%!PS-Adobe-2.0 %%Creator: dvips(k) 5.90a Copyright 2002 Radical Eye Software %%Title: Instant.dvi %%CreationDate: Mon Jun 05 14:07:01 2006 %%Pages: 17 %%PageOrder: Ascend %%BoundingBox: 0 0 596 842 %%DocumentFonts: CMBX12 CMR12 CMR8 CMR10 CMTI12 CMMI12 CMMI8 CMSY10 %%+ CMSY8 CMMI6 CMMIB10 %%EndComments %DVIPSWebPage: (www.radicaleye.com) %DVIPSCommandLine: dvips.exe -P pdf -G0 Instant %DVIPSParameters: dpi=8000, compressed %DVIPSSource: TeX output 2006.06.05:1406 %%BeginProcSet: tex.pro %! /TeXDict 300 dict def TeXDict begin/N{def}def/B{bind def}N/S{exch}N/X{S N}B/A{dup}B/TR{translate}N/isls false N/vsize 11 72 mul N/hsize 8.5 72 mul N/landplus90{false}def/@rigin{isls{[0 landplus90{1 -1}{-1 1}ifelse 0 0 0]concat}if 72 Resolution div 72 VResolution div neg scale isls{ landplus90{VResolution 72 div vsize mul 0 exch}{Resolution -72 div hsize mul 0}ifelse TR}if Resolution VResolution vsize -72 div 1 add mul TR[ matrix currentmatrix{A A round sub abs 0.00001 lt{round}if}forall round exch round exch]setmatrix}N/@landscape{/isls true N}B/@manualfeed{ statusdict/manualfeed true put}B/@copies{/#copies X}B/FMat[1 0 0 -1 0 0] N/FBB[0 0 0 0]N/nn 0 N/IEn 0 N/ctr 0 N/df-tail{/nn 8 dict N nn begin /FontType 3 N/FontMatrix fntrx N/FontBBox FBB N string/base X array /BitMaps X/BuildChar{CharBuilder}N/Encoding IEn N end A{/foo setfont}2 array copy cvx N load 0 nn put/ctr 0 N[}B/sf 0 N/df{/sf 1 N/fntrx FMat N df-tail}B/dfs{div/sf X/fntrx[sf 0 0 sf neg 0 0]N df-tail}B/E{pop nn A definefont setfont}B/Cw{Cd A length 5 sub get}B/Ch{Cd A length 4 sub get }B/Cx{128 Cd A length 3 sub get sub}B/Cy{Cd A length 2 sub get 127 sub} B/Cdx{Cd A length 1 sub get}B/Ci{Cd A type/stringtype ne{ctr get/ctr ctr 1 add N}if}B/CharBuilder{save 3 1 roll S A/base get 2 index get S /BitMaps get S get/Cd X pop/ctr 0 N Cdx 0 Cx Cy Ch sub Cx Cw add Cy setcachedevice Cw Ch true[1 0 0 -1 -.1 Cx sub Cy .1 sub]{Ci}imagemask restore}B/D{/cc X A type/stringtype ne{]}if nn/base get cc ctr put nn /BitMaps get S ctr S sf 1 ne{A A length 1 sub A 2 index S get sf div put }if put/ctr ctr 1 add N}B/I{cc 1 add D}B/bop{userdict/bop-hook known{ bop-hook}if/SI save N @rigin 0 0 moveto/V matrix currentmatrix A 1 get A mul exch 0 get A mul add .99 lt{/QV}{/RV}ifelse load def pop pop}N/eop{ SI restore userdict/eop-hook known{eop-hook}if showpage}N/@start{ userdict/start-hook known{start-hook}if pop/VResolution X/Resolution X 1000 div/DVImag X/IEn 256 array N 2 string 0 1 255{IEn S A 360 add 36 4 index cvrs cvn put}for pop 65781.76 div/vsize X 65781.76 div/hsize X}N /p{show}N/RMat[1 0 0 -1 0 0]N/BDot 260 string N/Rx 0 N/Ry 0 N/V{}B/RV/v{ /Ry X/Rx X V}B statusdict begin/product where{pop false[(Display)(NeXT) (LaserWriter 16/600)]{A length product length le{A length product exch 0 exch getinterval eq{pop true exit}if}{pop}ifelse}forall}{false}ifelse end{{gsave TR -.1 .1 TR 1 1 scale Rx Ry false RMat{BDot}imagemask grestore}}{{gsave TR -.1 .1 TR Rx Ry scale 1 1 false RMat{BDot} imagemask grestore}}ifelse B/QV{gsave newpath transform round exch round exch itransform moveto Rx 0 rlineto 0 Ry neg rlineto Rx neg 0 rlineto fill grestore}B/a{moveto}B/delta 0 N/tail{A/delta X 0 rmoveto}B/M{S p delta add tail}B/b{S p tail}B/c{-4 M}B/d{-3 M}B/e{-2 M}B/f{-1 M}B/g{0 M} B/h{1 M}B/i{2 M}B/j{3 M}B/k{4 M}B/w{0 rmoveto}B/l{p -4 w}B/m{p -3 w}B/n{ p -2 w}B/o{p -1 w}B/q{p 1 w}B/r{p 2 w}B/s{p 3 w}B/t{p 4 w}B/x{0 S rmoveto}B/y{3 2 roll p a}B/bos{/SS save N}B/eos{SS restore}B end %%EndProcSet %%BeginProcSet: alt-rule.pro %! % Patch by TVZ % Makes dvips files draw rules with stroke rather than fill. % Makes narrow rules more predictable at low resolutions % after distilling to PDF. % May have unknown consequences for very thick rules. % Tested only with dvips 5.85(k). 