɵɪɚɠɟɧɢɟ, ɪɟɝɭɥɹɪɧɨɟ ɩɪɢ ݒ՜Ͳ ɢ ɫɬɪɟɦɹɳɟɟɫɹ ɤ
ɧɭɥɸ ɩɪɢ ݒ՜λ
݂݀
݀ݒ
ൌെ
ሺݒͳ
ݒሺݒͳ
ሻ
ሻ
ሺ
ݒͳ
ஶ
ͳ
න
ሺ
ሻ
ఉ௩
భ
݁
݀ݒ
ݒ
ͳሻሺݒଵݒͳ
ଵ
ଵ
ሻ
ሺ
Ǥ
ͳͲǤͻ
ሻ
ሻ
ሻ
ሻ
ሻ
ሻ
ሻ
114
ɉɨɫɤɨɥɶɤɭ ɩɪɢ ߚൌͲ ɩɪɚɜɚɹ ɱɚɫɬɶ ɪɚɜɧɚ ɧɭɥɸ, ɦɨɠɧɨ ɜɵɪɚɡɢɬɶ ɩɟɪɜɨɟ ɫɥɚ-
ɝɚɟɦɨɟ ɜ ɩɪɚɜɨɣ ɱɚɫɬɢ ɱɟɪɟɡ ɢɧɬɟɝɪɚɥ ɩɪɢ ߚൌͲ, ɩɨɥɭɱɢɦ
݂݀
݀ݒ
ൌ
ͳ
ሺ
ݒͳ
ஶ
න
ሻ
ఉ௩
భ
൫݁
െͳ൯݀ݒ
ሺ
ݒ
ͳሻሺݒଵݒͳ
ଵ
ଵ
Ǥ
ሻ
ሺ
ͳͲǤͻͺ
ɉɪɢɧɢɦɚɹ ɜɨ ɜɧɢɦɚɧɢɟ
ͳ
ሺ
ͳሻሺݒଵݒͳ
ݒ
ଵ
ൌ
ሻ
ͳ
ݒ
ݒ
ͳ
ଵ
െ
ݒ
ݒͳ
ଵ
ͳ
൨ൌെ
ͳ
ݒ
݀ݒ
ݒ
݀
ଵ
ݒͳ
ଵ
ͳ
ݒ
ଵ
൨ሺͳͲǤͻͻ
ͳ
ɩɪɟɨɛɪɚɡɭɟɦ ɢɧɬɟɝɪɚɥ ɜ (10.98) ɩɨ ɱɚɫɬɹɦ (ɜɧɟ ɢɧɬɟɝɪɚɥɶɧɨɟ ɫɥɚɝɚɟɦɨɟ ɪɚɜɧɨ
ɧɭɥɸ):
݅ߚ
ஶ
න݀
ሻ
ଶ
ߞԜ
ஶ
ஶ
න݀
ሻ
݁
ߞߚ
ݒଵ݁
ߞԜ݁Ԝ
ൌ
ఉ௩
݅ߚ
భ
ଶԜ݁ିఉ
ሺ
ߟߚ
ݒ
ݒͳ
ଵ
ݒ
ߞߟߚ
ߞߚ
න
ሻ
൨Ǥ
ͳ
ଵ
, ɩɪɢ ɷɬɨɦ ߚ൏ߟ൏ߟ
ଵ
൨ൎ
ஶ
݀ݔ
݁௫Ǥ
ݔ
ఉ
ሺ
ͳͲǤͳͲͲ
ሺ
ͳͲǤͳͲͳ
ൌ
ݒሺݒͳ
݅ߚ
݂݀
݀ݒ
ȼɜɟɞɟɦ ɬɟɩɟɪɶ ɩɟɪɟɦɟɧɧɵɟ ߟൌߚݒ, ߞൌߚݒ
ߚاߟ
اͳ ɜ ɷɬɢɯ ɩɟɪɟɦɟɧɧɵɯ
݂݀
ൌ
ߚԜ
݀ߟ
ߟሺߟߚ
ଶ
ൎ
݅ߚ
ሺ
ߟߚ
ሻ
න݀
ɂɦɟɹ ɜ ɜɢɞɭ (10.89), ɩɨɥɭɱɢɦ ɪɚɜɟɧɫɬɜɨ
ିఉ
ߚܨ
ൎቆ
ଵ
ሺߟሻ
ߚ
ଶ
ܨ
ଶ
݅ߚ
ߟ
݅ߚԜ݁
ሺߟሻ
ൌ
ሺ
ߟߚ
ଶ
݅ߚ
െ
ቇሾെܿ݅ሺߚሻെ݅ݏ݅ሺߚ
ଶ
ߟ
ሾ
െܿ݅ሺߚሻെ݅ݏ݅ሺߚ
ሻ
ሻ
ሿ
ൎ
ሻ
ሿ
ǡ
ሺ
ͳͲǤͳͲʹ
ሻ
ሻ
ሻ
,
ሻ
ሻ
ɝɞɟ
ܿ݅ሺݔሻൌെන
௫
ஶ
ሻ
ሺݐ
Ԝ݀ݐǡ᩷᩷ݏ݅ሺݔሻൌെන
ݐ
௫
ஶ
ሺݐ
ݐ
ሻ
Ԝ݀ݐǡ
ሺ
ͳͲǤͳͲ͵
