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Example text

P Ë Ò Ð Ø ÓÙÔÐ z º ¿º½¾ Ó r cos ϑ sin ϑ cos λ ′ . M t− 3 4πρα r α ××ÙÑ × Ò Ð (x − x0 )/r = sin ϑ cos λ¸ Ò r¹ Ö Ø ÓÒ × Ò × Ò Ð uϑ = Ø × ÐÐ ÖÓØ Ø ÓÒ Ð ÑÓÑ Òغ ur = − ÁÒ ÓÒ
ÐÙ M (t) ´ ÙØ ÒÓÒ¹Þ ÖÓµº Ñ Ò× ÓÒ× Ó Ü¹Þ¹ÔÐ Ò º ÓÐÐÓÛ Ò x − z ¹ÔÐ Ò × ÓÛ× (y = 0) ¿º º ËÈÀ ÊÁ Ä Ï Î Ë ÊÇÅ ËÁÆ Ä ÇÊ ur 2ϑ cos λ ′ t− = − sin8πρα 3r M uϑ = uλ sin2 ϑ cos λ ′ 4πρβ 3 r M t− Æ r α ÁÈÇÄ ººº        r β ´¿º¾½µ       = 0. z S + - P - + º ¿º½¿ Ö Ð Ì Ö Ø Ó Ó Ø Ö Ø ÓÒ ´ ÓÖ ϑ Ò ÔÐ Ò × ÓØ ×ÔÐ Ñ ÒØ Ó ÓÙÔÐ º Ñ Ü ÑÙÑ S ¹Ö Ø ÓÒ ´ ÓÖ ϑ = √ = 450 µ × ÓÙØ ½¼¸ α ≈ β 3º Ì Ö Ø Ò cos λº z=0 ÈÐ Ò x=0 × ÓÒ ÓÒÐÝ ÓÖ Ì Ö¹ Ð × Ò Ð x y = 0 ÓÐÐÓÛ ÒÓ Ð ÔÐ Ò × ÖÓÑ Ø ÓÖ P¹ ÓÒ 900 µ × ÓÛÒ × Û ÐÐ × ÓÖ ×ÔÐ Ñ ÒØ× ÓÖ ÓÙ Ð ÓÙÔÐ ÒØ P¹ Ö Ø Ö ×Ø × Ý ÑÙÐØ ÔÐ Ø ÓÒ Û Ø S ¹Ö x−z ¹ÔÐ z P y r υ λ x ÓÙ Ð Ñ Ü ÑÙÑ Ø ÓÒ Ø ÓÒ Ø ÔÐ Ò Pº ¿º µ º ¿º½ ØÓ Ø Ö ÓÙÔÐ Ò Ø Ü¹Þ¹ÔÐ Ò º Ò Ö ´× Ü Ö × ¼ À ur = 2ϑ cos λ ′ − sin4πρα t− 3r M r α uϑ = 2ϑ cos λ M′ t − − cos4πρβ 3r r β uλ Ì cos ϑ sin λ ′ 4πρβ 3 r M = ÑÓÑ ÒØ ÙÒ
Ø ÓÒ Ò ´¿º¾¾µ × Ø ÓÙÔÐ º Ì Ö ÈÌ Ø ÓÒ ÓÒ Ø Ó r β t− Ó Ø Ç Ï Î         ØÛÓ Ò Ë ´¿º¾¾µ        .