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMR12 %!PS-AdobeFont-1.1: CMR12 1.0 %%CreationDate: 1991 Aug 20 16:38:05 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMR12) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR12 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 11 /ff put dup 12 /fi put dup 13 /fl put dup 14 /ffi put dup 16 /dotlessi put dup 18 /grave put dup 19 /acute put dup 20 /caron put dup 22 /macron put dup 34 /quotedblright put dup 39 /quoteright put dup 40 /parenleft put dup 41 /parenright put dup 43 /plus put dup 44 /comma put dup 45 /hyphen put dup 46 /period put dup 47 /slash put dup 48 /zero put dup 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put dup 53 /five put dup 54 /six put dup 55 /seven put dup 56 /eight put dup 57 /nine put dup 58 /colon put dup 59 /semicolon put dup 61 /equal put dup 64 /at put dup 65 /A put dup 66 /B put dup 67 /C put dup 68 /D put dup 69 /E put dup 70 /F put dup 71 /G put dup 72 /H put dup 73 /I put dup 74 /J put dup 75 /K put dup 76 /L put dup 77 /M put dup 78 /N put dup 79 /O put dup 80 /P put dup 81 /Q put dup 82 /R put dup 83 /S put dup 84 /T put dup 85 /U put dup 86 /V put dup 87 /W put dup 89 /Y put dup 91 /bracketleft put dup 92 /quotedblleft put dup 93 /bracketright put dup 94 /circumflex put dup 95 /dotaccent put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 104 /h put dup 105 /i put dup 106 /j put dup 107 /k put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 113 /q put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 118 /v put dup 119 /w put dup 120 /x put dup 121 /y put dup 122 /z put dup 123 /endash put dup 126 /tilde put readonly def /FontBBox{-34 -251 988 750}readonly def /UniqueID 5000794 def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA052A014267B7904EB3C0D3BD0B83D891 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cleartomark %%EndFont %%BeginFont: CMSY8 %!PS-AdobeFont-1.1: CMSY8 1.0 %%CreationDate: 1991 Aug 15 07:22:10 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMSY8) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.035 def /isFixedPitch false def end readonly def /FontName /CMSY8 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /minus put dup 3 /asteriskmath put readonly def /FontBBox{-30 -955 1185 779}readonly def /UniqueID 5000818 def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMSY10 %!PS-AdobeFont-1.1: CMSY10 1.0 %%CreationDate: 1991 Aug 15 07:20:57 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMSY10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.035 def /isFixedPitch false def end readonly def /FontName /CMSY10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /minus put dup 1 /periodcentered put dup 2 /multiply put dup 70 /F put dup 106 /bar put readonly def /FontBBox{-29 -960 1116 775}readonly def /UniqueID 5000820 def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMMI8 %!PS-AdobeFont-1.1: CMMI8 1.100 %%CreationDate: 1996 Jul 23 07:53:54 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.100) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMMI8) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMMI8 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 11 /alpha put dup 12 /beta put dup 31 /chi put dup 33 /omega put dup 59 /comma put dup 64 /partialdiff put dup 75 /K put dup 82 /R put dup 85 /U put dup 100 /d put dup 105 /i put dup 106 /j put dup 107 /k put dup 108 /l put dup 115 /s put dup 116 /t put readonly def /FontBBox{-24 -250 1110 750}readonly def /UniqueID 5087383 def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA0529731C99A784CCBE85B4993B2EEBDE 3B12D472B7CF54651EF21185116A69AB1096ED4BAD2F646635E019B6417CC77B 532F85D811C70D1429A19A5307EF63EB5C5E02C89FC6C20F6D9D89E7D91FE470 B72BEFDA23F5DF76BE05AF4CE93137A219ED8A04A9D7D6FDF37E6B7FCDE0D90B 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMMI12 %!PS-AdobeFont-1.1: CMMI12 1.100 %%CreationDate: 1996 Jul 27 08:57:55 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.100) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMMI12) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMMI12 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 11 /alpha put dup 12 /beta put dup 17 /eta put dup 26 /rho put dup 28 /tau put dup 31 /chi put dup 33 /omega put dup 58 /period put dup 59 /comma put dup 61 /slash put dup 62 /greater put dup 64 /partialdiff put dup 67 /C put dup 69 /E put