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ÐÀÑÑÅßÍÈÅ ÝËÅÊÒÐÎÌÀÃÍÈÒÍÛÕ ÂÎËÍ ÑÈÑÒÅÌÎÉ ÏËÎÑÊÈÕ ÐÅØÅÒÎÊ ÐÅÇÎÍÀÍÑÍÛÕ ÌÀÃÍÈÒÎÄÈÝËÅÊÒÐÈ×ÅÑÊÈÕ ÑÔÅÐ Ñ ÒÎÏÎËÎÃÈ×ÅÑÊÎÉ ÑÒÐÓÊÒÓÐÎÉ, ÇÀÄÀÂÀÅÌÎÉ ÃÅÎÌÅÒÐÈ×ÅÑÊÎÉ ÏÐÎÃÐÅÑÑÈÅÉ.........................................................................................7 ÔÅÄÎÒΠÔ.Â., ÍÅÐÓÕ À.Ã. ×ÈÑËÅÍÍÎÅ ÈÑÑËÅÄÎÂÀÍÈÅ ÂÇÀÈÌÎÄÅÉÑÒÂÈß ÝËÅÊÒÐÎÌÀÃÍÈÒÍÎÃÎ ÏÎËß Ñ ÍÅÑÒÀÖÈÎÍÀÐÍÛÌ ÄÈÝËÅÊÒÐÈ×ÅÑÊÈÌ ÑËÎÅÌ ÏÐÈ ÏÎÌÎÙÈ ÌÅÒÎÄÀ ÈÍÒÅÃÐÀËÜÍÛÕ ÓÐÀÂÍÅÍÈÉ ÂÎËÜÒÅÐÐÀ.................................................12 ÁÎÍÄÀÐÅÍÊÎ È.Í. Î ÍÅÊÎÒÎÐÛÕ ÂÎÇÌÎÆÍÎÑÒßÕ ÔÎÐÌÈÐÎÂÀÍÈß ÀÌÏËÈÒÓÄÍÎ- ×ÀÑÒÎÒÍÎÉ ÕÀÐÀÊÒÅÐÈÑÒÈÊÈ Â ÎÁÚÅÌÍÛÕ ÐÅÇÎÍÀÒÎÐÀÕ ÑÒÎß×ÈÕ ÂÎËÍ........................15 ÄÎÐÎØÅÍÊÎ Â.À., ÑÅÌÅÍÎÂÀ Å.Ê. ÐÀÑÑÅßÍÈÅ ÏËÎÑÊÎÉ ÝËÅÊÒÐÎÌÀÃÍÈÒÍÎÉ ÂÎËÍÛ ÍÅÇÀÌÊÍÓÒÛÌ ÊÎÍÈ×ÅÑÊÈÌ ÎÒÐÀÆÀÒÅËÅÌ..........................................................................18 ÈËÜߨÅÍÊÎ Ë.Í. ×ÈÑËÅÍÍÎÅ ÌÎÄÅËÈÐÎÂÀÍÈÅ ÏÐÈÇÌÅÍÍÛÕ ÝËÅÌÅÍÒΠÎÏÒÎÝËÅÊÒÐÎÍÍÛÕ ÏÐÈÁÎÐÎÂ...........................................................................................................................23 ÑÈÍÎÒÈÍ À.Ì. ÂËÈßÍÈÅ ÔÎÐÌÛ ÍÀÃÐÅÒÎÉ ÇÎÍÛ ÍÀ ÌÀÊÑÈÌÀËÜÍÛÉ ÏÅÐÅÃÐÅ ÀÏÏÀÐÀÒÀ............................................................................................................................................................................27 ÏÎÑÏÅËΠÁ.Á., ÃÐÓØÅÍÊÎ Ì.Â. ÀËÃÎÐÈÒÌ ÀÄÀÏÒÈÂÍÎÉ ÎÁÐÀÁÎÒÊÈ ÑÈÃÍÀËÎÂ È ÏÎÌÅÕ Â ÀÍÒÅÍÍÛÕ ÑÈÑÒÅÌÀÕ Ñ ÂÂÅÄÅÍÈÅÌ ÈÑÊÓÑÑÒÂÅÍÍÛÕ ØÓÌÎÂ..............................30 ÝËÅÊÒÐÎÍÈÊÀ OLEINICK A.I., AMATORE C.A., SVIR I.B. SIMULATION OF THE DOUBLE HEMICYLINDER ELECTRODE SYSTEM THROUGH CONFORMAL MAPPING. APPLICATION TO STEADY-STATE ELECTROGENERATED CHEMILUMINESCENCE..................................................................................................35 ÑÈÑÒÅÌÛ È ÏÐÎÖÅÑÑÛ ÓÏÐÀÂËÅÍÈß ÒÈÕÎÍΠÂ.À. ÎÁÎÁÙÅÍÍÀß ÌÎÄÅËÜ ÀÂÒÎÐÅÃÐÅÑÑÈÈ ÂÛÑØÈÕ ÐÀÍÃΠÍÅÃÀÓÑÑÎÂÛÕ ÏÐÎÖÅÑÑÎÂ......................................................................................................................................39 ÊÎÐÎËÜÊÎÂÀ Å.Å., ËÀÌÎÍÎÂÀ Í.Ñ., ÏËÈÑÑ È.Ï. ÌÎÄÈÔÈÖÈÐÎÂÀÍÍÛÅ ÀËÃÎÐÈÒÌÛ ÎÁÓ×ÅÍÈß ÍÅÉÐÎÍÍÛÕ ÑÅÒÅÉ ÂÅÊÒÎÐÍÎÃÎ ÊÂÀÍÒÎÂÀÍÈß..........................................................42 ÁÎÄßÍÑÊÈÉ Å.Â., ÏÎËÓØÊÈÍÀ Í.À. ÀËÃÎÐÈÒÌ ÎÁÓ×ÅÍÈß ÔÎÐÌÀËÜÍÎÃÎ ÍÅÉÐÎÍÀ Ñ ÏÎËÈÍÎÌÈÀËÜÍÎÉ ÀÊÒÈÂÀÖÈÎÍÍÎÉ ÔÓÍÊÖÈÅÉ...........................................................................45 ÔÈËÀÒΠÂ.À. ÎÁ ÎÄÍÎÌ ÏÎÄÕÎÄÅ Ê ÏÎÑÒÐÎÅÍÈÞ ÌÀÒÅÌÀÒÈ×ÅÑÊÎÉ ÌÎÄÅËÈ ÏÐÎÃÐÀÌÌÍÎÃÎ ÀÃÅÍÒÀ.............................................................................................................................................48 ÏÎËÎÍÑÊÈÉ À.Ä. ÑÈÍÒÅÇ ÈÍÂÀÐÈÀÍÒÍÛÕ ÊËÀÑÑÈÔÈÊÀÒÎÐÎÂ..................................................51 ÑÈÄÎÐΠÌ.Â. ÎÁ ÎÄÍÎÌ ÏÎÄÕÎÄÅ Ê ×ÈÑËÅÍÍÎÌÓ ÌÎÄÅËÈÐÎÂÀÍÈÞ ÊÎÍÂÅÊÒÈÂÍÛÕ ÂßÇÊÈÕ ÒÅ×ÅÍÈÉ Â ÎÄÍÎÑÂßÇÍÛÕ È ÌÍÎÃÎÑÂßÇÍÛÕ ÎÁËÀÑÒßÕ.............................................................................................................................................................................55 ÐÀÄÈÅÂÑÊÈÉ À.Å. ÇÀÄÀ×À ÄÈÍÀÌÈ×ÅÑÊÎÃÎ ÑÈÍÒÅÇÀ ÏÐÈ ÂÅÊÒÎÐÍÎÌ ÓÏÐÀÂËÅÍÈÈ.....................................................................................................................................................................57 ÃÐÅÁÅÍÍÈÊ È.Â., ËÀÏÊÎ Ä.À. 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Ðàññìàòðèâàåòñÿ ïðåîáðàçîâàíèå ïëîñêîãî ýëåêòðî- ìàãíèòíîãî ãàóññîâîãî èìïóëüñà, âûçâàííîå èçìåíå- íèåì âî âðåìåíè äèýëåêòðè÷åñêîé è ìàãíèòíîé ïðî- íèöàåìîñòåé ïîëóîãðàíè÷åííîé îáëàñòè â èçîòðîï- íîì ìàãíèòîäèýëåêòðè÷åñêîì ïðîñòðàíñòâå. Ìåòî- äîì èíòåãðàëüíûõ óðàâíåíèé ïîëó÷åíû òî÷íûå âûðà- æåíèÿ äëÿ êîìïîíåíòû ýëåêòðè÷åñêîãî ïîëÿ íà âñåé ïîëîæèòåëüíîé ïîëóîñè âðåìåíè â ëþáîé òî÷êå ïðî- ñòðàíñòâà. Ïðîâîäèòñÿ äåòàëüíûé àíàëèç ýâîëþöèè ýëåêòðè÷åñêîãî ïîëÿ âî âðåìåíè. 1. Ââåäåíèå Äàííàÿ ñòàòüÿ ÿâëÿåòñÿ ëîãè÷åñêèì ïðîäîëæåíèåì ðàáîòû [1], ãäå áûëè ïðèâåäåíû è ïðîàíàëèçèðîâà- íû ðåçóëüòàòû àíàëèòè÷åñêèõ èññëåäîâàíèé ïðåîá- ðàçîâàíèÿ êîìïîíåíòû ïëîñêîé ìîíîõðîìàòè÷åñ- êîé ýëåêòðîìàãíèòíîé âîëíû ÒÌ âèäà, èíèöèèðî- âàííîãî èçìåíåíèåì âî âðåìåíè ïàðàìåòðîâ ïîëó- ïðîñòðàíñòâà â áåçãðàíè÷íîé ìàãíèòîäèýëåêòðè- ÷åñêîé èçîòðîïíîé ñðåäå ïî çàêîíó ïðÿìîóãîëüíî- ãî èìïóëüñà.  íàñòîÿùåé ðàáîòå àíàëèòè÷åñêè èññëåäóåòñÿ âëè- ÿíèå èìïóëüñíîãî ñèíõðîííîãî èçìåíåíèÿ äèýëåê- òðè÷åñêîé ) t (  è ìàãíèòíîé ) t (  ïðîíèöàåìîñòåé â îáëàñòè ïîëóïðîñòðàíñòâà 0 x  íà ïðîõîæäåíèå ïëîñêîãî ýëåêòðîìàãíèòíîãî ãàóññîâà èìïóëüñà â ñâîáîäíîì ìàãíèòîäèýëåêòðèêå. Êàê è â äðóãèõ ñòàòüÿõ ïîäîáíîãî ïëàíà [1, 2-4], ïðàêòè÷åñêàÿ íåîáõîäèìîñòü ðàññìîòðåíèÿ ñêà÷êîîáðàçíûõ ôðîí- òîâ èìïóëüñîâ âîçìóùåíèÿ ñðåäû îáóñëîâëåíà òåì, ÷òî â îïòè÷åñêîì äèàïàçîíå èçìåíåíèå ïàðàìåòðîâ ñðåäû ìîæíî ñ÷èòàòü ñêà÷êîîáðàçíûì [5]. 