2 /β 2 Ð Ü r¸ Ø ÓÖ Ò S−Û ÓÖÑ Ó Ø × × ¿º ÓÖ ¸ S ¹Û P ¹Û Ú Ú Ø ÓÖ º × Ö × Ö Ø Ø Ò ´ ÙØ ¿º µº Ó Ø Ø Ø × Ò Ó Ø Ò × Ò Ð ÓÓ Ð×Ó ×ÓÐÙØ ÓÒ Ö ÕÙ Ö × Ø Ò Ø Ó Ø ¸ × Ø Ø × × Ò×Ú Ö× Ð P −Û Ö Ø Ö ×Ø × Ó × Ò Ð × ÓÒ º ÔÖ Ø Ð Ù× ÙØ Ø Ö ×ÔÐ Ñ ÒØ ×ÙÖ ÓÑÔÐ Ø Ú ¸Ø × Ú Ö Ò Ö Ñ ÛÓÖ Ú ÖÓ¹× × (ØÖ ÔÓ ÒØ ×ÓÙÖ Ù× Û ØÓ Ø Ï Ø ÓÑÔ Ö (ÐÓÒ Ö Ø Ö ×Ø × ´È¹ Ò Ó Ø ÓÒ Ö
Ð ÖÓÑ Ø t− r β Ø ÓÒ OP 1 ¸ ×Ø Ò OP 2 º r α Ø ÓÒ Ö Ø Ö ×Ø × Ë ¹Û Ú × Ú Ö Ö Ø ÓÒ Ò Ð ϑ Ø t− ×ÔÐ Ñ ÒØ× Ì º ¿º Ì Ý È ¹Û Ø Ø Ä Ö Ð× Ò´¿º½ µµ uϑ Ö ÊÇÅ ËÁÆ ÓÒÐÝ Ø ur Ì Ë ØÖ Ò×Ú Ö× Ψλ × ÒÓØ ÔÙÖ Ò uϑ Ò Ø Û Ä Ï Î ÒØÐÝ ÑÓÖ ÓÖ ÔÓ ÒØ ×ÓÙÖ ¸ ÑÓ Ð ÓÖ Ø ÓÖ ÜÔÐÓ× ÓÒ× ÓÒ× ØÓÒ Ø Ö Ø ÓÒ Ó ÓÑÔÐ Ø º Ø Ò Ø Ó ÙÖØ Ø Ô ÖÔ Ò ÖÓÔ Û ÐÓ× ØÓ Ø Ø× Ó ÖÑÓÖ ¸ Ø Ø ÙÐ Ö ÓÒ Ø Ø¸ Ü Ø Ø ÓÒ ×ÙÖ Ö º ×ÙÖ ¸ Ö Ò × ØÓ Ø À ÙÐй×Ô Ø ÑÓ Ò ¿¼ ØÓ Ð× ÓÖ ¼ Ö P ¹Û ÐР׸ Ö Ú × Ò S ¹Û ÓÖ ÈÌ Ê ¿º Ú ×¸ ÓÖ Ö Ç Ø ÓÒ Ï Î Ë Ò Ð × ×Ñ ÐÐ Ö ×Ñ Ðи Ö ×Ô Ø Ú Ðݺ Ü Ö × ¿º ÓÑÔÙØ Ø ÙÐ Ö¸ Ø ÓÑÔÐ Ø Ñ ÒØ Ú ØÓÖ Ú ¸ K(t) = K0 H(t) ¿º º¾ ÔÓÐ Ó Ø Ø ÓÖ Û º ¿º½¼ Ø ÓÙÔÐ ÜÔ Ø ×ÔÐ ¹ ÓÑÔÙØ ÓÖ (t > r/β).

0 Ò Ò ÛÖ ØØ Ò Ù× Ò τ 1 1 − r r1 U 1 (τ ) + α ÈÌ U 1 (ϑ)e ´¿º µ − rα (τ −ϑ) 1 × dϑ ´¿º½¾µ 0 × ×ÓÐÚ × Ø ÓÙÒ ÖÝ ÔÖÓ Ð Ñº ÔÔÐ Ø ÓÒ× ½º U1 (t) = U 0 δ t − r1 α Ì U0 Ø Ñ Ò× ÓÒ Ó × × ´× ÔÔ Ò U (r, t) = , . , U 1 (t) = U 0 δ(t). × Ø Ñ Ü Ø Ñ × Ð Ò Ø º r1 U 0 δ(τ ) + α r 1 1 − r r1 τ=0 (t=r/α) º ¿º¿ ¾º ʹָص × ÕÙ Ø ÓÒ ´¿º½¾µ × Ú Ð ÙÒ
Ø ÓÒ Ó Ø Ñ º r1 α , . U 1 (t) = U0 H(t). Ì U0 × Ð Ò Ø º U (r, t) = = = α e− r1 τ H(τ ) . τ U1 (t) = U0 H t − Ñ Ò× ÓÒ Ó Ð×Ó Ò µ r1 U0 H(τ ) + α r r1 U0 H(τ ) 1 + r r1 r1 U0 H(τ ) + r r ÖÓÑ ´¿º½¾µ¸ Ø ÓÐÐÓÛ× 1 1 − r r1 α e − r1 τ r1 rα ϑ e 1 α α r1 − 1 1 − e − r1 τ r r1 − rα τ 1− e 1 .

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