dup 70 /F put dup 75 /K put dup 78 /N put dup 80 /P put dup 84 /T put dup 85 /U put dup 86 /V put dup 87 /W put dup 97 /a put dup 99 /c put dup 101 /e put dup 102 /f put dup 103 /g put dup 105 /i put dup 106 /j put dup 107 /k put dup 109 /m put dup 110 /n put dup 112 /p put dup 115 /s put dup 116 /t put dup 117 /u put dup 118 /v put dup 120 /x put readonly def /FontBBox{-30 -250 1026 750}readonly def /UniqueID 5087386 def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMTI12 %!PS-AdobeFont-1.1: CMTI12 1.0 %%CreationDate: 1991 Aug 18 21:06:53 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMTI12) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMTI12 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 11 /ff put dup 12 /fi put dup 13 /fl put dup 44 /comma put dup 45 /hyphen put dup 46 /period put dup 49 /one put dup 51 /three put dup 58 /colon put dup 65 /A put dup 68 /D put dup 69 /E put dup 71 /G put dup 73 /I put dup 74 /J put dup 76 /L put dup 77 /M put dup 80 /P put dup 81 /Q put dup 82 /R put dup 83 /S put dup 87 /W put dup 90 /Z put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 103 /g put dup 104 /h put dup 105 /i put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 113 /q put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 118 /v put dup 119 /w put dup 120 /x put dup 121 /y put readonly def /FontBBox{-36 -251 1103 750}readonly def /UniqueID 5000829 def currentdict end currentfile eexec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All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMBX12) readonly def /FamilyName (Computer Modern) readonly def /Weight (Bold) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMBX12 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 12 /fi put dup 45 /hyphen put dup 46 /period put dup 48 /zero put dup 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put dup 53 /five put dup 54 /six put dup 55 /seven put dup 56 /eight put dup 57 /nine put dup 65 /A put dup 66 /B put dup 67 /C put dup 68 /D put dup 69 /E put dup 70 /F put dup 73 /I put dup 75 /K put dup 80 /P put dup 82 /R put dup 83 /S put dup 84 /T put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 104 /h put dup 105 /i put dup 107 /k put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 113 /q put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 118 /v put dup 119 /w put dup 120 /x put dup 121 /y put readonly def /FontBBox{-53 -251 1139 750}readonly def /UniqueID 5000769 def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMR10 %!PS-AdobeFont-1.1: CMR10 1.00B %%CreationDate: 1992 Feb 19 19:54:52 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.00B) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMR10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 40 /parenleft put dup 41 /parenright put dup 43 /plus put dup 44 /comma put dup 45 /hyphen put dup 46 /period put dup 48 /zero put dup 49 /one put dup 50 /two put dup 51 /three put dup 53 /five put dup 54 /six put dup 55 /seven put dup 57 /nine put dup 67 /C put dup 68 /D put dup 71 /G put dup 80 /P put dup 82 /R put dup 83 /S put dup 85 /U put dup 87 /W put dup 97 /a put dup 99 /c put dup 101 /e put dup 102 /f put dup 103 /g put dup 104 /h put dup 105 /i put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 118 /v put dup 120 /x put dup 121 /y put readonly def /FontBBox{-251 -250 1009 969}readonly def /UniqueID 5000793 def currentdict end currentfile eexec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All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. 