ɢɧɬɟɝɪɚɥɶɧɵɟ ɤɨɫɢɧɭɫ ɢ ɫɢɧɭɫ. Ⱦɥɹ ɧɚɫ ɜɚɠɧɵ ɫɥɟɞɭɸɳɢɟ ɩɪɟɞɟɥɶɧɵɟ ɡɧɚɱɟɧɢɹ
ܿ݅ሺݔሻൎሺߛݔሻǡ᩷ݏ݅ሺݔሻൎെ
ɝɞɟ ߛൌ݁
, ܥൌͲǡͷʹͳͷͶͻǥ – ɩɨɫɬɨɹɧɧɚɹ ɗɣɥɟɪɚ. Ɍɨɝɞɚ ɢɡ (10.102) ɩɨɥɭ-
ɱɚɟɦ, ɱɬɨ ɜ ɨɛɥɚɫɬɢ ߚ൏ߟ൏ߟ
ܨ
ߨ
ǡ᩷ɩɪɢ᩷ݔ՜Ͳǡ
ʹ
ɢɦɟɸɬ ɦɟɫɬɨ ɪɚɜɟɧɫɬɜɚ
݅
ͳ
ଵ
ሺߟሻ
ൌ
൬
ߟ
ߛߚ
݅
ߨ
ʹ
൰ǡ
ሺ
ͳͲǤͳͲͶ
ሺ
ͳͲǤͳͲͷ
115
ሻ
ሻ
ሻ
݅
ͳ
ሺߟሻ
ൌെ
ܨ
ଶ
൬
ଶ
ߟ
Ɉɠɢɞɚɟɦɵɣ ɝɥɚɜɧɵɣ ɱɥɟɧ ɜ (10.90) ɛɭɞɟɬ ɩɨɪɹɞɤɚ ߚ
ߛߚ
݅
ߨ
൰Ǥ
ʹ
ଶଶ
ߚ, ɞɥɹ ɟɝɨ ɨɩɪɟɞɟɥɟ-
ሺ
ͳͲǤͳͲ
ɧɢɹ ɫ ɥɨɝɚɪɢɮɦɢɱɟɫɤɨɣ ɬɨɱɧɨɫɬɶɸ ɞɨɫɬɚɬɨɱɧɨ ɜɵɪɚɠɟɧɢɹ (10.106) ɢ ɧɟɬ ɧɟɨɛɯɨ-
ሺߟሻ
ɞɢɦɨɫɬɢ ɧɚɯɨɞɢɬɶ ɜɵɪɚɠɟɧɢɟ ɞɥɹ ܨ
݁
ଶ
൬
݁
ൌ
ߪሺߗሻൌ݅
ଶ
ݒ߬ଶߚ
ߨ
ɜ ɨɛɥɚɫɬɢ ߟͳ
ଶ
ଶ
ଶ
ݒ߬ଶߚ
ߨ
ଵ
ͳ
݀ߟ
൰න
ߛߚ
ఉ
ஶ
න݀
ൎ
ߟ
ߟԜߟԜܨ
ଶ
݁
ݒ߬ଶߚ
ሺߟሻ
ൌ
ଶ
ଶ
ͳ
ଶ
ߨ
ǡ
ߚ
ሺ
ͳͲǤͳͲ
ሻ
ሻ
ɩɨɞɫɬɚɜɢɜ ߚൌʹߗ߬
, ɩɨɥɭɱɢɦ ɨɤɨɧɱɚɬɟɥɶɧɨ
ଶ
Ͷ
ɐሺߗሻൌ
݁ଶݒ߬
ߨ
ଶ
ଶଶ
ሺ
ߗ߬
ሻ
ߗ߬ଶǤ
ଶ
ሺ
ͳͲǤͳͲͺ
Ɍɚɤɢɦ ɨɛɪɚɡɨɦ, ɦɵ ɜ ɬɨɱɧɨɫɬɢ ɜɨɫɩɪɨɢɡɜɟɥɢ ɮɨɪɦɭɥɭ Ɇɨɬɬɚ (10.8), ɭɫɬɚɧɨ-
ɜɢɜ ɩɪɢ ɷɬɨɦ ɱɢɫɥɟɧɧɵɣ ɤɨɷɮɮɢɰɢɟɧɬ.
Ʌɢɬɟɪɚɬɭɪɚ
Ɋɟɡɭɥɶɬɚɬɵ ɷɬɨɣ ɝɥɚɜɵ ɛɵɥɢ ɩɨɥɭɱɟɧɵ ɜ ɪɚɛɨɬɟ Ⱥ. Ⱥ. Ⱥɛɪɢɤɨɫɨɜɚ ɢ ɂ. Ⱥ. Ɋɵɠ-
ɤɢɧɚ // ɀɗɌɎ. 1976. Ɍ. 71. ɋ. 12041224. Ɉɧɢ ɬɚɤ ɠɟ ɢɡɥɨɠɟɧɵ ɜ ɨɛɡɨɪɟ [7].