2. Ïîñòàíîâêà çàäà÷è Ïîëîæèì, ÷òî äî íóëåâîãî ìîìåíòà âðåìåíè èìå- åòñÿ îäíîðîäíàÿ è èçîòðîïíàÿ ìàãíèòîäèýëåêòðè- ÷åñêàÿ ñðåäà ñ äèýëåêòðè÷åñêîé è ìàãíèòíîé ïðî- íèöàåìîñòÿìè, ñîîòâåòñòâåííî ðàâíûìè 0  è 0  . Ïóñòü â îáëàñòè ïîëóïðîñòðàíñòâà 0 x  ïðîèñõî- äèò âîçìóùåíèå ïàðàìåòðîâ ñðåäû âî âðåìåíè ïî çàêîíàì         , ) t ( ) t ( ) t ( ) t ( ) t ( , ) t ( ) t ( ) t ( ) t ( ) t ( 1 0 1 0                                 (1) ãäå 1  è 1  – ñîîòâåòñòâåííî äèýëåêòðè÷åñêàÿ è ìàãíèòíàÿ ïðîíèöàåìîñòè îáëàñòè 0 x  â âîçìó- ùåííîì ñîñòîÿíèè íà èíòåðâàëå âðåìåíè ) ; 0 ( t   . Ïðè ýòîì â îáëàñòè 0 x  ñðåäà èìååò äèýëåêòðè- ÷åñêóþ è ìàãíèòíóþ ïðîíèöàåìîñòè, ñîîòâåòñòâåííî ðàâíûå 0  è 0  , íà âñåé îñè âðåìåíè ) ; ( t    . Òîãäà ïðåîáðàçîâàíèå ýëåêòðè÷åñêîé êîìïîíåíòû ïåðâè÷íîé âîëíû ) x , t ( E 0 â îáëàñòè 0 x  , âûç- âàííîå âîçìóùåíèåì ïàðàìåòðîâ ýòîé îáëàñòè ïî çàêîíó (1), ñîãëàñíî [6], áóäåò îïèñûâàòüñÿ èíòåã- ðàëüíûì óðàâíåíèåì Âîëüòåððà âòîðîãî ðîäà: , 0 t ), ' x , ' t ( E ) ' x , x , ' t , t ( K ' dx ' dt ) x , t ( E ) x , t ( E 0 t 0 0       (2) ãäå ) ' x , x , ' t , t ( K – ÿäðî èíòåãðàëüíîãî óðàâíåíèÿ (2), êîòîðîå èìååò ñëåäóþùèé âèä [1]:          ) ' x x ( ) ' t t ( ) ' t t ( ] m 1 [ ) ' x , x , ' t , t ( K 2 ) ' x x ) ' t t ( v ( t a 2 m a 1 0 2 2 2         . (3) Çäåñü , a 1 0    1 0 m    , 1  è 1  – âîçìóùåííûå çíà÷åíèÿ, ñîîòâåòñòâåííî, ìàãíèò- íîé è äèýëåêòðè÷åñêîé ïðîíèöàåìîñòåé; 1 1 1 c v    – ñêîðîñòü ðàñïðîñòðàíåíèÿ ñâåòà â ñâîáîäíîì ìàãíèòîäèýëåòðè÷åñêîì ïðîñòðàíñòâå ñ ìàãíèòíîé è äèýëåêòðè÷åñêîé ïðîíèöàåìîñòÿìè, ñîîòâåòñòâåííî ðàâíûìè 1  è 1  ; c – ñêîðîñòü ðàñïðîñòðàíåíèÿ ñâåòà â âàêóóìå. Èíòåãðàëüíîå óðàâíåíèå (1) ðåøàåòñÿ ìåòîäîì ðåçîëüâåíòû [7], ñîãëàñíî êîòîðîìó ðåøåíèå (2) íà èíòåðâàëå âðåìåíè ) ; 0 ( t   ìîæåò áûòü íàéäåíî ïîñðåäñòâîì ðàâåíñòâà  ) x , t ( E (4)         t 0 ), x , t ( E ) ' x , x , ' t , t ( R ' dx ' dt ) x , t ( E 0 0 t 0 0 , ãäå ) ' x , x , ' t , t ( R – ðåçîëüâåíòà èíòåãðàëüíîãî óðàâ- íåíèÿ (2), êîòîðàÿ, ñîãëàñíî [1], èìååò âèä                ) ' x x ( ) ' t t ( ) ' t t ( m 1 1 ) ' x , x , ' t , t ( R 2 ) ' x x ) ' t t ( v ( t m 2 m a 1 1 2 2 2         . 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Ýâîëþöèÿ ïîëÿ íà âñåé ïîëîæèòåëüíîé ïîëóîñè âðåìåíè Ïóñòü ïåðâè÷íîå ýëåêòðè÷åñêîå ïîëå ïðåäñòàâëÿåò ñîáîé ÒÌ âîëíó ñ êîìïîíåíòîé . v k , e ) x , t ( E 0 ) kx t ( 0 2       (6) Ïîäñòàâëÿÿ âûðàæåíèÿ (5) è (6) â êâàäðàòóðíóþ ôîðìóëó (4), ïîëó÷àåì, ÷òî âûðàæåíèå äëÿ ýëåêò- ðè÷åñêîé êîìïîíåíòû ïîëÿ ) x , t ( E â îáëàñòè ïî- ëóïðîñòðàíñòâà 0 x  íà èíòåðâàëå âðåìåíè ) ; 0 ( t   áóäåò èìåòü âèä  ) x , t ( E 1             2 2 ) kx t am ( 1 ) am kx t ( 1 1 e B e A ) v x t ( 2 ) kx t am ( 2 1 1 1 e } B ) v x t ( B ) v x t ( {            ,(7) ãäå am 1 a 2 A 1   , am 2 am 1 B 1    , am 1 ) am 1 ( 2 am B 2 2     . Ñðàâíèâàÿ âûðàæåíèÿ äëÿ ïðåîáðàçîâàííîãî (7) ñ ñîîòâåòñòâóþùèì âûðàæåíèåì äëÿ ïîëÿ ) x , t ( E 1 â ñòàòüå [1], ïðèõîäèì ê èäåíòè÷íîñòè êà÷åñòâåííîé êàðòèíû ïðåîáðàçîâàíèÿ ïîëÿ ) x , t ( E 0 íà ñêà÷êå ïàðàìåòðîâ â îáëàñòè ïîëóïðîñòðàíñòâà 0 x  (çäåñü è äàëåå, êàê è â ðàáîòå [1], óñëîâèìñÿ âîëíû èìåíîâàòü íàçâàíèåì èõ àìïëèòóäíûõ êîýôôèöè- åíòîâ): ñêà÷îê ñâîéñòâ îáëàñòè 0 x  â ìîìåíò âðåìåíè 0 t  ðàñùåïëÿåò ïåðâè÷íûé èìïóëüñ ) x , t ( E 0 íà ïðÿìîé  1 B (êîòîðûé äâèæåòñÿ â íàïðàâëåíèè ïåðâè÷íîãî) è ïðåäñòàâëåí òðåòüèì ñëàãàåìûì â âûðàæåíèè (7), à òàêæå îáðàòíûé  1 B (ðàñïðîñòðàíÿþùèéñÿ ê îáðàçîâàâøåéñÿ ãðàíèöå ðàçäåëà äâóõ ñðåä 0 x  ), êîòîðûé ïðåäñòàâëåí âòîðûì ñëàãàåìûì â âûðàæåíèè (7). Ôàçîâûå õàðàêòåðèñòèêè ýòèõ èìïóëüñîâ îòëè÷àþòñÿ îò ôàçîâûõ õàðàêòåðèñòèê ïåðâè÷íîãî èìïóëüñà ) x , t ( E 0 ; òàê, ñêîðîñòü ðàñïðîñòðàíåíèÿ äàííîé ïàðû èìïóëüñîâ ðàâíà 0 amv .  ìîìåíò âðåìåíè 1 v x t  íà÷èíàåò ñêàçûâàòüñÿ âëèÿíèå ãðàíèöû ðàçäåëà äâóõ ñðåä – åå äîñòèãàåò îáðàòíûé èìïóëüñ  1 B . Ðåçóëüòàòîì îòðàæåíèÿ ïîñëåäíåãî îò ãðàíèöû 0 x  ÿâëÿåòñÿ ïðÿìîé èìïóëüñ 2 B , êîòîðîìó â âûðàæåíèè (7) ñîîòâåò- ñòâóåò ÷åòâåðòîå ñëàãàåìîå. Êðîìå òîãî, â îáëàñòè 0 x  â ìîìåíò âðåìåíè 0 t  ïîÿâëÿåòñÿ ïðÿìîé èìïóëüñ 1 A , êîòîðûé ïî ñóòè ÿâëÿåòñÿ ïðîøåäøèì ïåðâè÷íûì èìïóëüñîì ñ íîâûìè â ñðàâíåíèè ñ ïîñëåäíèì ôàçîâûìè õàðàêòåðèñòèêàìè. Ñðàâíèâàÿ âûðàæåíèÿ äëÿ àìïëèòóä âîëí, îïèñû- âàåìûõ ôîðìóëîé (7), ñ ñîîòâåòñòâóþùèìè âûðà- æåíèÿìè ñòàòüè [1], ïðèõîäèì òàêæå ê èäåíòè÷- íîñòè êîëè÷åñòâåííîé ñòîðîíû ïðîöåññà ïðåîáðà- çîâàíèÿ ìîíîõðîìàòè÷åñêîé âîëíû è ãàóññîâîãî èìïóëüñà íà èíòåðâàëå âðåìåíè ) ; 0 ( t   . Ó÷èòûâàÿ êà÷åñòâåííóþ è êîëè÷åñòâåííóþ èäåí- òè÷íîñòü ïðåîáðàçîâàíèÿ ìîíîõðîìàòè÷åñêîé âîë- íû è ãàóññîâîãî èìïóëüñà íà ñêà÷êå ñâîéñòâ ïîëó- îãðàíè÷åííîé îáëàñòè ñâîáîäíîãî ìàãíèòîäèýëåê- òðè÷åñêîãî ïðîñòðàíñòâà, íåòðóäíî ïðåäïîëîæèòü î ñóùåñòâîâàíèè òàêîé æå ïîëíîé èäåíòè÷íîñòè ïîëÿ â íåâîçìóùåííîé îáëàñòè 0 x  íà èíòåðâàëå âðåìåíè ) ; 0 ( t   . Äåéñòâèòåëüíî, íåäîëãèå ïðåîá- ðàçîâàíèÿ ïðèâîäÿò ê ðàâåíñòâó     2 2 2 2 0 2 m a 2 m a 1 ) x , t ( E ) x , t ( E           )) ' t t ( v x , ' t ( E )) ' t t ( v x ( ' dt t 0 1 0 t 0   ) x , t ( E 0 } e C e C ){ v x t ( 2 2 2 2 ) kx t ( m a 2 ) kx t ( 1 0           ,(8) ãäå am 1 am 1 C 1     , 1 2 amC C  . Ðàññóæäàÿ ïî àíàëîãèè ñ àíàëèçîì ñîîòâåòñòâóþ- ùåãî âûðàæåíèÿ â ðàáîòå [1], ïðèõîäèì ê ñëåäóþ- ùåìó àíàëèçó (8).  