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cleartomark %%EndFont TeXDict begin 39158280 55380996 1000 8000 8000 (Instant.dvi) @start /Fa 185[915 70[{}1 1328.35 /CMMIB10 rf /Fb 140[388 8[340 5[427 14[557 9[681 75[{}5 664.176 /CMMI6 rf /Fc 146[1301 7[694 18[1149 8[563 73[{}4 1594.02 /CMR12 rf /Fd 252[470 2[732{}2 885.568 /CMSY8 rf /Fe 149[369 35[955 67[1033 369 1033{}5 1328.35 /CMSY10 rf /Ff 139[340 435 6[281 490 381 320 4[484 14[637 2[709 6[793 10[496 4[261 25[585 1[584 18[528 604 11[{}16 885.568 /CMMI8 rf /Fg 135[739 1[629 740 470 613 2[653 1[776 1138 1[676 538 444 1[623 641 603 1[560 1[683 9[1228 758 883 762 3[838 1[1042 2[1107 4[842 963 1[931 2[687 1[1012 650 1[361 361 24[810 1[815 2[565 1[671 8[642 4[734 827 11[{}38 1328.35 /CMMI12 rf /Fh 134[631 598 863 598 697 432 531 548 598 664 664 731 1063 332 2[399 664 598 1[598 664 598 598 664 6[797 2[1298 3[731 948 996 881 2[1165 815 1[682 501 1[1005 1[881 981 2[966 6[399 6[664 1[664 2[399 465 399 30[764 731 797 11[{}45 1328.35 /CMTI12 rf /Fi 134[1136 1136 1554 1136 1196 837 849 879 1136 1196 1076 1196 1793 598 1136 1[598 1196 1076 658 982 1196 956 1196 1046 23[801 3[1412 1649 1554 1530 11[1076 1076 1076 1076 1076 1076 3[717 32[1196 12[{}37 1912.83 /CMBX12 rf /Fj 134[584 584 1[584 615 430 437 434 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b(l)16003 14068 y Fn(=)369 b Fg(c)17944 13519 y Ff(R)17944 14396 y(ij)51 b(l)11 b(k)19925 14068 y Fn(=)368 b Fg(c)21865 13519 y Ff(R)21865 14396 y(k)24 b(l)11 b(ij)23911 14068 y Fg(;)1522 b(\021)26484 13519 y Ff(R)26436 14396 y(ij)51 b(k)24 b(l)28417 14068 y Fn(=)368 b Fg(\021)30487 13519 y Ff(R)30439 14396 y(j)51 b(ik)24 b(l)32420 14068 y Fn(=)369 b Fg(\021)34491 13519 y Ff(R)34443 14396 y(ij)51 b(l)11 b(k)36424 14068 y Fn(=)368 b Fg(\021)38494 13519 y Ff(R)38446 14396 y(l)11 b(k)24 b(ij)40492 14068 y Fg(:)7879 b Fn(\(5\))0 16712 y(Note)523 b(that)f(the)h(relation)g Fg(\021)14017 16230 y Ff(R)13969 17040 y(ij)51 b(k)24 b(l)16102 16712 y Fn(=)520 b Fg(\021)18324 16230 y Ff(R)18276 17040 y(k)24 b(l)11 b(ij)20411 16712 y Fn(has)523 b(not)f(y)-36 b(et)523 b(b)36 b(een)523 b(pro)-36 b(v)g(ed)522 b(to)h(hold)g (generally)-108 b(,)546 b(see,)f(e.g.,)0 18262 y(Carcione)332 b(\(2001\).)545 b(Relations)332 b(\(5\))f(reduce)g(the)f(n)-36 b(um)g(b)36 b(er)330 b(of)i(indep)36 b(enden)-36 b(t)329 b(comp)36 b(onen)-36 b(ts)330 b(of)i(elasticit)-36 b(y)0 19812 y(and)404 b(viscosit)-36 b(y)406 b(tensors)d(from)i(81)g(to)f (21.)569 b(Due)404 b(to)h(these)e(symmetries,)411 b(the)404 b(elasticit)-36 b(y)405 b(and)f(viscosit)-36 b(y)0 21361 y(tensors)433 b(can)h(b)36 b(e)433 b(also)i(expressed)e(as)h(symmetric) g(6)295 b Fe(\002)h Fn(6)434 b(matrices)f(\(V)-108 b(oigt)434 b(notation\).)1951 23856 y(Individual)g(quan)-36 b(tities)434 b(in)-36 b(tro)36 b(duced)432 b(in)h(this)h(section)f(are)h(expressed)f (in)h(the)f(follo)-36 b(wing)436 b(units:)0 25406 y(displacemen)-36 b(t)414 b(comp)36 b(onen)-36 b(ts)415 b Fg(u)15639 25605 y Ff(i)16430 25406 y Fn(in)g(meters)f(\()p Fg(m)p Fn(\),)k(the)d (particle-v)-36 b(elo)36 b(cit)-36 b(y)416 b(comp)36 b(onen)-36 b(ts)415 b Fg(v)44514 25605 y Ff(i)45304 25406 y Fn(in)g Fg(m)221 b(s)48775 24924 y Fd(\000)p Fm(1)50032 25406 y Fn(,)0 26955 y(densit)-36 b(y)386 b Fg(\032)h Fn(in)f Fg(k)45 b(g)269 b(m)9719 26473 y Fd(\000)p Fm(3)10976 26955 y Fn(,)396 b(the)386 b(comp)36 b(onen)-36 b(ts)386 b(of)h(b)36 b(o)g(dy)387 b(forces)g Fg(f)29965 27154 y Ff(i)30728 26955 y Fn(in)f(newton)g(p)36 b(er)386 b(cubic)g(meter)g (\()p Fg(N)30 b(=m)49285 26473 y Fm(3)49811 26955 y Fn(;)0 28505 y Fg(k)45 b(g)269 b(m)2751 28023 y Fd(\000)p Fm(2)4008 28505 y Fg(s)4621 28023 y Fd(\000)p Fm(2)5879 28505 y Fn(\),)573 b(the)544 b(strain)h(comp)36 b(onen)-36 b(ts)544 b Fg(e)21349 28704 y Ff(ij)22702 28505 y Fn(are)h(dimensionles,)573 b(the)545 b(stress)g(comp)36 b(onen)-36 b(ts)544 b Fg(\034)46946 28704 y Ff(ij)48298 28505 y Fn(and)0 30055 y(elasticit)-36 b(y)572 b(mo)36 b(duli)570 b Fg(c)10704 29573 y Ff(R)10704 30383 y(ij)51 b(k)24 b(l)12886 30055 y Fn(are)571 b(in)f(pascals)h(\()p Fg(P)181 b(a)p Fn(;)221 b Fg(k)45 b(g)269 b(m)26945 29573 y Fd(\000)p Fm(1)28203 30055 y Fg(s)28816 29573 y Fd(\000)p Fm(2)30073 30055 y Fn(\),)605 b(and)569 b(the)h(viscosit)-36 b(y)572 b(mo)36 b(duli)571 b Fg(\021)47176 29573 y Ff(R)47128 30383 y(ij)51 b(k)24 b(l)49310 30055 y Fn(in)0 31719 y(pascal)434 b Fe(\002)g Fn(second)f(\()p Fg(P)181 b(a)221 b(s)p Fn(;)g Fg(k)45 b(g)269 b(m)15949 31237 y Fd(\000)p Fm(1)17207 31719 y Fg(s)17820 31237 y Fd(\000)p Fm(1)19077 31719 y Fn(\).)0 37532 y Fi(3)2152 b(Energy-balance)716 b(equation)g(for)h(w)-60 b(a)g(v)g(e)717 