Ʉɨɧɬɪɨɥɶɧɵɟ ɜɨɩɪɨɫɵ
1. Ʉɚɤɨɜ ɦɟɯɚɧɢɡɦ ɩɪɨɜɨɞɢɦɨɫɬɶ ɥɨɤɚɥɢɡɨɜɚɧɧɵɯ ɷɥɟɤɬɪɨɧɨɜ ɜ ɩɟɪɟɦɟɧɧɨɦ
ɩɨɥɟ?
2. ɑɟɦɭ ɪɚɜɧɚ ɷɧɟɪɝɢɹ, ɩɨɝɥɨɳɚɟɦɚɹ ɜ ɟɞɢɧɢɰɭ ɜɪɟɦɟɧɢ ɧɚ ɟɞɢɧɢɰɟ ɞɥɢɧɵ
ɨɛɪɚɡɰɚ ɜ ɩɟɪɟɦɟɧɧɨɦ ɷɥɟɤɬɪɢɱɟɫɤɨɦ ɩɨɥɟ ɫ ɱɚɫɬɨɬɨɣ ȳ?
3. ɉɪɨɞɟɥɚɬɶ ɜɵɜɨɞ ɭɪɚɜɧɟɧɢɹ (10.55) ɞɥɹ ɩɪɨɢɡɜɨɞɹɳɟɣ ɮɭɧɤɰɢɢ ܤሺݔǡݐ
ɪɟɤɭɪɪɟɧɬɧɵɯ ɫɨɨɬɧɨɲɟɧɢɣ (10.37).
ሻ
4. ɑɟɦɭ ɪɚɜɧɨ ɩɪɟɞɟɥɶɧɨɟ ɡɧɚɱɟɧɢɟ ɩɪɨɜɨɞɢɦɨɫɬɢ ɐሺߗ
ɫɬɪɟɦɥɟɧɢɢ ɱɚ-
ɩɪɢ
ɫɬɨɬɵ ߗ ɤ ɧɭɥɸ?
5. Ʉɚɤɨɜɚ ɮɢɡɢɱɟɫɤɚɹ ɩɪɢɱɢɧɚ ɩɨɹɜɥɟɧɢɹ ɦɧɨɠɢɬɟɥɹ
ଶ
ߗ߬
ɜ ɜɵɪɚɠɟɧɢɢ ɞɥɹ
ଶ
ɩɪɨɜɨɞɢɦɨɫɬɢ (10.108)?
ሻ
ɢɡ
ሻ
116
Ƚɥɚɜɚ 11
ɉȿɊȿɏɈȾ ɉȺɃȿɊɅɋȺ ɆȿɌȺɅɅ
ȾɂɗɅȿɄɌɊɂɄ
ȼ ɈȾɇɈɆȿɊɇɈɃ ɎȿɊɆɂ-ɋɂɋɌȿɆȿ
ɉɟɪɟɯɨɞɨɦ ɉɚɣɟɪɥɫɚ ɧɚɡɵɜɚɸɬ ɫɬɪɭɤɬɭɪɧɵɣ ɮɚɡɨɜɵɣ ɩɟɪɟɯɨɞ ɦɟɬɚɥɥ
ɞɢɷɥɟɤɬɪɢɤ ɜ ɨɞɧɨɦɟɪɧɵɯ ɫɨɟɞɢɧɟɧɢɹɯ, ɩɪɢ ɤɨɬɨɪɨɦ ɮɨɪɦɢɪɭɟɬɫɹ ɩɟɪɢɨɞɢɱɟ-
ɫɤɢɟ ɜ ɩɪɨɫɬɪɚɧɫɬɜɟ ɫɦɟɳɟɧɢɹ ɢɨɧɨɜ ɢɡ ɢɯ ɩɨɥɨɠɟɧɢɹ ɪɚɜɧɨɜɟɫɢɹ ɜ ɦɟɬɚɥɥɢɱɟɫɤɨɣ ɮɚɡɟ. ȼ ɪɟɡɭɥɶɬɚɬɟ ɷɬɢɯ ɫɦɟɳɟɧɢɣ ɩɪɨɢɫɯɨɞɢɬ ɭɞɜɨɟɧɢɟ ɩɟɪɢɨɞɚ ɪɟɲɟɬɤɢ ɢ
ɩɟɪɟɪɚɫɩɪɟɞɟɥɟɧɢɟ ɷɥɟɤɬɪɨɧɧɨɣ ɩɥɨɬɧɨɫɬɢ, ɤɨɬɨɪɨɟ ɩɪɢɜɨɞɢɬ ɤ ɜɨɡɧɢɤɧɨɜɟɧɢɸ
ɬɚɤ ɧɚɡɵɜɚɟɦɨɣ «ɜɨɥɧɵ ɡɚɪɹɞɨɜɨɣ ɩɥɨɬɧɨɫɬɢ».