íóëåâîé ìîìåíò âðåìåíè íàðÿäó ñ ôðîíòîì ïîëÿ t v x 1  îò ãðàíèöû 0 x  îòäåëÿåòñÿ è äâèæåòñÿ ñî ñêîðîñòüþ 0 v â íàïðàâ- ëåíèè, ïðîòèâîïîëîæíîì íàïðàâëåíèþ ïåðâè÷íî- ãî èìïóëüñà, ôðîíò ïîëÿ t v x 0   . Ýòîò ôðîíò îãðàíè÷èâàåò ñëåâà îáëàñòü âíåøíåãî ïîëÿ, âîç- áóæäàåìîãî îòðàæåííûì ïåðâè÷íûì èìïóëüñîì 1 C (âòîðîå ñëàãàåìîå â âûðàæåíèè (8)), ïðîøåä- øèì ÷åðåç ãðàíèöó ðàçäåëà äâóõ ñðåä 0 x  , à òàêæå èìïóëüñîì  1 B , êîòîðûé â îáëàñòè 0 x  èìååò àìïëèòóäó 2 C (òðåòüå ñëàãàåìîå â âûðàæåíèè (8)).  îáëàñòè 0 x  òàêæå ñóùåñòâóåò ïåðâè÷íûé èì- ïóëüñ. Ñêîðîñòè ðàñïðîñòðàíåíèÿ èìïóëüñîâ 1 C è 2 C ñîâïàäàþò ñî ñêîðîñòüþ ðàñïðîñòðàíåíèÿ ïåð- âè÷íîãî èìïóëüñà, ôàçîâûå õàðàêòåðèñòèêè ïåð- âè÷íîãî èìïóëüñà è èìïóëüñà 1 C òàêæå ñîâïàäàþò. 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Êàê âèäíî èç âûðàæåíèÿ (10), ýëåêòðè÷åñêàÿ êîìïîíåíòà ïîëÿ, îáðàçîâàâøàÿñÿ ïîñëå âîçâðàò- íîãî ñêà÷êà ñâîéñòâ îáëàñòè 0 x  (êàê ìîæíî áûëî ïðåäïîëîæèòü), ïðåäñòàâèìà â âèäå ñóïåðïîçèöèè ÷åòûðåõ ïàð ïðÿìûõ è îáðàòíûõ èìïóëüñîâ, à òàêæå ïåðâè÷íîãî èìïóëüñà, êîòîðûé «ïðîäîëæàåò ïîñòó- ïàòü» èç áåñêîíå÷íîñòè. Êàê è â ñëó÷àå ïåðâè÷íîé ìîíîõðîìàòè÷åñêîé âîëíû, àìïëèòóäû îáðàçîâàâ- øèõñÿ èìïóëüñîâ ïðåäñòàâëÿþò ñîáîé ðàçðûâíûå ôóíêöèè ïðîñòðàíñòâåííî-âðåìåííîé îáëàñòè. Äàííàÿ îáëàñòü, ãäå îòëè÷íû îò íóëÿ ýòè àìïëèòó- äû, à çíà÷èò è îáëàñòü ñóùåñòâîâàíèÿ ñîîòâåòñòâó- þùèõ èì èìïóëüñîâ, îãðàíè÷åíà ïëîñêîïàðàëëåëü- íûìè ïëîñêîñòÿìè, äâèæóùèìèñÿ âäîëü îñè x è íàîáîðîò ñî ñêîðîñòÿìè, ðàâíûìè 0 v . Êðîìå òîãî, îáðàçîâàâøèåñÿ ïàðû èìïóëüñîâ õàðàêòåðíû íàëè- ÷èåì íàáåãîâ ôàç, ÷òî îáóñëîâëåíî íåïðåðûâíîñ- òüþ ñïåêòðà ãàóññîâîãî èìïóëüñà. Óêàçàííûå âûøå äâèæóùèåñÿ ïëîñêîïàðàëëåëüíûå îáëàñòè ñóùå- ñòâîâàíèÿ îáðàçîâàâøèõñÿ èìïóëüñîâ â ìîìåíò âðåìåíè   t áûñòðî óõîäÿò íà áåñêîíå÷íîñòü, ïîýòîìó, êàê è â ñëó÷àå ïåðâè÷íîé ìîíîõðîìàòè- ÷åñêîé âîëíû [1], ïðåäñòàâëÿåò èíòåðåñ ïîëó÷åíèå ïîçäíåâðåìåííîãî ïðèáëèæåíèÿ âûðàæåíèÿ (10), ò.å. îïðåäåëåíèå ñîñòàâà èìïóëüñîâ, îñòàþùèõñÿ â êîíå÷íîé îáëàñòè ïðîñòðàíñòâà, ñ ïîñëåäóþùèì îïðåäåëåíèåì âûðàæåíèÿ äëÿ èõ àìïëèòóä. Àíàëèç àðãóìåíòîâ ôóíêöèé Õåâèñàéäà, îïðåäåëÿ- þùèõ îáëàñòè ñóùåñòâîâàíèÿ èìïóëüñîâ â âûðàæå- íèè (10), ïîçâîëÿåò çàïèñàòü ïîçäíåâðåìåííîå ïðèáëèæåíèå äëÿ ýòîãî âûðàæåíèÿ â âèäå 2 ) ) 1 am ( kx t ( 2 2 0 e am 4 m a 1 ) x , t ( E ) x , t ( E           . (11) Êàê âèäíî èç (11), â ñëó÷àå ïåðâè÷íîãî ãàóññîâîãî èìïóëüñà, êàê è â ñëó÷àå ïåðâè÷íîé ìîíîõðîìàòè- ÷åñêîé âîëíû [1], ïîçäíåâðåìåííîå ïðèáëèæåíèå âûðàæåíèÿ äëÿ ýëåêòðè÷åñêîé êîìïîíåíòû ïîëÿ ïðåäñòàâèìî â âèäå ñóììû ïåðâè÷íîé è îáðàòíîé âîëí, ñ ÷èñëîâûìè õàðàêòåðèñòèêàìè ïåðâè÷íîé âîëíû. Òåì íå ìåíåå î÷åâèäíû êîëè÷åñòâåííûå ðàçëè÷èÿ: îòëè÷àþòñÿ âûðàæåíèÿ äëÿ àìïëèòóä îáðàòíûõ âîëí, â ÷àñòíîñòè, àìïëèòóäà îáðàòíîé âîëíû â ñëó÷àå ïåðâè÷íîãî ãàóññîâîãî èìïóëüñà ÿâíî íå ñîäåðæèò äëèòåëüíîñòè èìïóëüñà âîçìóùå- íèÿ ñâîéñòâ ñðåäû, òàê êàê ïîñëåäíÿÿ ïî ïðè÷èíå íåïðåðûâíîñòè ñïåêòðà ãàóññîâîãî èìïóëüñà âõî- äèò ìíîæèòåëåì â íàáåã ôàç îáðàòíîãî èìïóëüñà. Îäíàêî â îáîèõ óêàçàííûõ ñëó÷àÿõ àìïëèòóäà ñòðåìèòñÿ ê íóëþ ïðè 1 a  è 1 m  , ÷òî ñîîòâåò- ñòâóåò ñëó÷àþ îòñóòñòâèÿ âîçìóùåíèÿ ñâîéñòâ ñðå- äû. Ïîñëåäíèé ôàêò ìîæåò ñëóæèòü íåîáõîäèìûì êðèòåðèåì ïðàâèëüíîñòè ïîëó÷åííûõ âûðàæåíèé äëÿ ïðåîáðàçîâàííûõ ýëåêòðè÷åñêèõ êîìïîíåíò ïîëÿ. Èç âûðàæåíèé (10), (11) âèäíî, ÷òî ïðåîáðàçîâàíèå ãàóññîâîãî ýëåêòðîìàãíèòíîãî èìïóëüñà â ïîëóîã- ðàíè÷åííîé ìàãíèòîäèýëåêòðè÷åñêîé ñðåäå ïðè èçìåíåíèè âî âðåìåíè å¸ ñâîéñòâ ïî çàêîíó ïðÿìî- óãîëüíûõ èìïóëüñîâ èíâàðèàíòíî îòíîñèòåëüíî èçìåíåíèé ïðîíèöàåìîñòåé, â òî âðåìÿ êàê àíàëî- ãè÷íîå âîçìóùåíèå ïàðàìåòðîâ â ñëó÷àå ïåðâè÷íîé ìîíîõðîìàòè÷åñêîé ýëåêòðîìàãíèòíîé âîëíû íå õàðàêòåðèçóåòñÿ îïèñàííîé âûøå èíâàðèàíòíîñ- òüþ. 4. Çàêëþ÷åíèå Ðàññìîòðåíî ïðåîáðàçîâàíèå ãàóññîâîãî ýëåêòðî- ìàãíèòíîãî èìïóëüñà ïðè èçìåíåíèè âî âðåìåíè ñâîéñòâ ïîëóîãðàíè÷åííîé ìàãíèòîäèýëåêòðè÷åñ- êîé îáëàñòè ïî çàêîíó ïðÿìîóãîëüíîãî èìïóëüñà. Íàó÷íàÿ íîâèçíà ïîëó÷åííûõ ðåçóëüòàòîâ ñîñòîèò â ñëåäóþùåì: 1.  ïðîñòðàíñòâå îáðàçóåòñÿ ïîëå ñëîæíîé ñòðóê- òóðû, êîòîðîå ïðåäñòàâèìî â âèäå ñóììû ïðÿìûõ è îáðàòíûõ ïëîñêèõ èìïóëüñîâ ñ ÷èñëîâûìè õàðàê- òåðèñòèêàìè ïåðâè÷íîé âîëíû. 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}<{3> |V'^Tx/(/0_z iGicXۢyOcZRI} u?VZ~/=ϙ1+O]:GԸ?a~5[gw)qϘ.qdϗJ3={r)SGȮ2:x'5^׭!nэ͝щ~`쉰5#]:iiK8薮9s`U==Դe?@߬'gtޘaSԳ4Qϙ*9zޭ,U}6{f(gY'~Hld\NhS{hh#_i[7)?8w8~@޿O+rWΖܒp͚8 ę,esا0yHgOM.V^oft>LBp0<;Hid&3L~>F^uy"JS&GNܣqqEǨ}|G3 +W$=m!nMޠv7 mkhNF4҇BW#oTĢ%h{[?Vhi/͛7]=})og^|meGT[9|%.+*U@0NSu7f 9ٿӝ&b'uZsb0xlw{R.T\5,^!L-ṽ>h[As?xr.hw {b{ś_L׵YXMxc=;Ji7VpKZJj8mVWRk^Iz%Gan[̡y; ^X[$t\ХE_ ^-Flmѐzs!7khXB6Ļ6jiÆۜyUXrRp˼Rpk{0x[o0x ;4zy/Z9鲚iso;Zik%aj76Mȭc ,o.PpCZe7[8+7$\Vp )͞Jʕ4f!+ʕ4[Vp1**^\Jn܃^5|r8f͊OweCy;{X_wO: 쨬x]*epCo]ܟ$o5 GC&-:oJnj~_'CSyW;B@Gy; ORx'&x{Ry(tAŠzXJ:-bXY[m^QnJ&]ǹe^|-dxy?Q6[rF}ŵ5FfVdF~:^J)_ G>CQB^m^k^B=7*+M}*Q Vxc[6Jm6D.sxkS|E_oqA-f_4{T?|Dl{} wua΍-vt8Nu눐{Pwx{I鄼nv;>#בVy ,g׭3G%}~>xmUź=ċ˔.&Fy+y2xs 8 ^Qf$_+7+TJ7WQ\ky}EFa/ jWJ g]5ܗu؞uLILɒ]pn2J>G HJ\+ar Ơuw>U+y xՉaDLռ"M^o8nmր-r*=g;ET%y ~֫w;Q@Z{elz  Qw0%x{< ޡ ޡ 4jy{؞훗{]|;0ERz-~{5Ǵ-oG*oBfKx]> Yr` U\Xo/sIWV!_@ ޞnvJgkC^IF`z݋o!J藣XNE'{ }QGo[ՖʻXFy9+z 庼y֎nO_,g՛MlW%%/ ]6 Y2՛ݞM>mUV vgsq\~E掠re$oq='WzQ]R.+!gE.1'y5*yKƳ \ͫ`CeZ毿jVԅA 9QM+^5y[nБ.W6 9і<ԡd[Pȉ2 FtԭQS^ﳖ潮Sf{'ϡz7Xz{!J.ZV6h6Qk%,ށ ^OR}q0%^ˍAm,];4:'Ҟ@mvcCΑU[Ctdqy5xKQM9ѻ= xƈBNyiT{o^si³o6K|si?zQJIiH xFyUWãw=mՀW -eQj"Uz2y\l/3*€Wg ]d1D[9Im`b&/z(mk3K3`q>vQ|>F?3`Ȣ| 猋i..rT.b$aM^aN"^w#AHN'xU?