b(\014elds)3228 40454 y(with)g(arbitrary)f(time)h(dep)60 b(endence)0 44264 y Fn(In)381 b(this)f(section,)392 b(w)-36 b(e)382 b(consider)e(the)h(v)-36 b(ectors)381 b Fg(v)22772 44463 y Ff(i)23529 44264 y Fn(and)f Fg(f)26646 44463 y Ff(i)27022 44264 y Fn(,)392 b(and)380 b(tensors)h Fg(\034)35186 44463 y Ff(ij)35993 44264 y Fn(,)392 b Fg(e)37349 44463 y Ff(ij)38537 44264 y Fn(to)381 b(b)36 b(e)381 b(real)g(v)-72 b(alued)381 b(and)0 45814 y(time)434 b(dep)36 b(enden)-36 b(t.)576 b(Multiplying)434 b(\(3\))f(b)-36 b(y)434 b Fe(\000)p Fg(v)22400 46013 y Ff(i)23209 45814 y Fn(and)f(\(2\))h(b)-36 b(y)433 b Fe(\000)p Fg(\034)31238 46013 y Ff(ij)32045 45814 y Fn(,)h(and)f(adding)h(b)36 b(oth,)433 b(w)-36 b(e)434 b(obtain)15469 48310 y(\()p Fe(\000)p Fg(v)17637 48509 y Ff(i)18012 48310 y Fg(\034)18577 48509 y Ff(ij)19384 48310 y Fn(\))19890 48509 y Ff(;j)21007 48310 y Fn(=)369 b Fe(\000)p Fg(\032v)24721 48509 y Ff(i)25290 48310 y Fn(_)-554 b Fg(v)25726 48509 y Ff(i)26396 48310 y Fe(\000)296 b Fg(\034)28290 48509 y Ff(ij)29290 48310 y Fn(_)-554 b Fg(e)29700 48509 y Ff(ij)30802 48310 y Fn(+)295 b Fg(v)32738 48509 y Ff(i)33113 48310 y Fg(f)33754 48509 y Ff(i)34564 48310 y Fg(:)13807 b Fn(\(6\))0 50805 y(Equation)585 b(\(6\))f(represen)-36 b(ts)583 b(a)h(v)-36 b(ery)585 b(general)g(form)g(of)g(the)e(energy-balance)i(equation.)1031 b(Ph)-36 b(ysical)0 52355 y(meaning)481 b(of)g(all)h(terms)e(in)h (\(6\),)493 b(except)480 b Fe(\000)p Fg(\034)22107 52554 y Ff(ij)23108 52355 y Fn(_)-554 b Fg(e)23518 52554 y Ff(ij)24805 52355 y Fn(is)481 b(the)g(same)g(as)g(in)f(p)36 b(erfectly)482 b(elastic)f(media)g(and)0 53905 y(will)314 b(b)36 b(e)312 b(explained)h(later.)538 b(The)312 b(term)g Fe(\000)p Fg(\034)20317 54104 y Ff(ij)21318 53905 y Fn(_)-554 b Fg(e)21728 54104 y Ff(ij)22847 53905 y Fn(dep)36 b(ends)311 b(considerably)i(on)g(the)e(stress-strain)h(relation)0 55455 y(under)426 b(consideration.)576 b(It)427 b(can)g(b)36 b(e)427 b(separated)f(in)-36 b(to)427 b(t)-36 b(w)g(o)428 b(terms;)h(one)e(corresp)36 b(onding)426 b(to)h(the)g(strain)0 57004 y(energy)434 b(and)f(the)g(other)g(to)h(the)f(dissipated)g (energy)-108 b(.)578 b(This)434 b(separation)g(is)g(not)f(necessarily)i (unique.)1951 59499 y(The)343 b(situation)h(simpli\014es)g(if)g(w)-36 b(e)344 b(consider)f(Kelvin-V)-108 b(oigt)344 b(stress-strain)f (relation)h(\(4\).)548 b(Then)343 b(the)0 61049 y(separation)434 b(of)g(the)f(term)g Fe(\000)p Fg(\034)14720 61248 y Ff(ij)15721 61049 y Fn(_)-554 b Fg(e)16131 61248 y Ff(ij)17371 61049 y Fn(b)36 b(ecomes)434 b(unique)f(and)g(trivial.)580 b(Inserting)433 b(\(4\))h(in)-36 b(to)433 b(\(6\))h(yields)10962 63545 y(\()p Fe(\000)p Fg(v)13130 63744 y Ff(i)13506 63545 y Fg(\034)14071 63744 y Ff(ij)14878 63545 y Fn(\))15384 63744 y Ff(;j)16501 63545 y Fn(=)368 b Fe(\000)p Fg(\032v)20214 63744 y Ff(i)20784 63545 y Fn(_)-555 b Fg(v)21219 63744 y Ff(i)21890 63545 y Fe(\000)295 b Fg(c)23778 62996 y Ff(R)23778 63873 y(ij)51 b(k)24 b(l)25583 63545 y Fn(_)-554 b Fg(e)25993 63744 y Ff(ij)26800 63545 y Fg(e)27403 63744 y Ff(k)24 b(l)28558 63545 y Fe(\000)295 b Fg(\021)30576 62996 y Ff(R)30528 63873 y(ij)51 b(k)24 b(l)32333 63545 y Fn(_)-554 b Fg(e)32743 63744 y Ff(ij)33743 63545 y Fn(_)g Fg(e)34153 63744 y Ff(k)24 b(l)35309 63545 y Fn(+)294 b Fg(v)37244 63744 y Ff(i)37620 63545 y Fg(f)38261 63744 y Ff(i)39070 63545 y Fg(:)9301 b Fn(\(7\))0 66040 y(This)434 b(can)f(b)36 b(e)434 b(expressed)f(in)h(a)f(condensed)g(form:)16082 68536 y(div)19 b Fk(F)369 b Fn(=)g Fe(\000)21896 68013 y Ff(@)p 21726 68226 888 45 v 21726 68994 a(@)52 b(t)22746 68536 y Fn(\()p Fg(K)390 b Fn(+)295 b Fg(U)139 b Fn(\))295 b Fe(\000)h Fg(P)30046 68735 y Ff(d)30881 68536 y Fn(+)f Fg(P)33026 68735 y Ff(s)33950 68536 y Fg(;)14421 b Fn(\(8\))0 71032 y(where)11310 72491 y Fg(F)12152 72690 y Ff(j)13008 72491 y Fn(=)369 b Fe(\000)p Fg(v)16051 72690 y Ff(i)16426 72491 y Fg(\034)16991 72690 y Ff(ij)18232 72491 y Fg(;)2873 b(K)464 b Fn(=)24551 71968 y Fm(1)p 24551 72181 471 45 v 24551 72949 a(2)25154 72491 y Fg(\032v)26454 72690 y Ff(i)26829 72491 y Fg(v)27458 72690 y Ff(i)28268 