11.1. Ɉɞɧɨɦɟɪɧɵɣ ɷɥɟɤɬɪɨɧɧɵɣ ɝɚɡ
ɜ ɩɨɥɟ ɞɟɮɨɪɦɚɰɢɢ ɪɟɲɟɬɤɢ.
ɍɫɥɨɜɢɟ ɪɚɜɧɨɜɟɫɢɹ
Ɋɚɫɫɦɨɬɪɢɦ ɨɞɧɨɦɟɪɧɵɣ ɜɵɪɨɠɞɟɧɧɵɣ ɷɥɟɤɬɪɨɧɧɵɣ ɝɚɡ ɫ ɢɦɩɭɥɶɫɨɦ Ɏɟɪɦɢ
. Ɋɚɫɫɦɨɬɪɢɦ ɜɡɚɢɦɨɞɟɣɫɬɜɢɟ ɷɥɟɤɬɪɨɧɨɜ ɫ ɮɨɧɨɧɧɨɣ ɦɨɞɨɣ, ɢɦɟɸɳɟɣ ɜɨɥɧɨ-
ɜɨɣ ɜɟɤɬɨɪ ʹ
ɝɞɟ
ɤɨɦɩɥɟɤɫɧɚɹ ɜɟɥɢɱɢɧɚ, ɡɚɞɚɸɳɚɹ ɤɚɤ ɚɦɩɥɢɬɭɞɭ, ɬɚɤ ɢ ɮɚɡɭ ɮɨɧɨɧɧɨɣ ɦɨɞɵ.
ȼ ɩɪɨɰɟɫɫɟ ɷɬɨɝɨ ɜɡɚɢɦɨɞɟɣɫɬɜɢɹ ɷɥɟɤɬɪɨɧ, ɢɦɟɸɳɢɣ ɢɦɩɭɥɶɫ ɜ ɨɤɪɟɫɬɧɨ-
ɫɬɢ ɬɨɱɤɢ
Ⱥɦɩɥɢɬɭɞɚ ɷɬɨɝɨ ɩɪɨɰɟɫɫɚ ɜ ɬɟɪɦɢɧɚɯ ɨɩɟɪɚɬɨɪɨɜ (8.9), (8.10) ɢɦɟɟɬ ɜɢɞ
ሺݖሻ
ɝɞɟ ߰
ɪɚɬɧɵɣ ɩɪɨɰɟɫɫ ɩɟɪɟɯɨɞɚ ɷɥɟɤɬɪɨɧɚ ɢɡ ɨɤɪɟɫɬɧɨɫɬɢ ɬɨɱɤɢ െ
ɨɩɢɫɵɜɚɟɬɫɹ ɚɦɩɥɢɬɭɞɨɣ
ݑሺݖሻൌȟሺݖሻ݁
ȟሺݖሻൌȟ
, ɛɭɞɟɬ ɪɚɫɫɟɢɜɚɬɶɫɹ ɮɨɧɨɧɧɨɣ ɦɨɞɨɣ ɜ ɨɤɪɟɫɬɧɨɫɬɶ ɬɨɱɤɢ െ
ା
כ
ሺݖሻ
ሺݖሻ
ȟ
߰
ଶ
ା
ሺݖሻ
ɢ ߰
ɬɟ ɠɟ ɞɜɭɯɤɨɦɩɨɧɟɧɬɧɵɟ ɨɩɟɪɚɬɨɪɵ, ɱɬɨ ɢ ɜ (8.12, 8.13). Ɉɛ-
ା
כ
ሺݖሻ
ሺݖሻ
ȟ
߰
ଵ
ଵ
߰
߰
ሺݖሻ
ଵ
ଶ
ሺݖሻ
ሺݖሻ
ଶబ௭ȟכ
ሺݖሻ
݅ȟ
ଶ
ା
ሺݖሻ
ൌ߰
ା
ሺݖሻ൫ȟሺݖሻ
ൌ߰
ሺݖሻ
ൌȁȟሺݖ
כ
ሺݖሻ
൫ȟ
݁
ߪො
ߪො
ିଶబ௭
ఞሺ௭
ሻ
ȁ
݁
ሺିሻ
൯߰
ሺାሻ
൯߰
ǡ
ሻ
ሺݖሻ
ǡ
ɜ ɨɤɪɟɫɬɧɨɫɬɶ
ሺݖሻ
Ǥ
ሺ
ͳͳǤͳ
ሺ
ͳͳǤʹ
.