[%b?G]}%Hm^&6-ky6 Û N$j*,dm%HޚJ[SIux"S#՞:_5iwh&4Sjp )mVHM%a,MBT:8 y/g!oO//cH{+R7Qo-dTtzv;X&st-B)އ[|uފyQ1ʹrz*{Nzmūx*]mռݡ śqA}-l7[]o.fŪz_ռ'$ =reռWة)NU8(woyPޏ'y+Du$y{h »;(]YڼIފS:zՄ %-tߞPR)IPIy+ޚռ) j0c~?oIcREjj0EGGXwBűN<|^YʏԬwΎ &3Zp8Tqq;vAѹqs*Ot|HeG]P6]- > tB% d:UE!UsȻ_8+F E_ -=͉щΖyFKڸD=I,,7 ÐÈ, 2003, ¹ 4 2. Ïîñëå ñíÿòèÿ âîçìóùåíèÿ ñâîéñòâ ñðåäû àñèì- ïòîòè÷åñêè â êîíå÷íîé îáëàñòè ïðîñòðàíñòâà îñòà- þòñÿ ïåðâè÷íûé è îáðàòíûé åìó èìïóëüñû. 3. Ïðåîáðàçîâàíèå ïëîñêèõ ìîíîõðîìàòè÷åñêèõ ýëåêòðîìàãíèòíûõ âîëí â ïîëóîãðàíè÷åííîé ìàã- íèòîäèýëåêòðè÷åñêîé ñðåäå ïðè èçìåíåíèè âî âðå- ìåíè å¸ ñâîéñòâ ïî çàêîíó ïðÿìîóãîëüíûõ èìïóëü- ñîâ íå èíâàðèàíòíî, à ïðåîáðàçîâàíèå ãàóññîâîãî ýëåêòðîìàãíèòíîãî èìïóëüñà èíâàðèàíòíî îòíîñè- òåëüíî èçìåíåíèé ïðîíèöàåìîñòåé. Ëèòåðàòóðà: 1. Ñëèï÷åíêî Í.È., Øóëüãà Ë.Í., Ðûáèí O.H. Ïðåîáðàçîâàíèå ýëåêòðîìàãíèòíîãî èìïóëüñà âðåìåííûì âîçáóæäåíèåì ñðåäû â ïîëóïðîñòðàíñòâå / / Ðàäèîýëåêòðîíèêà è èíôîðìàòèêà. 1998. ¹1. C. 31-34. 2. Áîëîòîâñêèé Á.M., Ïëèñ À.È., Ñòîëÿðîâ C.H. Ðàñïðî- ñòðàíåíèå èìïóëüñíîãî èçëó÷åíèÿ â íåñòàöèîíàðíûõ ñðåäàõ // Èçâ. âóçîâ. Ðàäèîôèçèêà. 1976. T. 19, ¹4. C. 567-573. 3. Áîðèñîâ Â.Â. Òðàíñôîðìàöèÿ ýëåêòðîìàã- íèòíîãî ïîëÿ ïðè èçìåíåíèè ïðîâîäèìîñòè ñðåäû âî âðåìåíè // Ãåîìàãíåòèçì è àýðîíîìèÿ. 1989. T.29, ¹5. C. 730-737. 4. Àôàíàñüåâ C.Â. Èçëó÷åíèå ìîäåëè ñâåðõ- ñâåòîâîãî èñòî÷íèêà â íåñòàöèîíàðíîé ñðåäå // Èçâ. âóçîâ. Ðàäèîôèçèêà. 1991. T. 34,¹5. C. 605-607. 5. Áîëî- òîâñêèé Á.Ì., Äàâûäîâ Â.À., Ðîê Â.Å. Îá èçëó÷åíèè ýëåêòðîìàãíèòíûõ âîëí ïðè ìãíîâåííîì èçìåíåíèè ñîñòîÿíèÿ èçëó÷àþùåé ñèñòåìû // ÓÔÍ. 1978. Ò.126. Ñ. 311-321. 6. Íåðóõ À.Ã., Õèæíÿê Í.À. Ñîâðåìåííûå ïðî- áëåìû íåñòàöèîíàðíîé ìàêðîñêîïè÷åñêîé ýëåêòðî- äèíàìèêè. Õ.: ÍÏÎ Òåñò-Ðàäèî, 1991. 280 c. 7. Âëàäèìè- ðîâ Â.Ñ. Óðàâíåíèÿ ìàòåìàòè÷åñêîé ôèçèêè. Ì.: Íà- óêà, 1976. 527 c. Ïîñòóïèëà â ðåäêîëëåãèþ 12.07.2003 Ðåöåíçåíò: ä-ð òåõí. íàóê, ïðîô. Àñååâ Ã.Ã. Ñëèï÷åíêî Íèêîëàé Èâàíîâè÷, êàíä. òåõí. íàóê, äî- öåíò, ïðîðåêòîð ïî íàó÷íîé ðàáîòå ÕÍÓÐÝ. Íàó÷íûå èíòåðåñû: ðàäèîôèçèêà è ýëåêòðîíèêà. Àäðåñ: Óêðàè- íà, 61166, Õàðüêîâ, ïð. Ëåíèíà, 14, òåë. (0572) 40-90-20. Ðûáèí Îëåã Íèêîëàåâè÷, êàíä. ôèç.-ìàò. íàóê, ñòàð- øèé íàó÷íûé ñîòðóäíèê êàôåäðû ìèêðîýëåêòðîíèêè, ïîëóïðîâîäíèêîâûõ ïðèáîðîâ è óñòðîéñòâ ÕÍÓÐÝ. Íàó÷íûå èíòåðåñû: ìàòåìàòè÷åñêîå ìîäåëèðîâàíèå â ðàäèîôèçèêå. Àäðåñ: Óêðàèíà, 61166, Õàðüêîâ, ïð. Ëå- íèíà, 14, òåë. (0572) 40-93-62. Øóëüãà Ëàðèñà Íèêàíäðîâíà, àñïèðàíòêà êàôåäðû ìèêðîýëåêòðîíèêè, ïîëóïðîâîäíèêîâûõ ïðèáîðîâ è óñòðîéñòâ ÕÍÓÐÝ. Íàó÷íûå èíòåðåñû: ìàòåìàòè÷åñ- êîå ìîäåëèðîâàíèå â ðàäèîôèçèêå. Àäðåñ: Óêðàèíà, 61166, Õàðüêîâ, ïð. Ëåíèíà, 14, òåë. (0572) 40-93-62. Íîâèêîâ Íèêîëàé Èâàíîâè÷, êàíä. âîåííûõ íàóê, 1-é çàìåñòèòåëü íà÷àëüíèêà ôàêóëüòåòà ïî ó÷åáíîé è íàó÷íîé ðàáîòå ÕÈÂÂÑ. Íàó÷íûå èíòåðåñû: ðàäèî- ôèçèêà è ýëåêòðîíèêà. Àäðåñ: Óêðàèíà, 61000, Õàðü- êîâ, óë. Êëî÷êîâñêàÿ, 228, òåë. 30-82-84. ÓÄÊ 621.371.3 ÐÀÑÑÅßÍÈÅ ÝËÅÊÒÐÎÌÀÃÍÈÒÍÛÕ ÂÎËÍ ÑÈÑÒÅÌÎÉ ÏËÎÑÊÈÕ ÐÅØÅÒÎÊ ÐÅÇÎÍÀÍÑÍÛÕ ÌÀÃÍÈÒÎÄÈÝËÅÊÒÐÈ×ÅÑÊÈÕ ÑÔÅÐ Ñ ÒÎÏÎËÎÃÈ×ÅÑÊÎÉ ÑÒÐÓÊÒÓÐÎÉ, ÇÀÄÀÂÀÅÌÎÉ ÃÅÎÌÅÒÐÈ×ÅÑÊÎÉ ÏÐÎÃÐÅÑÑÈÅÉ ÊÎÇÀÐÜ À.È. Îïèñûâàåòñÿ ðåøåíèå çàäà÷è î ðàññåÿíèè ýëåêòðî- ìàãíèòíûõ âîëí ñèñòåìîé ïëîñêèõ ðåøåòîê ðåçîíàí- ñíûõ ìàãíèòîäèýëåêòðè÷åñêèõ ñôåð, ïðîñòðàíñòâåí- íîå ðàñïðåäåëåíèå êîòîðûõ ïîä÷èíåíî ãåîìåòðè÷åñ- êîé ïðîãðåññèè. Çàäà÷à ðåøàåòñÿ ñ ïîìîùüþ èíòåã- ðàëüíûõ óðàâíåíèé ýëåêòðîäèíàìèêè. Ïðèâîäÿòñÿ ôîðìóëû äëÿ âíóòðåííèõ ïîëåé ñôåð è ðàññåÿííûõ ïîëåé. Ââåäåíèå Öåëüþ ðàáîòû ÿâëÿåòñÿ ðåøåíèå çàäà÷è î ðàññåÿíèè ýëåêòðîìàãíèòíûõ âîëí ñèñòåìîé ïëîñêèõ îäíî- ñëîéíûõ ðåøåòîê, ïîñòðîåííûõ èç ìàëûõ îäíîðîä- íûõ ðåçîíàíñíûõ ìàãíèòîäèýëåêòðè÷åñêèõ ñôåð [1]. Èõ ïðîñòðàíñòâåííîå ðàñïðåäåëåíèå ïîä÷èíåíî ãåîìåòðè÷åñêîé ïðîãðåññèè, îáùèé âèä ÷ëåíà êî- òîðîé äëÿ ñëó÷àÿ, êîãäà ïåðâûé ÷ëåí ïðîãðåññèè 1 a1  , ïðåäñòàâèì â âèäå n1|t| n10 aaq(q1)   , (1) ãäå çíàìåíàòåëü ïðîãðåññèè – 0 qq1;  q0,1,2,3,;n|t|1;|t|0,1,2,3,   .  çàäà÷å äëèíà ðàññåèâàåìîé âîëíû ìîæåò áûòü ñîèçìåðèìà ñ ïîñòîÿííûìè ðåøåòêè, ÷òî ïîçâîëÿåò èçó÷èòü âëèÿíèå ðåøåòî÷íûõ ñòðóêòóðíûõ ðåçî- íàíñîâ ýëåêòðîìàãíèòíîãî âçàèìîäåéñòâèÿ ñôåð íà âíóòðåííèå ðåçîíàíñû ñôåð ðåøåòêè è èõ òîíêóþ ñòðóêòóðó. Ýòî ðåøåíèå îïèñûâàåò îáëàñòè àíî- ìàëüíîé äèñïåðñèè ðåøåòîê. Ïîñòàíîâêà è ðåøåíèå çàäà÷è Ðàññìîòðèì â äåêàðòîâîé ñèñòåìå êîîðäèíàò ïî- ðîæäàþùåå ïðîñòðàíñòâåííóþ ñèñòåìó ïðîñòûõ ïëîñêèõ îäíîñëîéíûõ ðåøåòîê êîîðäèíàòíîå ïðåä- ñòàâëåíèå (ðèñ. 1) ss1 p,sp,s0 |t| tt1 p,tp,t0 p0p0 x[s{(1)1}0.5]d(1)x (s0,1,2,...,[(q1)1]), y[t{(1)1}0.5]h(1)y (t0,1,2,), zzlzpl (p0,1,2,),             (2) ãäå âåëè÷èíû d, h îïðåäåëÿþòñÿ óñëîâèÿìè x0,xd;y0,yh,  à p,s0p,t0p x,y,z  – êîîð- äèíàòû óçëîâ, íàõîäÿùèõ- ñÿ âíóòðè îáëàñòè: p,s0 p,t0 p 0xd, 0yh, zzz.      (3) Ðèñ. 1 ;x{pU{n Bx&\PhxD@bBGmL@iH V>iJy(/ d(DˣbE*XA-"2"&wÝB̗ۙﵿs`~":A/||&4iL"k_M;ɗ)F.2Ȫi契hNRHVrB+7'I7AZɢHN$׾7̏mM1i&&L_rDdsKc9Ǫx^Cb7]83V1;޲*Nc_Gɲ%4)O[dEՇO;Fwd:YKt##(mq wz9b/L7ݡ턈mߎىkjS&)|ж.sso檏wrJ VH[c7ByɶXNk1uܻRՅqk֣3Ȫpz)S˜&wU^s6|e[8Nݣ݉j `7 sKU~>^pGc+XMd}Bss⛥^u B BSp!y؛-_|u='S$=.me&yˑÕ9rbh)ps߅8 k ٥& 6rgX?&iٛ:l+YGA^=Z֬{@s|qR󯇩U=^||?kفX'q^*ϿYs۰tbNIc,)Y 1?