72491 y Fg(;)1107 b(U)508 b Fn(=)32640 71968 y Fm(1)p 32640 72181 V 32640 72949 a(2)33244 72491 y Fg(c)33804 72009 y Ff(R)33804 72819 y(ij)51 b(k)24 b(l)35416 72491 y Fg(e)36019 72690 y Ff(ij)36826 72491 y Fg(e)37429 72690 y Ff(k)g(l)38722 72491 y Fg(;)11310 74417 y(P)12148 74616 y Ff(d)13057 74417 y Fn(=)369 b Fg(\021)15128 73935 y Ff(R)15080 74745 y(ij)51 b(k)24 b(l)16885 74417 y Fn(_)-555 b Fg(e)17294 74616 y Ff(ij)18295 74417 y Fn(_)h Fg(e)18705 74616 y Ff(k)24 b(l)19998 74417 y Fg(;)1107 b(P)22304 74616 y Ff(s)23164 74417 y Fn(=)369 b Fg(v)25174 74616 y Ff(i)25549 74417 y Fg(f)26190 74616 y Ff(i)27000 74417 y Fg(:)48732 73401 y Fn(\(9\))24221 77652 y(183)p eop end %%Page: 184 6 TeXDict begin 184 5 bop 0 1328 a Fn(The)486 b(v)-36 b(ector)486 b Fk(F)h Fn(and)e(the)h(scalars)h Fg(K)23 b(;)221 b(U)-6 b(;)221 b(P)21247 1527 y Ff(d)22274 1328 y Fn(and)486 b Fg(P)25694 1527 y Ff(s)26671 1328 y Fn(are)g(real)g(v)-72 b(alued)486 b(and)g(time)g(dep)36 b(enden)-36 b(t.)734 b(The)0 2878 y(meaning)473 b(of)h(individual)g(quan)-36 b(tities)474 b(in)f(\(9\))g(is)h(as)f(follo)-36 b(ws:)660 b Fk(F)474 b Fn(is)g(energy)f(\015ux)g(v)-36 b(ector)474 b(\(also)g(called)0 4428 y(energy-\015o)-36 b(w)405 b(v)-36 b(ector,)411 b(p)36 b(o)-36 b(w)g(er-\015o)g(w)405 b(v)-36 b(ector,)411 b(P)-36 b(o)g(yn)g(ting)405 b(v)-36 b(ector,)411 b(P)-36 b(o)g(yn)g(ting-Umo)g(v)405 b(v)-36 b(ector\).)569 b(It)405 b(repre-)0 5978 y(sen)-36 b(ts)392 b(the)g(amoun)-36 b(t)392 b(of)i(energy)f(passing)f(through)g(a)h(unit)f(area)h(p)36 b(erp)g(endicular)392 b(to)g(it)h(p)36 b(er)392 b(unit)g(time.)0 7527 y Fg(K)524 b Fn(is)429 b(the)g(densit)-36 b(y)429 b(of)h(kinetic)f(energy)-108 b(,)431 b Fg(U)568 b Fn(is)429 b(the)g(densit)-36 b(y)429 b(of)h(strain)e(energy)i(\(also)g(called)g (p)36 b(oten)-36 b(tial)0 9077 y(energy\),)424 b Fg(P)5850 9276 y Ff(d)6811 9077 y Fn(is)d(the)f(densit)-36 b(y)421 b(of)h(dissipated)e(energy)i(p)36 b(er)420 b(second)h(and)f Fg(P)36294 9276 y Ff(s)37206 9077 y Fn(is)i(the)e(densit)-36 b(y)421 b(of)g(energy)0 10627 y(supplied)433 b(b)-36 b(y)433 b(in)-36 b(ternal)433 b(sources)h(p)36 b(er)433 b(second.)1951 13121 y(Equation)653 b(\(8\))f(is)g(the)g (energy-balance)g(equation)h(for)g(visco)36 b(elastic)654 b(anisotropic)f(media)f(for)0 14671 y(arbitrary)556 b(time)g(dep)36 b(enden)-36 b(t)554 b(real-v)-72 b(alued)556 b(w)-36 b(a)g(v)g(e)557 b(\014eld.)944 b(It)556 b(immediately)h(yields)g(the)e (w)-36 b(ell-kno)g(wn)0 16221 y(energy-balance)420 b(equation)g(for)g (p)36 b(erfectly)420 b(elastic)g(media,)j(in)c(whic)-36 b(h)419 b Fg(P)35672 16420 y Ff(d)36581 16221 y Fn(=)369 b(0.)574 b(See)41990 15885 y(\024)41845 16221 y(Cerv)-36 b(en)18 b(\023)-668 b(y)420 b(\(2001,)0 17771 y(eq.)579 b(2.1.38\).)1951 20265 y(Since)439 b(the)f(densit)-36 b(y)439 b(of)g(strain)g(energy)g Fg(U)578 b Fn(should)439 b(b)36 b(e)438 b(p)36 b(ositiv)-36 b(e,)442 b(and)c(the)h(densit)-36 b(y)438 b(of)i(dissipated)0 21815 y(energy)477 b(p)36 b(er)477 b(second,)488 b Fg(P)11970 22014 y Ff(d)12509 21815 y Fn(,)h(non-negativ)-36 b(e,)488 b(the)476 b(6)325 b Fe(\002)g Fn(6)477 b(V)-108 b(oigt)478 b(elasticit)-36 b(y)478 b(matrix)f(m)-36 b(ust)477 b(b)36 b(e)476 b(p)36 b(ositiv)-36 b(e-)0 23365 y(de\014nite,)433 b(and)g(the)g(6)296 b Fe(\002)f Fn(6)434 b(V)-108 b(oigt)434 b(viscosit)-36 b(y)435 b(matrix)f(p)36 b(ositiv)-36 b(e)435 b(de\014nite)d(or)i(zero.) 1951 25859 y(Individual)510 b(energy)h(quan)-36 b(tities)510 b(in)g(\(9\))g(ha)-36 b(v)g(e)510 b(the)g(follo)-36 b(wing)512 b(dimensions:)731 b Fg(U)650 b Fn(and)509 b Fg(K)605 b Fn(are)511 b(ex-)0 27409 y(pressed)412 b(in)g(pascals)h(\()p Fg(P)181 b(a)p Fn(;)221 b Fg(k)45 b(g)270 b(m)15985 26927 y Fd(\000)p Fm(1)17242 27409 y Fg(s)17855 26927 y Fd(\000)p Fm(2)19112 27409 y Fn(\),)417 b Fg(P)21234 27608 y Ff(d)22187 27409 y Fn(and)412 b Fg(P)25533 27608 y Ff(s)26436 27409 y Fn(in)g(pascals)i(p)36 b(er)412 b(second)g(\()p Fg(P)181 b(a)221 b(s)41808 26927 y Fd(\000)p Fm(1)43065 27409 y Fn(;)g Fg(k)45 b(g)270 b(m)46399 26927 y Fd(\000)p Fm(1)47656 27409 y Fg(s)48269 26927 y Fd(\000)p Fm(3)49527 