ሺ
ͳͳǤ͵
ሺ
ͳͳǤͶ
ሻ
ሻ
ሻ
ሻ
ɍɱɢɬɵɜɚɹ, ɱɬɨ ɝɚɦɢɥɶɬɨɧɢɚɧ ɫɜɨɛɨɞɧɵɯ ɷɥɟɤɬɪɨɧɨɜ ɞɚɟɬɫɹ ɮɨɪɦɭɥɨɣ (8.14),
ɝɚɦɢɥɶɬɨɧɢɚɧ ɷɥɟɤɬɪɨɧɨɜ, ɜɡɚɢɦɨɞɟɣɫɬɜɭɸɳɢɯ ɫ ɮɨɧɨɧɧɨɣ ɦɨɞɨɣ (11.1), ɦɨɠɧɨ
ɡɚɩɢɫɚɬɶ ɜ ɫɥɟɞɭɸɳɟɦ ɜɢɞɟ
ܪൌන߰
ା
ሺݖሻ
൬െ݅ݒ
ߪොଷ
μ
ȟሺݖሻԜߪො
μݖ
ሺାሻ
ȟ
ሺିሻ
כ
ሺݖሻ
൰߰
ሺݖሻ
൨݀ݖǤ
Ԝߪො
ሺ
ͳͳǤͷ
117
ሻ
Ɇɵ ɜɤɥɸɱɢɥɢ ɞɥɹ ɭɩɪɨɳɟɧɢɹ ɮɨɪɦɭɥ ɤɨɧɫɬɚɧɬɭ ɷɥɟɤɬɪɨɧ-ɮɨɧɨɧɧɨɝɨ ɜɡɚɢ-
ɦɨɞɟɣɫɬɜɢɹ ݃ ɜ ɜɟɥɢɱɢɧɭ ȟ, ɩɪɢ ɷɬɨɦ, ɨɧɚ ɜɨɡɧɢɤɧɟɬ ɜ ɩɥɨɬɧɨɫɬɢ ɷɧɟɪɝɢɢ ɮɨɧɨɧ-
ɧɨɣ ɦɨɞɵ:
ଶ
ൌන
ȁȟȁ
݃
݀ݖǤ
ଶ
ሺ
ͳͳǤ
ሺȟሻ
ܧ
ɋɜɨɛɨɞɧɚɹ ɷɧɟɪɝɢɹ ɷɥɟɤɬɪɨɧɧɨɝɨ ɝɚɡɚ ɩɪɢ ɬɟɦɩɟɪɚɬɭɪɟ ܶൌͳȀߚ ɞɚɟɬɫɹ ɫɬɚ-
ɬɢɫɬɢɱɟɫɤɨɣ ɫɭɦɦɨɣ
ሺሻ
ିఉா
ሺȟሻ
ൌെܶ݁
ܨ
ሺȟሻ – ɷɧɟɪɝɟɬɢɱɟɫɤɢɣ ɫɩɟɤɬɪ ɷɥɟɤɬɪɨɧɧɨɝɨ ɝɚɡɚ ɩɪɢ ɡɚɞɚɧɧɨɣ ɤɨɧɮɢɝɭɪɚ-
ɝɞɟ ܧ
ɰɢɢ ɮɨɧɨɧɧɨɣ ɦɨɞɵ ȟሺݖ
ሻ
:
ǡ
ሺ
ͳͳǤ
ሻ
ሻ
ܪȁ݊ۧൌܧ
ሺȟሻ
ȁ݊ۧǤ
ሺ
ͳͳǤͺ
ɉɨɥɧɚɹ ɫɜɨɛɨɞɧɚɹ ɷɧɟɪɝɢɹ ɫɢɫɬɟɦɵ ɟɫɬɶ
ଶ
ܨሺȟሻൌܨ
ሺȟሻ
ܨ
ൌන
ȁȟȁ
݀ݖെܶ݁
ଶ
݃
ሺȟሻ
ିఉா
ሺሻ
ሺ
ǡ
ͳͳǤͻ
ɦɵ ɩɪɟɧɟɛɪɟɝɥɢ ɷɧɬɪɨɩɢɟɣ ɮɨɧɨɧɧɨɣ ɦɨɞɵ. Ɋɚɜɧɨɜɟɫɧɚɹ ɤɨɧɮɢɝɭɪɚɰɢɹ ɮɨɧɨɧɧɨɣ ɦɨɞɵ ɨɩɪɟɞɟɥɹɟɬɫɹ ɦɢɧɢɦɭɦɨɦ ɫɜɨɛɨɞɧɨɣ ɷɧɟɪɝɢɢ, ɩɨɷɬɨɦɭ ɜ ɪɚɜɧɨɜɟɫɢɢ
ɢɦɟɟɦ
ሻ
ߜܨሺȟ
ൌͲǡ
ߜȟሺݖሻ
ɩɪɢ ɜɚɪɢɚɰɢɢ ɫɜɨɛɨɞɧɨɣ ɷɧɟɪɝɢɢ ɜɟɥɢɱɢɧɵ ȟሺݖሻ ɢ ȟ