|5'#]A&Hq7ѿLϽ1ށsw/=W22O>CS]_WI as͔D.3/_X˭g84>̔DysE%DI#~w#KU_1^$ha># V7ʱ~,/C>rdu38n3EUu}C; U MHŮ/骞lFfh1p%9{G t_Ʀ1FSԳbǢ+; 'TMe9FYU^jZ"ϔC=灁` xA:|X}w9Eoς5Y_roRHa0f cSΫFx o U}\n9mc`?pY"_ʚ rx \D(1X rSaB& X?w/J"7F&9m&ۛT;$RUWeۏ2K-]p9V+ v1H= ׼x]p[ִ?6>Ǧp --p%vr:)`|ՓkS!RwV$l-/ޫ]ygΤ9{}6~t{5C Gw#Cy_PXpJxW؊Q ' ͷq;_x%kG7sj+Չ<;YSAdg3_S2ova0ֺ(%0TҐ 2rXG< ٣aם94|y ;-ez^J 6[-GDc-uvuZK}Z+DSbvZjÇ+;:ÿq6lwz-W׺ dWO} Gy|.^>\ŘQ D>yYUSTmUJfgTW#¿\{ewk*s$dxe;9|/Xx7|oڵjMcզAf6io^;:ޯ.c_]P̺|.UcԖ?(~]>w &uū+x~ČgRL~]|\k}/OPAvA23nP q}؆34fm:2q܅U߉i6msGf}Ov}Wm̴ c̻ɴ.^+͵j;uAKA{ ?BM/v7mܗ~WV @Xa +V @Xa +\? 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Êàæäîìó óçëó ðåøåòîê (2) ñîïîñòàâëÿåòñÿ óïîðÿäî÷åííàÿ òðîéêà ÷èñåë c(p,s,t)  , âûäåëåííûé óçåë ðåøåòîê áóäåì îáî- çíà÷àòü c'(p',s',t')  . Çíà÷åíèå èíäåêñà ð îïðåäåëÿåò íîìåð ïðîñòîé ïëîñ- êîé ðåøåòêè (ðèñ.2), à çíà÷åíèÿ èíäåêñîâ s, t çàäàþò ïîëîæåíèå óçëà ð-é ïðîñòîé ïëîñêîé ðåøåòêè. Íóæíûé òèï ýëåìåíòàðíîé ÿ÷åéêè ñëîæíîé ïëîñ- êîé îäíîñëîéíîé ðåøåòêè ôîðìèðóþò èç ð ïîðîæ- äàþùèõ ïðîñòûå ïëîñêèå ðåøåòêè óçëîâ âíóòðè îáëàñòè (3), êîòîðóþ ïîâòîðèò çà åå ïðåäåëàìè êîîðäèíàòíîå ïðåäñòàâëåíèå (2) â âèäå îïðåäåëåí- íîé ñëîæíîé ïëîñêîé îäíîñëîéíîé ðåøåòêè. Ðèñ. 2 Çàäàâàÿ ìàêñèìàëüíûå çíà÷åíèÿ ÷èñåë p,s,t , ìîæíî ðàññìàòðèâàòü êîíå÷íûå è áåñêîíå÷íûå ðåøåòêè. Íà ðèñ. 2 èçîáðàæåíà ñèñòåìà ïðîñòûõ ïëîñêèõ îäíîñëîéíûõ ðåøåòîê, êîãäà ïîðîæäàþùèé óçåë ðåøåòêè íàõîäèòñÿ â öåíòðå îáëàñòè (3), äëÿ ñëó÷àÿ – q=1; ð=0, 1, 2; t=0,±1,...,±5. Íà ðèñ. 3 ïðåäñòàâëåíà ñèñòåìà ðåøåòîê, êîãäà ïîðîæäàþùèé óçåë ðåøåòêè íàõîäèòñÿ â öåíòðå îáëàñòè (3), äëÿ ñëó÷àåâ – q=0, p=0, t =0,±1,...,±5 è q=1, p=1; q =2, p=2; q=3, p=3; q=4, p=4 ïðè t=0, ±1, ±2. Ñëó÷àé q=0, p=0, t=0,±1,...,±5 ñîîòâåòñòâó- åò ëèíåéíîé öåïî÷êå óçëîâ. Ðàñïðåäåëåíèå ñôåð â ïëîñêîñòè ðåøåòêè âäîëü îñè õ ïîä÷èíåíî ÷ëåíàì ïðîãðåññèè (1). Êàæäîé òðîéêå ÷èñåë   p,s0,t  ïëîñêîñòè p,tp y,z ñòàâèòñÿ â ñîîòâåòñòâèå ÷ëåí ïðîãðåññèè, íàïðèìåð, ïðè 1  q òî÷êå   p0,s0,t5  – ÷èñëî 32, à òî÷êå   p2,s0,t3  – ÷èñëî 8 (ðèñ. 2). Ðèñ. 3 Åñëè q =const, òî ìîæíî èçó÷àòü ñèñòåìó îäèíàêî- âûõ ïëîñêèõ ðåøåòîê (ñì. ðèñ. 2). Ìåæäó òðîéêîé ÷èñåë   p,s0,t  ïëîñêîñòè p,tp y,z è ÷ëåíàìè ïðîãðåññèè ñóùåñòâóåò ñâÿçü, îïðåäåëÿ- åìàÿ ôóíêöèåé (1): |t| (q1)  . Òîãäà ÷èñëà s, îïðåäåëÿþùèå êîîðäèíàòû óçëîâ âäîëü îñè õ (2), çàäàþòñÿ ïîñëåäîâàòåëüíîñòüþ |t| 0,1,2,3,,[(q1)1]   . Åñëè èçìåíÿòü êîîðäèíàòû óçëîâ, íàõîäÿùèõñÿ â ïðåäåëàõ îáëàñòè (3), òî, â ñîîòâåòñòâèè ñ êîîðäè- íàòíûì ïðåäñòàâëåíèåì (2), ïîëîæåíèÿ óçëîâ ðå- øåòîê âíå îáëàñòè (3) áóäóò òàêæå ñîîòâåòñòâóþ- ùèì îáðàçîì ñìåùàòüñÿ, ÷òî ïîçâîëÿåò ïåðåñòðàè- âàòü ïðîñòðàíñòâåííóþ êîíôèãóðàöèþ ðåøåòîê. Ïîëàãàåì, ÷òî â óçëû ðåøåòîê (2) ïîìåùàþòñÿ öåíòðû N= p p N  ìàëûõ îäíîðîäíûõ ðåçîíàíñíûõ ìàãíèòîäèýëåêòðè÷åñêèõ ñôåð ñ ïðîíèöàåìîñòÿìè cc ,  è ðàäèóñàìè c a(cN)  , ãäå p N – ÷èñëî ñôåð ð-é ïëîñêîé ðåøåòêè. Ïðîíèöàåìîñòè çàïîëíåíèÿ ïðîñòðàíñòâà âíå ñôåð – 00 ,  . Ïîëÿ áóäåì çàïèñûâàòü â âèäå it E(r,t)E(r)e     , it H(r,t)H(r)e     . Ðàññòîÿíèå ìåæäó öåíòðàìè ñôåð c è c  ïðåäñòà- âèì (2) 222 ccp,sp,sp,tp,tpp r(xx)(yy)(zz).   (4) Ñ÷èòàåì, ÷òî âíå ñôåð a/1  , íî âíóòðè ñôåð âîçìîæåí ðåçîíàíñíûé ñëó÷àé g a/~1  , ãäå  – äëèíà âîëíû âíå ñôåð, à g  – äëèíà âîëíû â ñôåðå. Äëÿ ðåøåíèÿ çàäà÷è èñïîëüçóåì èíòåãðàëüíûå óðàâíåíèÿ [2] è ðåøàòü åå áóäåì â äâà ýòàïà. Íà ïåðâîì ýòàïå íàéäåì âíóòðåííåå ïîëå ðàññåèâàþ- ùèõ ñôåð, à íà âòîðîì – ïîëå, ðàññåÿííîå ïðîñòðàí- ñòâåííîé ðåøåòêîé ñôåð. 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Åñëè ó÷åñòü, ÷òî äëÿ âíåøíèõ òî÷åê ñôåðû (rr)   èíòåãðàë ïî îáúåìó ñôåðû îò ôóíêöèè Ãðèíà äëÿ ñâîáîäíîãî ïðîñòðàíñòâà (7) èìååò âèä 00 1 ikrr V ikr 111 3 1 e W(r)dV rr 4e (sinkakacoska), r k             (8) ãäå 100 kk,k2/  , r – ðàññòîÿíèå îò öåíòðà äî âíåøíèõ òî÷åê ñôåðû, òî ìîæíî ïîñòðîèòü óðàâíåíèÿ äëÿ âíóòðåííèõ ïîëåé c  ñôåðû, èñ- ïîëüçóÿ ðåøåíèÿ, ïðåäëîæåííûå â [3,4], â âèäå     1с' 0(p,s,t) 2 c' эф01c'c'эф1c'c'эф0 0 (p,s,t) i 0 cэф 2 э0 00cc(p,s,t) pst 0 (p,s,t)(p,s,t) cэф 0 0 E(r,t) 2i2 E(r,t) 3e 1 (k)1W(r)E(r,t) 4 1 ik,1 4                                       м0 cc(p,s,t) W(r)H(r,t),                  1с' 0(p,s,t) 2 c' эф01c'c'эф1c'c'эф0 0 (p,s,t) i 0 cэф 2 м0 00cc(p,s,t) pst 0 (p,s,t)(p,s,t) cэф 0 0 H(r,t) 2i2 H(r,t) 3e 1 (k)1W(r)H(r,t) 4 1 ik,1 4                                       э0 cc(p,s,t) W(r)E(r,t),              ãäå 0(p,s,t)0(p,s,t) E(r,t),H(r,t)     è 00 (p,s,t)(p,s,t) E(r,t),H(r,t)    – ïîëå ïàäàþùåé âîëíû â ñôåðå è âíóòðåííåå ïîëå ñôåð ñîîòâåòñòâåííî, 222 1c'c'00 ka.  