27409 y Fn(\),)0 28959 y(and)455 b Fk(F)h Fn(in)f(w)-36 b(atts)455 b(p)36 b(er)455 b(square)h(meter)f(\()p Fg(W)181 b(m)22188 28477 y Fd(\000)p Fm(2)23445 28959 y Fn(;)221 b Fg(k)45 b(g)269 b(s)26253 28477 y Fd(\000)p Fm(3)27510 28959 y Fn(\).)644 b(Th)-36 b(us,)460 b(all)d(terms)e(in)g(the)g (energy-balance)0 30509 y(equation)434 b(\(8\))g(are)f(expressed)h(in)f (pascals)h(p)36 b(er)434 b(second.)0 36327 y Fi(4)2152 b(Energy-balance)716 b(equations)g(for)g(real-v)-120 b(alued)3228 39249 y(time-harmonic)717 b(w)-60 b(a)g(v)g(e)718 b(\014elds)0 43059 y Fn(Let)363 b(us)g(no)-36 b(w)363 b(use)g(the)g(energy-balance)g(equation)h(\(8\))f(for)h(time-harmonic)f (w)-36 b(a)g(v)g(es)364 b(with)g(a)f(frequency)0 44609 y Fg(!)701 b(>)653 b Fn(0.)1079 b(In)601 b(Secs.)1079 b(4.1)602 b(and)e(4.2,)643 b(w)-36 b(e)601 b(use)f(t)-36 b(w)g(o)601 b(sp)36 b(eci\014cations)601 b(of)g(time-harmonic)g(real-v) -72 b(alued)0 46159 y(expressions)434 b(for)g Fg(v)9394 46358 y Ff(i)9770 46159 y Fg(;)221 b(f)10993 46358 y Ff(i)11369 46159 y Fg(;)g(\034)12516 46358 y Ff(ij)13324 46159 y Fg(;)g(e)14509 46358 y Ff(ij)15317 46159 y Fn(,)433 b(describing)h(the)f(time-harmonic)g(w)-36 b(a)g(v)g(es.)0 51303 y Fl(4.1)1793 b(Energy-balance)599 b(equation)f(for)g Fc(Re)p Fl(F)29760 50724 y Fj(+)0 54655 y Fn(W)-108 b(e)434 b(express)f(the)g(real-v)-72 b(alued)434 b(time-dep)36 b(enden)-36 b(t)431 b(quan)-36 b(tities)434 b Fg(v)31505 54854 y Ff(i)31880 54655 y Fg(;)221 b(f)33103 54854 y Ff(i)33480 54655 y Fg(;)g(\034)34627 54854 y Ff(ij)35435 54655 y Fg(;)g(e)36620 54854 y Ff(ij)37861 54655 y Fn(as)434 b(follo)-36 b(ws:)9810 57470 y Fg(v)10439 57669 y Ff(i)11184 57470 y Fn(=)12697 56947 y Fm(1)p 12697 57160 471 45 v 12697 57928 a(2)13300 57470 y Fn(\()p Fg(V)14564 57669 y Ff(i)14940 57470 y Fg(e)15543 56988 y Fm(i)p Ff(!)32 b(t)17111 57470 y Fn(+)295 b Fg(V)19465 56988 y Fd(\003)19176 57798 y Ff(i)19991 57470 y Fg(e)20594 56988 y Fd(\000)p Fm(i)p Ff(!)32 b(t)22599 57470 y Fn(\))434 b Fg(;)1355 b(e)25858 57669 y Ff(ij)27034 57470 y Fn(=)28548 56947 y Fm(1)p 28548 57160 V 28548 57928 a(2)29151 57470 y Fn(\()p Fg(E)30620 57669 y Ff(ij)31427 57470 y Fg(e)32030 56988 y Fm(i)p Ff(!)32 b(t)33598 57470 y Fn(+)295 b Fg(E)35946 56988 y Fd(\003)35868 57798 y Ff(ij)36675 57470 y Fg(e)37278 56988 y Fd(\000)p Fm(i)p Ff(!)32 b(t)39283 57470 y Fn(\))433 b Fg(;)9810 59396 y(f)10451 59595 y Ff(i)11196 59396 y Fn(=)12709 58873 y Fm(1)p 12709 59086 V 12709 59854 a(2)13313 59396 y Fn(\()p Fe(F)14774 59595 y Ff(i)15149 59396 y Fg(e)15752 58914 y Fm(i)p Ff(!)32 b(t)17320 59396 y Fn(+)295 b Fe(F)19714 58914 y Fd(\003)19582 59725 y Ff(i)20239 59396 y Fg(e)20842 58914 y Fd(\000)p Fm(i)p Ff(!)32 b(t)22847 59396 y Fn(\))434 b Fg(;)1107 b(\034)25820 59595 y Ff(ij)26996 59396 y Fn(=)28510 58873 y Fm(1)p 28510 59086 V 28510 59854 a(2)29113 59396 y Fn(\()p Fg(T)30381 59595 y Ff(ij)31188 59396 y Fg(e)31791 58914 y Fm(i)p Ff(!)32 b(t)33360 59396 y Fn(+)294 b Fg(T)35609 58914 y Fd(\003)35428 59725 y Ff(ij)36236 59396 y Fg(e)36839 58914 y Fd(\000)p Fm(i)p Ff(!)32 b(t)38844 59396 y Fn(\))433 b Fg(:)48081 58380 y Fn(\(10\))0 61992 y(Here)339 b Fg(V)3734 62191 y Ff(i)4110 61992 y Fg(;)221 b(F)5534 62191 y Ff(i)5910 61992 y Fg(;)g(E)7455 62191 y Ff(ij)8601 61992 y Fn(and)339 b Fg(T)11798 62191 y Ff(ij)12944 61992 y Fn(are)g(time-indep)36 b(enden)-36 b(t,)355 b(complex-v)-72 b(alued,)358 b(frequency-dep)36 b(enden)-36 b(t)337 b(quan-)0 63541 y(tities.)579 b(W)-108 b(e)433 b(no)-36 b(w)434 b(insert)f(\(10\))h(in)-36 b(to)433 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Fd(\003)19593 68490 y Ff(ij)20696 68161 y Fn(+)295 b Fg(V)23050 67613 y Fd(\003)22761 68490 y Ff(i)23576 68161 y Fg(T)24338 68360 y Ff(ij)25145 68161 y Fn(\))g Fe(\000)27408 67638 y Fm(1)p 27408 67851 V 27408 68619 a(4)28011 68161 y Fn([)p Fg(V)29130 68360 y Ff(i)29506 68161 y Fg(T)30268 68360 y Ff(ij)31076 68161 y Fg(e)31679 67613 y Fm(2i)p Ff(!)32 b(t)33718 68161 y Fn(+)294 b Fg(V)36072 67613 y Fd(\003)35782 68490 y Ff(i)36598 68161 y Fg(T)37541 67613 y Fd(\003)37360 68490 y Ff(ij)38167 68161 y Fg(e)38770 