ɦɵɦɢ. Ɏɚɤɬɢɱɟɫɤɢ ɞɜɚ ɭɪɚɜɧɟɧɢɹ (11.9) ɨɤɚɡɵɜɚɸɬɫɹ ɷɤɜɢɜɚɥɟɧɬɧɵ. ȼɵɱɢɫɥɹɹ
ɜɚɪɢɚɰɢɨɧɧɭɸ ɩɪɨɢɡɜɨɞɧɭɸ ɩɨ ȟ
ȟ
݁
ଶ
݃
ఉிሺሻ
כ
ሺݖሻ, ɩɨɥɭɱɚɟɦ
ିఉா
݁
ߜܨሺȟ
כ
ߜȟ
ሺሻ
ሺݖሻ
ሻ
ቆ
ൌͲǡ
ߜܧ
ߜȟ
כ
ሺݖሻ ɫɱɢɬɚɸɬɫɹ ɧɟɡɚɜɢɫɢ-
ሺȟሻ
ቇൌͲǤ
כ
ሺݖሻ
ሺ
ͳͳǤͳͲ
ሺ
ͳͳǤͳͳ
ɂɫɩɨɥɶɡɭɟɦ ɮɨɪɦɭɥɭ Ƚɟɥɶɦɚɧɚ Ɏɟɣɧɦɚɧɚ [1, ɫɬɪ. 53]
ߜܧ
ߜȟ
ሺȟሻ
ൌۃ݊ቤ
כ
ሺݖሻ
ߜܪ
ߜȟ
ሺȟሻ
כ
ሺݖሻ
ቤ݊ۄǡ
ሺ
ͳͳǤͳʹ
ɬɨɝɞɚ
ۃ
ߜܪ
ߜȟ
ሺȟሻ
כ
ሺݖሻ
ۄ
ǡ
்
ሺ
ͳͳǤͳ͵
ɝɞɟ ɫɤɨɛɤɢ
ሺȟሻ
ߜܧ
ିఉா
ሺሻ
ሻ
ఉிሺ
݁
݁
ۃǥۄ
ɨɡɧɚɱɚɸɬ ɭɫɪɟɞɧɟɧɢɟ ɩɨ ɝɢɛɛɫɨɜɫɤɨɦɭ ɚɧɫɚɦɛɥɸ ɫ ɬɟɦɩɟɪɚɬɭ-
்
ቆ
ߜȟ
כ
ሺݖሻ
ቇൌ
ɪɨɣ ܶ.
ሻ
ሻ
ሻ
ሻ
ሻ
ሻ
118
Ɍɨɝɞɚ
ȟ
ۃ
ଶ
݃
ߜȟ
כ
ሺݖሻ
ۄ
்
ൌͲǤ
ሺ
ͳͳǤͳͶ
ሺȟሻ
ߜܪ
ɂɡ (11.5) ɫɥɟɞɭɟɬ
ሺȟሻ
ߜܪ
ߜȟ
כ
ሺݖሻ
ൌ߰
ା
ሺݖሻ
ሺିሻ
ሺݖሻ
ߪො
߰
ǡ
ሺ
ͳͳǤͳͷ
ɨɬɤɭɞɚ ɦɵ ɩɨɥɭɱɚɟɦ ɭɫɥɨɜɢɟ, ɨɩɪɟɞɟɥɹɸɳɟɟ ɪɚɜɧɨɜɟɫɧɭɸ ɤɨɧɮɢɝɭɪɚɰɢɸ ɮɨɧɨɧɧɨɣ ɦɨɞɵ
ȟሺሻ
ଶ
݃
ۃ߰
ሺିሻ
ା
ሺݖሻ
ሺݖሻ
ߪො
ۄ
߰
்
ൌͲǤ
ሺ
ͳͳǤͳ
11.2. Ɋɚɜɧɨɜɟɫɧɚɹ ɞɟɮɨɪɦɚɰɢɹ ɪɟɲɟɬɤɢ,
ɬɟɦɩɟɪɚɬɭɪɚ ɩɟɪɟɯɨɞɚ
ɋɪɟɞɧɟɟ ɨɬ ɷɥɟɤɬɪɨɧɧɵɯ ɨɩɟɪɚɬɨɪɨɜ ɪɨɠɞɟɧɢɹ ɢ ɭɧɢɱɬɨɠɟɧɢɹ ɩɪɢ ɬɟɦɩɟɪɚ-
ɬɭɪɟ ܶ ɦɨɠɧɨ ɜɵɪɚɡɢɬɶ ɱɟɪɟɡ ɦɚɰɭɛɚɪɨɜɫɤɭɸ ɮɭɧɤɰɢɸ Ƚɪɢɧɚ ɷɥɟɤɬɪɨɧɨɜ (3.36),
ɤɨɬɨɪɚɹ ɜ ɞɚɧɧɨɦ ɫɥɭɱɚɟ ɹɜɥɹɟɬɫɹ ɦɚɬɪɢɰɟɣ ʹൈʹ ɩɨ ɢɧɞɟɤɫɚɦ ͳ ɢ ʹ
ܩ