Âåëè÷èíû эм cccc W(r),W(r)   èìåþò âèä (4), (8), (9): 1cc 1cc ikr э cc1c1c1c 3 cc 1 ikr м cc1c1c1c 3 cc 1 4e W(r)(sinkakacoska), r k 4e W(r)(sinkakacoska), r k               à âåëè÷èíû сэфсэф ,  çàïèøåì [4,5,6] сэфcccc сэфcccc F(ka), F(ka),   (11) ãäå ccc ccccccccc 22 cccccccccccc F(ka) 2(sinkakacoska) . (ka1)sinkakacoska     Íà ðèñ. 4 ïðåäñòàâëåíû îñîáåííîñòè ïîâåäåíèÿ ReF()  (ñïëîøíàÿ êðèâàÿ) è ImF()  (ïóíêòèðíàÿ êðèâàÿ) â çàâèñèìîñòè îò  Re ïðè ðàçíûõ çíà÷å- íèÿõ òàíãåíñà óãëà äèýëåêòðè÷åñêèõ ïîòåðü tg   (êðèâàÿ 1 –   tg =0; êðèâàÿ 2 –   tg =0,05; êðèâàÿ 3 –   tg =0,1) è c 1  , çäåñü ccc ka  . Ïåðâûå ñëàãàåìûå ñïðàâà â óðàâíåíèÿõ (9) ñâÿçàíû ñ âíóòðåííèì ïîëåì âûäåëåííîé ñôåðû, à âòîðûå ñëàãàåìûå ó÷èòûâàþò âëèÿíèå íà âûäåëåííóþ ñôåðó âñåõ îñòàëüíûõ ñôåð. Äëÿ îïðåäåëåíèÿ âíóòðåííåãî ïîëÿ, èìååì àëãåáðà- è÷åñêóþ ñèñòåìó 2N âåêòîðíûõ íåîäíîðîäíûõ óðàâíåíèé (9) èëè æå äëÿ x,y,z  ñîñòàâëÿþùèõ 6N óðàâíåíèé ñ 6N íåèçâåñòíûìè. Îñíîâíàÿ ìàòðèöà àëãåáðàè÷åñêîé ñèñòåìû óðàâíå- íèé (9) ñîäåðæèò èíôîðìàöèþ î õàðàêòåðå ýëåêò- ðîìàãíèòíîãî âçàèìîäåéñòâèÿ ìåæäó ñôåðàìè äàí- íîãî òèïà ðåøåòîê. Äëÿ ñôåðû c' ðåøåíèå ñèñòåìû óðàâíåíèé (9) èìååò âèä: 0 (p,s,t) эсэс c0(p,s,t)c0(p,s,t) эм c E(r,t) 1 (gE(r,t)H(r,t)),               (12) 0 (p,s,t) мсмс c0(p,s,t)c0(p,s,t) эм c H(r,t) 1 (H(r,t)gE(r,t)),               ãäå эм  – äåòåðìèíàíò îñíîâíîé ìàòðèöû ñèñòåìû óðàâíåíèé (9). Åñëè ïðåäïîëîæèòü, ÷òî ó âñåõ ñôåð ñ îäèíàêîâûì èíäåêñîì p âíóòðåííåå ïîëå ðàâíî, íàïðèìåð, âíóòðåííåìó ïîëþ ñôåðû (p,s,0,t0)  , òî ñèñòåìó óðàâíåíèé (9) ìîæíî ñâåñòè ê ñèñòåìå   22p1  óðàâíåíèé.  ñëó÷àå, êîãäà âñå ñôåðû ðåøåòêè (9) (10) BIx tEo}%!$kX$ ;AM@YGETpJB#[l*jb}7fΙ=%[u֭UݪsD"R*HKx6'Z$>I*DUBk iE8K X^0X-pETqIȫ*^ g~ttx*`\ 8GDgi}xf[`:arxN]&Ui#W*[0U[xV' %=t3*ndn8I2D^I^&(˽JN)c y g押ST"n[))dɣ暞4,sk+G(yR >sqky7Q v˺:Lx飲 DZaD6Β%KjKK▓*9\:*|'v?`>r:eK~6[/=%usd.&xM]`TY ~NzvԢ|P_TtUg3rl&mPCn܏ݣMe*|& E]iK2=SꐟaNKs5LuYOO/)nRD :]Z7LrI7/MiIk. 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Åñëè ýëåêòðîìàãíèòíûì âçàèìîäåéñòâèåì ìåæäó ñôåðàìè ðåøåòêè ìîæíî ïðåíåáðå÷ü, òî âûðàæåíèÿ äëÿ âíóòðåííåãî ïîëÿ ïðîèçâîëüíîé ñôåðû ðåøåò- êè (12) áóäóò èìåòü âèä 1 i 0 0 coc 2 эф01эф1эф0 3e Е(r,t)Е(r,t), (2)i(2)       1 i 0 0 coc 2 эф01эф1эф0 3e Н(r,t)Н(r,t), (2)i(2)       . Ïîòåíöèàëû Ãåðöà э П  è м П  ðàññåÿííîãî ñèñòåìîé ïëîñêèõ ðåøåòîê ïîëÿ ïî èçâåñòíîìó âíóòðåííåìó ïîëþ (12) îòäåëüíûõ ðàññåèâàòåëåé ïðåäñòàâèì â âèäå ñóïåðïîçèöèè ïîòåíöèàëîâ Ãåðöà îòäåëüíûõ ñôåð ðåøåòîê (6): 1(p,s,t) 1(p,s,t) э 1c1c1c 3 pst 1 ikr cэф 0 c 0(p,s,t) м 1c1c1c 3 pst 1 ikr cэф 0 c 0(p,s,t) 1 П(r,t)(sinkakacoska) k e 1E(r,t), r 1 П(r,t)(sinkakacoska) k e 1H(r,t). r                               (13)  âûðàæåíèÿõ (13) 222 (p,s,t)p,sp,tp r(xx)(yy)(zz),  (14) ãäå êîîðäèíàòû (x,y,z) îïðåäåëÿþò âíå ñôåð òî÷êó íàáëþäåíèÿ ïîëÿ, ðàññåÿííîãî ñèñòåìîé ïëîñêèõ ðåøåòîê, à êîîðäèíàòû p,sp,tp (x,y,z) – òî÷êó íàõîæäåíèÿ öåíòðà ðàññåèâàþùåé ñôåðû ðåøåòêè. Òîãäà, ó÷èòûâàÿ (13), èç (5) íàéäåì ðàññåÿííîå ñèñòåìîé ïëîñêèõ ðåøåòîê ïîëå 1c расс1c1c1c 3 c 1 c эфсэф 00 cc0cc 00 i(tkr) расс1c1c1c 3 c 1 сэф 0 cc 0 1 E(r,t)(sinkakacoska) k 1LE(r)ik1(1)PH(r) e, 1 H(r,t)(sinkakacoska) k 1(1)LH                                   1c сэф 0 0cc 0 i(tkr) (r)ik1PE(r) e,                  ãäå c L  è c P  – ôóíêöèîíàëüíûå ìàòðèöû, êîòîðûå èìåþò âèä 0 zcyc xxcxycxzc 0 cyxcyycyzcczcxc 0 zxczyczzc ycxc 0 L ; P0 . 0                    (16) Âåëè÷èíû, âõîäÿùèå â ôóíêöèîíàëüíûå ìàòðèöû (16), çàïèøåì â âèäå (2), (14): Ðèñ. 4 (15) 8&xX pVv/y IT`BXP H  ha@F-:`""h((e A^rw#8:|9g=sD/1 LJq<hN40͚~?)dx<0/mt +zaix,H9z8Hs7-:"kLQY%\;i: @9Yv`\Jn- T½˙; U/p-09r2w\NnXg;8-qn2ܻE:nS!>7m)߈fȠoZnJcQ}P,ƋR||k)AYul'rݕj DID|C*~~_\F9诧eԫur@|w|6gxD7&n* s^ΧF!6A)8DL|;:_N;+}jS9D[T-nc鄫v66PGP4` 2^_qa"dEqϳ 5QuCjJvZMFy9yxA':T MC(gu;\9.4YskdWyB,;ж\bgh Bo5Nj$uh[~I3<5ȕ4Q.21Ĭ9&<1>x- d\䨭5~kw[ggpXH=acy{j7S,ZܢB{s tE7{6nh8vg'}0Zr=p9$%D q(:kJiqs%oH rU|*J~u` Z'I_8AT"Q lK=*ЇsDO/$Rz}~NBtd)M:Ǥ3a po6 [(0[9'!ėFw9ɶ"uϦ/9@w"DrD}6 T{Q{|OA{ $≺-4=iOvNb; vWо aY[[[~ꊫDWI$On{)!gO8[?+sg&hϭuo!\͸3geh!FOdrl:){LchuSLR/WlDw yJOd'F]#AƫdBǧ.gt ޿ob=Vsȣ䖛hL+eq;LSkb3 wk& ,gZ`rhˬd8ǧ]kS$1^ӔyIB5k=k-z>x(7X]ufHlKxĽ(֞Pt_S +ݕr#20ИіsbTo$9{-cw2Mqz(cXXbnQy{YifZěS3Dzz:mcECp^%gy( :9Du\8gg!^aY&u4a̓dP:t|>wTj& ؽ!ڗ%#4+ Ko6gݕDn{u&}S&[~K{om\mqu%%= ʔlDT|ᨨcss8{朹3]X~Xߵ^{}- `Y mtQ=<AnAI0w%>A_МBvG -. 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Èç äåòåðìèíàíòà óðàâíåíèé (9) íàõîäèì ðåçîíàí- ñíûå óñëîâèÿ äëÿ ñëó÷àÿ, êîãäà cg a/~1  âíóòðè ñôåð. Åñëè cc ,  ñôåð ðåøåòîê äåéñòâèòåëüíû, òî ðåçîíàíñíûå óñëîâèÿ íàõîäèì èç âûðàæåíèÿ [6]: i detRe0 j  , ðàçðåøàÿ åãî îòíîñèòåëüíî ôóíêöèè ccc F(ka)  (11) (ñì. ðèñ. 4), ãäå i j  – îñíîâíàÿ ìàòðèöà ñèñòåìû óðàâíåíèé (9). Çàêëþ÷åíèå Âïåðâûå ðàññìîòðåíî ðàññåÿíèå ýëåêòðîìàãíèòíûõ âîëí íà ñèñòåìå ïëîñêèõ ðåøåòîê ðåçîíàíñíûõ ñôåð, ïðîñòðàíñòâåííàÿ ñòðóêòóðà êîòîðûõ îïðåäå- ëÿåòñÿ ãåîìåòðè÷åñêîé ïðîãðåññèåé. Ðåçóëüòàòû ðåøåíèÿ äàííîé çàäà÷è ìîãóò áûòü ïîëåçíû äëÿ ñîçäàíèÿ ðàçëè÷íîãî ðîäà óñòðîéñòâ ïî óïðàâëåíèþ ïîëåì èçëó÷åíèÿ ýëåêòðîìàãíèòíûõ èçëó÷àòåëåé [7], â êîòîðûõ ìîãóò èñïîëüçîâàòüñÿ òîíêèå ýô- ôåêòû âçàèìîäåéñòâèÿ ðåøåòî÷íûõ ñòðóêòóðíûõ ðåçîíàíñîâ è âíóòðåííèõ ðåçîíàíñîâ ñôåð ðåøåòêè. Ëèòåðàòóðà: 1. Êîçàðü À.È., Õèæíÿê Í.À. Îòðàæåíèå ýëåêòðîìàãíèòíûõ âîëí îò ðåçîíàíñíîé äèýëåêòðè- ÷åñêîé ñôåðû â âîëíîâîäå // Óêð. ôèç. æóðí. 1970. Ò. 15. Ñ. 847-849. 2. Õèæíÿê Í.À. Èíòåãðàëüíûå óðàâíå- íèÿ ìàêðîñêîïè÷åñêîé ýëåêòðîäèíàìèêè. Êèåâ: Íàóê. äóìêà. 1986. 279 ñ. 3. Êîçàðü À.È. Ðàññåÿíèå ýëåêòðîìàã- íèòíûõ âîëí ñèñòåìîé ïëîñêèõ îäíîñëîéíûõ ðåøåòîê ðåçîíàíñíûõ ìàãíèòîäèýëåêòðè÷åñêèõ ñôåð // Ðàäèî- òåõíèêà. 2002. Âûï. 130. Ñ. 42–53. 4. Ëåâèí Ë. Ñîâðåìåí- íàÿ òåîðèÿ âîëíîâîäîâ. Ì.: Èçä-âî èíîñòð. ëèò. 1954. 216 ñ. 5. Êîçàðü À.