ఈఉ
ሺ
߬
ଵെ߬ଶ
ǡݖଵǡݖ
ሻ
ൌെۃ
ଶ
ఛ߰ఈ
ሺ
߬
ெ
ɉɨɥɭɱɢɦ
ȟሺሻ
ଶ
݃
ൣܩ
ெ
ሺ
െͲǡݖǡݖሻߪො
ɉɟɪɟɣɞɟɦ ɤ ɮɭɪɶɟ-ɤɨɦɩɨɧɟɧɬɚɦ ɮɭɧɤɰɢɢ Ƚɪɢɧɚ ɩɨ ɞɢɫɤɪɟɬɧɵɦ ɱɚɫɬɨɬɚɦ
(3.48)
ஶ
ெ
ܩ
ൌߨܶሺʹݏͳሻ, ɬɨɝɞɚ ɭɪɚɜɧɟɧɢɟ (11.18) ɩɪɢɦɟɬ ɜɢɞ
ɝɞɟ ߱
௦
μ
ଷ
μݖ
ȟሺݖሻ
ଶ
݃
ெ
ሺ
െȟሺݖሻԜߪො
Ɏɭɧɤɰɢɹ Ƚɪɢɧɚ ܩ
݅߱
݅ݒߪො
௦
ᇱ
ሺ
ሻ
߬ǡݖǡݖ
ൌܶܩ
ஶ
ܶൣߪො
௦ୀିஶ
ᇱ
݅ߞ
ሻ
ɭɞɨɜɥɟɬɜɨɪɹɟɬ ɭɪɚɜɧɟɧɢɸ
ǡݖǡݖ
௦
ሺାሻ
כ
ሺݖሻ
െȟ
௦ୀିஶ
ሺିሻ
Ԝߪො
ܩ
ሺିሻ
ெ
ሺ
݅߱
ெ
ሺ
݅߱
௦
ெ
ሺ
൨ܩ
ɡɞɟɫɶ ͳ– ɟɞɢɧɢɱɧɚɹ ɦɚɬɪɢɰɚ.
ଵǡݖଵ
ሺିሻ
ǡݖǡݖ
௦
ǡݖǡݖሻ݁
݅߱
௦
ା
ሻ
߰
ఉ
൧ൌͲǤ
ᇱ
ሻ
݁
ೞఋ
ǡݖǡݖ
ሺ
ሻ
ۄ
߬
ଵǡݖଶ
்
ିೞఛ
ǡ
൧
ൌͲǤ
ᇱ
ሻ
ൌͳߜሺݖെݖ
ሺ
Ǥ
ᇱ
ሻǡሺ
ͳͳǤͳ
ሺ
ͳͳǤͳͺ
ሺ
ͳͳǤͳͻ
ሺ
ͳͳǤʹͲ
ͳͳǤʹͳ
ሻ
ሻ
ሻ
ሻ
ሻ
ሻ
ሻ
ሻ
119
Ȼɭɞɟɦ ɪɟɲɚɬɶ ɭɪɚɜɧɟɧɢɟ (11.20) ɜ ɩɪɟɞɩɨɥɨɠɟɧɢɢ, ɱɬɨ ɚɦɩɥɢɬɭɞɚ ɮɨɧɨɧɧɨɣ
ɦɨɞɵ ɩɨɫɬɨɹɧɧɚ ȟሺݖሻൌȟൌ. ȼ ɷɬɨɦ ɫɥɭɱɚɟ ɮɭɧɤɰɢɹ Ƚɪɢɧɚ ܩ
ɛɭɞɟɬ ɡɚɜɢɫɟɬɶ ɬɨɥɶɤɨ ɨɬ ɪɚɡɧɨɫɬɢ ɤɨɨɪɞɢɧɚɬ ݖെݖ
ᇱ
, ɢ ɜ ɷɬɨɦ ɫɥɭɱɚɟ ɦɨɠɧɨ ɩɟ-
ெ
ሺ
݅߱
ǡݖǡݖ
௦
ᇱ
ሻ
ɪɟɣɬɢ ɤ ɮɭɪɶɟ-ɤɨɦɩɨɧɟɧɬɚɦ ɩɨ ɢɦɩɭɥɶɫɭ
ܩ
ɩɪɢ ɷɬɨɦ ɮɭɧɤɰɢɹ ܩ
ൣ݅߱
ெ
ሺ
݅߱
ெ
ሺ
௦െݒ
ሻ
ൌන
ሺାሻ
݀݇
ʹߨ
െȟכԜߪො
ᇱ
ǡݖെݖ
௦
ሻ
݅ߞ
ǡ݇
ɭɞɨɜɥɟɬɜɨɪɹɟɬ ɭɪɚɜɧɟɧɢɸ
௦
݇ߪොଷെȟԜߪො
݁
ሺ௭ି௭
ሺିሻ
൧ܩ
ᇲ
ሻ
ெ
ሺ
ሻሺ
݅߱
ܩ
ெ
ሺ
݅ߞ
ǡ݇
௦
ǡ݇ሻൌͳ
௦
ͳͳǤʹʹ
ሺ
ͳͳǤʹ͵