È. Ðàññåÿíèå ýëåêòðîìàãíèòíûõ âîëí â âîëíîâîäå ñ îäíîðîäíûìè ìàãíèòîäèýëåêòðè- ÷åñêèìè ñôåðàìè // Ðàäèîôèçèêà è ýëåêòðîíèêà. Õàðü- êîâ: Èí-ò ðàäèîôèçèêè è ýëåêòðîíèêè ÍÀÍ Óêðàèíû. 2002. 7. Ñïåö. âûïóñê. Ñ. 183-189. Ïîñòóïèëà â ðåäêîëëåãèþ 31.03.2003 Ðåöåíçåíò: ä-ð òåõí. íàóê, ïðîô. Êðèâóëÿ Ã.Ô. Êîçàðü Àíàòîëèé Èâàíîâè÷, êàíä. ôèç.-ìàò. íàóê, ïðî- ôåññîð êàôåäðû ôèçèêè ÕÍÓÐÝ. Íàó÷íûå èíòåðåñû: ðàäèîôèçèêà. 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+V*SYEqdz[BY%8.n冲HGSH"AG=bl,,12 ÐÈ, 2003, ¹ 4 ÓÄÊ 537.87:621.371 ×ÈÑËÅÍÍÎÅ ÈÑÑËÅÄÎÂÀÍÈÅ ÂÇÀÈÌÎÄÅÉÑÒÂÈß ÝËÅÊÒÐÎÌÀÃÍÈÒÍÎÃÎ ÏÎËß Ñ ÍÅÑÒÀÖÈÎÍÀÐÍÛÌ ÄÈÝËÅÊÒÐÈ×ÅÑÊÈÌ ÑËÎÅÌ ÏÐÈ ÏÎÌÎÙÈ ÌÅÒÎÄÀ ÈÍÒÅÃÐÀËÜÍÛÕ ÓÐÀÂÍÅÍÈÉ ÂÎËÜÒÅÐÐÀ ÔÅÄÎÒΠÔ.Â., ÍÅÐÓÕ À.Ã. Ðàññìàòðèâàåòñÿ ÷èñëåííîå èññëåäîâàíèå âçàèìîäåé- ñòâèÿ ýëåêòðîìàãíèòíîãî ïîëÿ ñ íåñòàöèîíàðíûì äèýëåêòðè÷åñêèì ñëîåì ïðè ïîìîùè ìåòîäà èíòåã- ðàëüíûõ óðàâíåíèé Âîëüòåððà (ÌÈÓÂ). Ðàçðàáàòûâà- åòñÿ ñïåöèàëüíîå ïðîãðàììíîå îáåñïå÷åíèå äëÿ àâòî- ìàòèçàöèè ìîäåëèðîâàíèÿ òàêèõ âçàèìîäåéñòâèé íà îñíîâå ðàññìîòðåííîãî ìåòîäà. 1. Ââåäåíèå Ïàðàìåòðè÷åñêèå ÿâëåíèÿ â àêòèâíûõ ñðåäàõ âû- çûâàþò èíòåðåñ â ñâÿçè ñ âîçìîæíîñòüþ ãåíåðàöèè, óñèëåíèÿ è ïðåîáðàçîâàíèÿ ýëåêòðîìàãíèòíûõ âîëí è èìåþò âàæíîå çíà÷åíèå â ñîâðåìåííîé ðàäèîòåõ- íèêå è îïòîýëåêòðîíèêå [1,2]. Èñïîëüçîâàíèå ìå- òîäîâ, îñíîâàííûõ íà ïðåîáðàçîâàíèÿõ Ôóðüå, ìàëîýôôåêòèâíî è çàòðóäíèòåëüíî äëÿ äàííîãî êëàññà çàäà÷ âñëåäñòâèå øèðîêîãî ñïåêòðà èññëå- äóåìûõ ñèãíàëîâ è çàâèñèìîñòè ïàðàìåòðîâ ñðåäû îò âðåìåíè [3].  íàñòîÿùåå âðåìÿ íàèáîëåå ïîïóëÿðíûì ÷èñëåí- íûì ìåòîäîì äëÿ èññëåäîâàíèÿ ýëåêòðîäèíàìè÷åñ- êèõ çàäà÷ âî âðåìåííîé îáëàñòè ÿâëÿåòñÿ ìåòîä êîíå÷íûõ ðàçíîñòåé (FDTD)[4]. Îäíàêî ïðè ðåøå- íèè çàäà÷ ñ ïåðåìåííûìè âî âðåìåíè ïàðàìåòðàìè ñðåäû äàííûì ìåòîäîì âîçíèêàþò ïðîáëåìû ñî ñòàáèëüíîñòüþ ðåøåíèÿ [5,6].  äàííîé ñòàòüå ïðåäëàãàåòñÿ ìåòîä, îñíîâàííûé íà èíòåãðàëüíûõ óðàâíåíèÿõ Âîëüòåððà âî âðåìåííîé îáëàñòè (ÌÈÓÂ) [3,5], êîòîðûå ýêâèâàëåíòíû ýëåê- òðîäèíàìè÷åñêèì óðàâíåíèÿì Ìàêñâåëëà. Îòëè- ÷èòåëüíîé îñîáåííîñòüþ ÌÈÓ ÿâëÿåòñÿ òî, ÷òî íà÷àëüíûå è ãðàíè÷íûå óñëîâèÿ íåÿâíî ó÷èòûâà- þòñÿ â èíòåãðàëüíîì óðàâíåíèè çàäà÷è [5]. Äàííûé ïîäõîä äëÿ ðÿäà çàäà÷ ïðåâîñõîäèò ïî òî÷íîñòè ìåòîä FDTD [6] è îáëàäàåò âûñîêîé ñòàáèëüíîñòüþ ïðè ìîäåëèðîâàíèè íåñòàöèîíàðíûõ ñðåä. 2. Ïîñòàíîâêà çàäà÷è Ðàññìîòðèì îäíîðîäíóþ íåîãðàíè÷åííóþ ñðåäó ñ îòíîñèòåëüíîé äèýëåêòðè÷åñêîé ïðîíèöàåìîñòüþ  , â êîòîðîé â îïðåäåëåííûé íà÷àëüíûé ìîìåíò âðåìåíè 0 t0  îáðàçóåòñÿ íåñòàöèîíàðíûé ñëîé â îáëàñòè 0xL  .  ìîìåíò ñîçäàíèÿ ñëîÿ â ôîíî- âîé ñðåäå, ïåðïåíäèêóëÿðíî ê âîçíèêàþùåìó ñëîþ, ðàñïðîñòðàíÿåòñÿ ýëåêòðîìàãíèòíûé ñèãíàë, çà- äàííûé ôóíêöèåé   0 Et,x .  îáùåì ñëó÷àå ñðåäà â ñëîå îïèñûâàåòñÿ ôóíêöèåé ïîëÿðèçàöèè   Pt,x , ïàðàìåòðû êîòîðîé, ïîä âîçäåéñòâèåì ñòîðîííèõ ôàêòîðîâ, èçìåíÿþòñÿ âî âðåìåíè ñî ñêîðîñòüþ, ñîèçìåðèìîé ñî ñêîðîñòüþ èçìåíåíèÿ ýëåêòðè÷åñ- êîãî ïîëÿ. Ýëåêòðè÷åñêîå ïîëå äëÿ äàííîé çàäà÷è âíóòðè è âíå ñëîÿ îïèñûâàåòñÿ ñëåäóþùèì óðàâíå- íèåì Âîëüòåððà II ðîäà [5]:         tL 0 00 0 1 Et,xEt,xdt'dx' 2vt xx' 11 tt'Pt',x'Et',x' v              ãäå vc/  , c – ñêîðîñòü ñâåòà â âàêóóìå;   t  – äåëüòà – ôóíêöèÿ Äèðàêà; L – ãëóáèíà ñëîÿ; 0  – äèýëåêòðè÷åñêàÿ ïîñòîÿííàÿ. Äëÿ ðåàëèçàöèè ÷èñëåííîãî ñ÷åòà ïåðåéäåì ê áåçðàçìåðíûì êîîð- äèíàòàì, íîðìèðîâàííûì íà ãëóáèíó ñëîÿ: vt/L  , x/L  . Òîãäà ïîñëå âû÷èñëåíèÿ âíóòðåííåãî èíòåãðàëà ïî ïðîñòðàíñòâåííîé êîîðäèíàòå  óðàâíåíèå ñâîäèòñÿ ê âèäó:       0 1 E,E,HL 2    , çäåñü   0' 0'1 HP',(')d'      ,   0' 0'1 LP',(')d'      . Äâà èíòåãðàëà L è H â óðàâíåíèè íà ïðîñòðàíñòâåí- íî-âðåìåííîé ïëîñêîñòè    ñîîòâåòñòâóþò äâóì, ñõîäÿùèìñÿ ê âû÷èñëÿåìîé òî÷êå äèàãîíàëüíûì ëèíèÿì. Äëÿ äèýëåêòðè÷åñêîé ñðåäû ôóíêöèÿ P  â èíòåã- ðàëàõ çàäà¸òñÿ ñëåäóþùèì ñîîòíîøåíèåì:       1 , P,1E,        . 3. Ðåàëèçàöèÿ àëãîðèòìà ÷èñëåííîãî ñ÷åòà Äëÿ ÷èñëåííîãî ïîñòðîåíèÿ ðåøåíèÿ óðàâíåíèÿ öåëåñîîáðàçíî èñïîëüçîâàòü ïðÿìîóãîëüíóþ ïðî- ñòðàíñòâåííî-âðåìåííóþ ñåòêó, â êîòîðîé øàã ïî âðåìåíè â áåçðàçìåðíûõ âåëè÷èíàõ âäâîå ìåíüøå øàãà ïî ïðîñòðàíñòâó [6]. Ðåøåíèå, ïîñòðîåííîå íà òàêîé ñåòêå, îáëàäàåò ëó÷øåé ñòàáèëüíîñòüþ è áîëåå âûñîêîé òî÷íîñòüþ, ïî ñðàâíåíèþ ñ àíàëî- ãè÷íîé êâàäðàòíîé ñåòêîé [6]. Äëÿ ñîâìåùåíèÿ ëèíèé èíòåãðèðîâàíèÿ ñ óçëàìè ñåòêè óçëû íà íå÷åòíûõ âðåìåííûõ ñëîÿõ ñìåùåíû íà ïîëøàãà ïî ïðîñòðàíñòâåííîé êîîðäèíàòå. Ñòðóê- òóðà òàêîé ñåòêè èçîáðàæåíà íà ðèñ.1. Äëÿ ðåøåíèÿ óðàâíåíèÿ âûïîëíåíà äèñêðåòèçàöèÿ ïðîèçâîäíîé êîíå÷íîé ðàçíîñòüþ, à èíòåãðàë ðàññ÷èòàí ïî ôîðìóëå òðàïåöèé. 