ɪɟɲɟɧɢɟ ɤɨɬɨɪɨɝɨ ɢɦɟɟɬ ɜɢɞ
ெ
ሺ
݅ߞ
ܩ
ǡ݇ሻൌെ
௦
݅߱
௦ݒ
߱
݇ߪොଷȟԜߪො
ଶ
ሺݒ݇
௦
ሺାሻ
ଶ
ሻ
ȁȟ
ɉɨɞɫɬɚɜɥɹɹ ɷɬɨ ɪɟɲɟɧɢɟ ɜ (11.22), ɚ ɡɚɬɟɦ (11.20) ɢ ɭɱɢɬɵɜɚɹ ൣߪො
ȟכԜߪො
ଶ
ȁ
ሺିሻ
Ǥ
ሺାሻߪොሺିሻ
ሺ
ͳͳǤʹͶ
൧ൌͳ,
ɢ ɫɨɤɪɚɳɚɹ ɧɚ ȟ, ɩɨɥɭɱɢɦ
ͳ
ൌܶන
ଶ
݃
௦ୀିஶ
݀݇
ʹߨ
߱
ଶ
ሺݒ݇
௦
ͳԜ
ଶ
ሻ
ȁȟ
Ǥ
ଶ
ȁ
ሺ
ͳͳǤʹͷ
ஶ
Ɇɵ ɩɨɥɭɱɢɥɢ ɭɪɚɜɧɟɧɢɟ, ɨɩɪɟɞɟɥɹɸɳɟɟ ɪɚɜɧɨɜɟɫɧɭɸ ɤɨɧɮɢɝɭɪɚɰɢɸ ɮɨɧɨɧ-
ɧɨɣ ɦɨɞɵ, ɢɡ ɤɨɬɨɪɨɝɨ ɜɢɞɧɨ, ɱɬɨ ɨɧɨ ɨɩɪɟɞɟɥɹɟɬ ɬɨɥɶɤɨ ɦɨɞɭɥɶ ɮɨɧɨɧɧɨɣ ɦɨɞɵ
ȁȟȁ
, ɨɫɬɚɜɥɹɹ ɮɚɡɭ ߯ ɧɟɨɩɪɟɞɟɥɟɧɧɨɣ. ɋɭɦɦɚ ɩɨ ɱɚɫɬɨɬɚɦ ɜ (11.25) ɜɵɱɢɫɥɹɟɦ ɩɨ
ɮɨɪɦɭɥɟ
ɨɛɨɡɧɚɱɚɹ ߳ሺ݇ሻൌ
ሺ
ݒ
ඥ
ஶ
ߨ
ୀିஶ
ଶ
ሻ
݇
ȁȟ
ஶ
ܶ
௦ୀିஶ
ͳ
ଶ
ሺʹ݊ͳሻଶܽ
ଶ
ȁ
, ɩɨɥɭɱɢɦ
ͳԜ
ଶ
ଶ
௦
߳
ሺ݇ሻ
߱
ൌ
ൌ
ଶ
ͳ
ʹ߳ሺ݇
ሻ
ͳ
ʹܽ
߳ሺ݇
ʹܶ
ܽ
ǡ
ʹ
ሻ
Ǥ
ሺ
ͳͳǤʹ
ሺ
ͳͳǤʹ
ሻ
ሻ
ሻ
ሻ
ሻ
ሻ
ɉɨɞɫɬɚɜɢɜ ɷɬɨ ɜ ɢɧɬɟɝɪɚɥ (11.25) ɢ ɩɟɪɟɯɨɞɹ ɤ ɩɟɪɟɦɟɧɧɨɣ ߝൌݒ
ఌ
ಷ
ʹɎݒ
݃
ଶ
ൌන
݀ߝ
ଶ
ȁȟ
ȁ
ඥ
ߝ
Ɉɩɪɟɞɟɥɢɦ ɜ ɩɟɪɜɭɸ ɨɱɟɪɟɞɶ ɜɟɥɢɱɢɧɭ
, ɜ ɷɬɨɦ ɫɥɭɱɚɟ ɬɚɧɝɟɧɫ ɪɚɜɟɧ ɟɞɢɧɢɰɟ, ɬɨɝɞɚ ɫ ɭɱɟɬɨɦ ȟاߝ
ɤɚɤ ȟ
ఌ
ಷ
ʹɎݒ
݃
ଶ
ൌන
݀ߝ
ଶ
ඥ
ߝ
ο
ଶ
ଶ
ඥ
ଶ
ȁȟȁ
ɩɪɢ ܶൌͲ, ɤɨɬɨɪɭɸ ɨɛɨɡɧɚɱɢɦ
ൌ
ଶ
ȁ
ȁȟ
ߝ
ʹܶ
ʹߝ
ȟ
Ǥ
ி
ǡ
120
݇
ሺ
ሻ
ͳͳǤʹͺ
ி
ሺ
ሻ
ͳͳǤʹͻ
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