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Ó÷èòûâàÿ, ÷òî ïóòè èíòåãðèðîâà- íèÿ íà ðàçíûõ âðåìåííûõ èòåðàöèÿõ íàêëàäûâàþò- ñÿ äðóã íà äðóãà, ìîæíî èçáåæàòü ïîâòîðíûõ îïåðàöèé ñóììèðîâàíèÿ äëÿ âû÷èñëåíèÿ èíòåãðà- ëîâ è îïòèìèçèðîâàòü ÷èñëåííûé ðàñ÷åò. Äëÿ ýòîãî äîñòàòî÷íî çàïîìèíàòü âû÷èñëåííóþ èíòåãðàëü- íóþ ñóììó íà òåêóùåé èòåðàöèè è èñïîëüçîâàòü å¸ ïîçæå äëÿ âû÷èñëåíèÿ ñîîòâåòñòâóþùåãî èíòåãðà- ëà íà ñëåäóþùåì âðåìåííîì øàãå.   1  i+2  i+1  i  i-1  i-2  i-3  n-1  n-2  n-4  n-3  n Ðèñ.1. Ïðÿìîóãîëüíàÿ ïðîñòðàíñòâåííî-âðåìåííàÿ ñåòêà. Øàã ïî âðåìåíè âäâîå ìåíüøå ïðîñòðàí- ñòâåííîãî øàãà. Óçëû ñåòêè íà íå÷åòíûõ âðåìåííûõ ñëîÿõ ñìåùåíû íà ïîëøàãà ïî ïðîñòðàíñòâåííîé êîîðäèíàòå äëÿ ñîâìåùåíèÿ ñ ëèíèÿìè èíòåãðèðî- âàíèÿ 4. Òåñòèðîâàíèå àëãîðèòìà Äëÿ ïðîâåðêè òî÷íîñòè è ñòàáèëüíîñòè ïðåäëîæåí- íîãî ìåòîäà ïðîèçâåäåì ÷èñëåííûé ðàñ÷åò ñïåöè- àëüíîé òåñòîâîé çàäà÷è è ñðàâíèì ðåçóëüòàò ñ èçâåñòíûì òî÷íûì àíàëèòè÷åñêèì ðåøåíèåì. Ðàñ- ñìîòðèì ñëó÷àé, êîãäà âîçíèêàþùèé ñëîé, ïîñëå ñîçäàíèÿ, èìååò ñòàöèîíàðíûå ïàðàìåòðû ñðåäû, çíà÷èòåëüíî îòëè÷àþùèåñÿ îò ôîíîâîé ñðåäû, â ÷àñòíîñòè, äèýëåêòðè÷åñêàÿ ïðîíèöàåìîñòü ñðåäû â îáëàñòè ñëîÿ ðåçêî èçìåíÿåòñÿ â ìîìåíò âðåìåíè t0  îò  äî 1  . Ñóùåñòâóþùåå â ôîíîâîé ñðåäå ïîëå ÿâëÿåòñÿ ïëîñêîé âîëíîé è îïèñûâàåòñÿ ñîîò- íîøåíèåì     0 Ecos2L/  , ãäå  – äëèíà âîëíû â ôîíîâîé ñðåäå. Òî÷íîå ðåøåíèå äëÿ äàííîé çàäà÷è èçâåñòíî [5]. Ñëåäóþùèå çíà÷åíèÿ ïàðàìåò- ðîâ èñïîëüçîâàíû äëÿ òåñòèðîâàíèÿ àëãîðèòìà: 9  , 1 11  , íîðìèðîâàííàÿ ÷àñòîòà èñõîäíîé âîëíû 1  . Ñðàâíåíèå òî÷íîãî ðåøåíèÿ äàííîé çàäà÷è ñ ÷èñ- ëåííûìè ðåøåíèÿìè, ïîëó÷åííûìè ìåòîäàìè ÌÈÓ è FDTD, ïðåäñòàâëåíî íà ðèñ.2. Ðåçóëüòèðóþùåå ýëåêòðè÷åñêîå ïîëå â ñëîå ñîäåðæèò ñêà÷îê àìïëè- òóä, ïîÿâèâøèéñÿ âñëåäñòâèå îáðàçîâàíèÿ ñëîÿ. Íà ðèñ.3 ïðèâåäåíû ðåçóëüòàòû àïïðîêñèìàöèè ðåøå- íèÿ ìåòîäîì ÌÈÓ ïðè ðàçëè÷íûõ âåëè÷èíàõ øàãà äèñêðåòèçàöèè. Exact FDTD VIETD  1,5 1 0,5 1 0 -1  0,6 0,58 0,56 0,54 0,52 1,05 1 0,95 0,9 0,85 0,8 FDTD МИУВ Точное решение Ðèñ.2. Ñðàâíåíèå ÷èñëåííûõ ðåçóëüòàòîâ, ðàññ÷è- òàííûõ ìåòîäàìè FDTD è ÌÈÓÂ, ñ òî÷íûì àíàëè- òè÷åñêèì ðåøåíèåì òåñòîâîé çàäà÷è. Ðàçðûâ ïîëÿ âûçâàí ðåçêèì èçìåíåíèåì ïðîíèöàåìîñòè ñðåäû â íóëåâîé ìîìåíò âðåìåíè Exact 0,0005 0,001 0,01  0,65 0,6 0,55 0,5 1 0,9 0,8 Точное решение Ðèñ. 3. Ñðàâíåíèå ÷èñëåííûõ ðåçóëüòàòîâ ìåòîäà ÌÈÓÂ, ðàññ÷èòàííûõ ñ ðàçíûì øàãîì äèñêðåòèçà- öèè h, ñ òî÷íûì ðåøåíèåì çàäà÷è âáëèçè òî÷êè ðàçðûâà. Êà÷åñòâî àïïðîêñèìàöèè çàâèñèò îò âåëè- ÷èíû øàãà h Êàê âèäíî èç ïðèâåäåííûõ ãðàôèêîâ, ÌÈÓ ïðåâîñõîäèò ïî òî÷íîñòè ìåòîä FDTD, à òàêæå îáëàäàåò áîëåå âûñîêîé ñòàáèëüíîñòüþ. Äàæå ïðè âûáîðå äîâîëüíî ãðóáîãî øàãà ìîäåëèðîâàíèÿ, êàê ïîêàçàíî íà ðèñ. 3, ÷èñëåííîå ðåøåíèå, ïîëó÷åííîå ìåòîäîì ÌÈÓÂ, âñ¸ ðàâíî ñõîäèòñÿ ê òî÷íîìó ðåøåíèþ, íî ñîäåðæèò ìåíüøå äåòàëåé â îáëàñòè ñêà÷êà ïîëÿ. Ìåòîä æå FDTD ïðîèçâîäèò ðåçóëü- òàò, ñîäåðæàùèé óãàñàþùèé íåôèçè÷åñêèé ÷èñ- ëåííûé øóì (ñì. ðèñ. 2), âîçíèêàþùèé âñëåäñòâèå ÷èñëåííîãî äèôôåðåíöèðîâàíèÿ ïîëÿ â ðåçêî ìå- íÿþùåéñÿ ôóíêöèè ñðåäû. z+x{pUǿ'I. 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Ðàñïðîñòðàíåíèå ïëîñêîé ýëåêòðîìàãíèòíîé âîëíû â ñëîå ñ ìîäóëèðóåìîé âî âðåìåíè äèýëåêòðè÷åñêîé ïðîíèöàåìîñòüþ Ðàññìîòðèì ïðåîáðàçîâàíèå ïëîñêîé ýëåêòðîìàã- íèòíîé âîëíû ñ íîðìèðîâàííîé àìïëèòóäîé â äèýëåêòðè÷åñêîì ñëîå, äèýëåêòðè÷åñêàÿ ïðîíèöà- åìîñòü êîòîðîãî ìåíÿåòñÿ âî âðåìåíè ïîä âîçäåé- ñòâèåì ñòîðîííèõ ñèë ïî ãàðìîíè÷åñêîìó çàêîíó:   1 /11/9msin2  , ãäå L/  – íîðìèðî- âàííàÿ ÷àñòîòà ìîäóëÿöèè,  – äëèíà âîëíû óïðàâëÿþùåãî ïîëÿ è m – ãëóáèíà ìîäóëÿöèè. Èç- ìåíåíèå ñïåêòðà ñèãíàëà ïîñëå ïðîõîæäåíèÿ ñëîÿ ïðåäñòàâëåíî íà ðèñ. 4 è 5. Частота (THz) глубина - m=1/3 Нормированная частота модуляции  =0,5 0 100 200 300 400 А м п л и т у д а 0,0 0,2 0,4 0,6 0,8 1,0 Ðèñ. 4. Èçìåíåíèå ñïåêòðà ñèãíàëà ïîñëå âçàèìîäåé- ñòâèÿ ñî ñëîåì ñ ãàðìîíè÷åñêè èçìåíÿþùèìèñÿ âî âðåìåíè ïàðàìåòðàìè: Q=0,5 Äîïîëíèòåëüíûå êðàòíûå ãàðìîíèêè ïîÿâëÿþòñÿ â ñèãíàëå è ðàñòóò ñ óâåëè÷åíèåì ãëóáèíû ìîäóëÿ- öèè. ×àñòîòà äîïîëíèòåëüíûõ ãàðìîíèê çàâèñèò îò ÷àñòîòû ìîäóëÿöèè ñðåäû. Äëèòåëüíîñòü èíòåðâàëà ìîäåëèðîâàíèÿ max  ïðî- öåññà ðàâíà 100,0 (â áåçðàçìåðíûõ âåëè÷èíàõ, ÷òî ñîîòâåòñòâóåò 1 ps), øàã èíòåãðèðîâàíèÿ h0.01  . Äëÿ NX=100 è NT=10 000 ìèíèìàëüíî íåîáõîäèìûé îáú¸ì îïåðàòèâíîé ïàìÿòè ðàâåí 6 Ká, è îáùèé îáú¸ì ïàìÿòè äëÿ õðàíåíèÿ ðåçóëüòàòîâ – 8 Má. Âðåìÿ ìîäåëèðîâàíèÿ ðàâíî ~ 2,8 ñ íà ÏÊ ñ ïðîöåññîðîì Intel Celeron ñ òàêòîâîé ÷àñòîòîé 700 MÃö. 0,0 0,2 0,4 0,6 0,8 1,0 0 100 200 300 400 500 Частота (THz) А м п л и т у д а глубина - m=1/3 Нормированная частота модуляции  =1,25 Ðèñ. 5. Èçìåíåíèå ñïåêòðà ñèãíàëà ïîñëå âçàèìîäåé- ñòâèÿ ñî ñëîåì ñ ãàðìîíè÷åñêè èçìåíÿþùèìèñÿ âî âðåìåíè ïàðàìåòðàìè: Q=1,25 Ðàññìîòðèì ïîäîáíóþ çàäà÷ó, íî ñî ñðåäîé, ïðîíè- öàåìîñòü êîòîðîé ãàðìîíè÷åñêè ìåíÿåòñÿ â ïðî- ñòðàíñòâå è âî âðåìåíè â ñîîòâåòñòâèè ñ     1 /11/9msin2  . Ðåçóëüòàòû âëèÿ- íèÿ ñðåäû â äàííîì ñëó÷àå ïðåäñòàâëåíû íà ðèñ.6 (à, á): 0 100 200 300 400 600 Частота (THz) 0,0 0,2 0,4 0,6 0,8 700 500 глубина - m=1/3 Нормированная частота модуляции  =0,5 à 0,0 0,2 0,4 0,6 0,8 0 200 400 600 800 Частота (THz) 1200 1000 глубина - m=1/3 Нормированная частота модуляции  1,25 á Ðèñ. 7. Èçìåíåíèå ñïåêòðà ñèãíàëà ïîñëå âçàèìîäåé- ñòâèÿ ñî ñëîåì ñ ãàðìîíè÷åñêè ìîäóëèðóåìûìè âî âðåìåíè è â ïðîñòðàíñòâå ïàðàìåòðàìè  ñðàâíåíèè ñ ïðåäñòàâëåííûì ðàíåå ñëó÷àåì äëÿ ìîäóëèðóåìîé òîëüêî âî âðåìåíè ñðåäû â äàííîì ñëó÷àå êîëè÷åñòâî äîïîëíèòåëüíûõ ãàðìîíèê è èõ àìïëèòóäû çíà÷èòåëüíî âûøå. Ýòî ìîæåò áûòü îáúÿñíåíî äîïîëíèòåëüíîé ôàçîâîé ìîäóëÿöèåé ïîëÿ â êàæäîé òî÷êå ñëîÿ. 6. Çàêëþ÷åíèå Ïðåäñòàâëåí ÷èñëåííî-àíàëèòè÷åñêèé ïîäõîä äëÿ ìîäåëèðîâàíèÿ âçàèìîäåéñòâèÿ ýëåêòðîìàãíèòíî- ãî ñèãíàëà ïðîèçâîëüíîé ôîðìû ñ ïëîñêèì äèýëåê- òðè÷åñêèì ñëîåì, ïàðàìåòðû êîòîðîãî ìåíÿþòñÿ âî âðåìåíè è/èëè â ïðîñòðàíñòâå. 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