67613 y Fd(\000)p Fm(2i)p Ff(!)32 b(t)41246 68161 y Fn(])13304 70247 y(=)1106 b Fe(\000)16588 69724 y Fm(1)p 16588 69937 V 16588 70705 a(2)17191 70247 y Fn(Re)q(\()p Fg(V)20280 69698 y Fd(\003)19991 70575 y Ff(i)20806 70247 y Fg(T)21568 70446 y Ff(ij)22375 70247 y Fn(\))295 b Fe(\000)24637 69724 y Fm(1)p 24637 69937 V 24637 70705 a(2)25241 70247 y Fn(Re\()p Fg(V)28040 70446 y Ff(i)28416 70247 y Fg(T)29178 70446 y Ff(ij)29985 70247 y Fg(e)30588 69698 y Fm(2i)p Ff(!)32 b(t)32332 70247 y Fn(\))433 b Fg(:)14449 b Fn(\(11\))0 72781 y(Since)537 b Fg(v)4129 72980 y Ff(i)5042 72781 y Fn(and)g Fg(\034)8240 72980 y Ff(ij)9585 72781 y Fn(are)h(real)g(v)-72 b(alued,)564 b(the)537 b(expression)h(for)g Fe(\000)p Fg(v)31530 72980 y Ff(i)31905 72781 y Fg(\034)32470 72980 y Ff(ij)33815 72781 y Fn(is)g(also)g(real-v)-72 b(alued.)891 b(The)537 b(\014rst)0 74331 y(term)580 b(on)h(the)f(righ)-36 b(t-hand)579 b(side)h(of)i(\(11\))f(is)g(time-indep)36 b(enden)-36 b(t,)615 b(and)580 b(the)g(second)g(term)h(is)g(time)24221 77652 y(184)p eop end %%Page: 185 7 TeXDict begin 185 6 bop 0 1328 a Fn(harmonic,)443 b(with)f(frequency)g (2)p Fg(!)48 b Fn(.)601 b(W)-108 b(e)442 b(can)f(immediately)h(see)g (that)f(the)g(\014rst)f(term)h(represen)-36 b(ts)440 b(the)0 2878 y(time)434 b(a)-36 b(v)g(erage)434 b(o)-36 b(v)g(er)434 b(one)g(p)36 b(erio)g(d.)578 b(F)-108 b(or)433 b(suc)-36 b(h)433 b(time-a)-36 b(v)g(eraging,)435 b(the)e(second)g (term)g(is)h(zero.)1951 5373 y(W)-108 b(e)434 b(in)-36 b(tro)36 b(duce)432 b(complex-v)-72 b(alued)434 b(frequency-dep)36 b(enden)-36 b(t)432 b(v)-36 b(ectors)434 b Fk(F)36678 4891 y Fm(+)37465 5373 y Fg(;)38414 5033 y Fn(\026)38270 5373 y Fk(F)f Fn(and)42316 5033 y(~)42172 5373 y Fk(F)p Fn(,)21321 8218 y Fk(F)22260 7669 y Fm(+)23417 8218 y Fn(=)24942 7878 y(\026)24797 8218 y Fk(F)296 b Fn(+)27483 7878 y(~)27338 8218 y Fk(F)434 b Fg(;)19009 b Fn(\(12\))0 11177 y(where)3902 10838 y(\026)3757 11177 y Fk(F)434 b Fn(and)7804 10838 y(~)7659 11177 y Fk(F)g Fn(are)g(giv)-36 b(en)434 b(b)-36 b(y)434 b(relations)14775 13686 y(\026)14480 14022 y Fg(F)15322 14221 y Ff(j)16178 14022 y Fn(=)368 b Fe(\000)18724 13499 y Fm(1)p 18724 13712 471 45 v 18724 14480 a(2)19328 14022 y Fg(V)20375 13474 y Fd(\003)20086 14351 y Ff(i)20901 14022 y Fg(T)21663 14221 y Ff(ij)22904 14022 y Fg(;)25515 13686 y Fn(~)25221 14022 y Fg(F)26063 14221 y Ff(j)26919 14022 y Fn(=)g Fe(\000)29465 13499 y Fm(1)p 29465 13712 V 29465 14480 a(2)30069 14022 y Fg(V)30827 14221 y Ff(i)31203 14022 y Fg(T)31965 14221 y Ff(ij)32772 14022 y Fg(e)33375 13474 y Fm(2i)p Ff(!)32 b(t)35552 14022 y Fg(:)12168 b Fn(\(13\))0 17107 y(Here)3236 16768 y(\026)3091 17107 y Fk(F)455 b Fn(is)f(the)g(complex-v)-72 b(alued)454 b(time-a)-36 b(v)g(eraged)455 b(energy-\015ux)f(v)-36 b(ector,)36812 16768 y(~)36668 17107 y Fk(F)454 b Fn(is)h(the)e (complex-v)-72 b(alued)0 18657 y(time-dep)36 b(enden)-36 b(t)536 b(energy-\015ux)i(v)-36 b(ector,)565 b(and)538 b Fk(F)24146 18175 y Fm(+)25472 18657 y Fn(is)g(the)g(complete)g (complex-v)-72 b(alued)539 b(energy-\015ux)0 20207 y(v)-36 b(ector.)579 b(Then)433 b(\(11\))h(can)f(b)36 b(e)434 b(expressed)f(as)h(follo)-36 b(ws:)20547 23052 y Fe(\000)p Fg(v)22209 23251 y Ff(i)22585 23052 y Fg(\034)23150 23251 y Ff(ij)24326 23052 y Fn(=)369 b(Re)p Fg(F)28265 22504 y Fm(+)28084 23381 y Ff(j)29485 23052 y Fg(:)18235 b Fn(\(14\))0 25897 y(Quite)433 b(analogous)i(expressions)f(can)f(b)36 b(e)433 b(deriv)-36 b(ed)434 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Ff(s)22176 39681 y Fn(and)24998 39345 y(\026)24705 39681 y Fg(P)25543 39880 y Ff(d)26517 39681 y Fn(are)433 b(giv)-36 b(en)435 b(b)-36 b(y)433 b(relations:)15117 42358 y(\026)14769 42694 y Fg(K)464 b Fn(=)17853 42171 y Fm(1)p 17853 42384 V 17853 43152 a(4)18456 42694 y Fg(\032V)19885 42893 y Ff(i)20261 42694 y Fg(V)21309 42212 y Fd(\003)21019 43022 y Ff(i)22268 42694 y Fg(;)23958 42358 y Fn(\026)23736 42694 y Fg(U)508 b Fn(=)26641 42171 y Fm(1)p 26641 42384 V 26641 43152 a(4)27244 42694 y Fg(c)27804 42212 y Ff(R)27804 43022 y(ij)51 b(k)24 b(l)29416 42694 y Fg(E)30379 42893 y Ff(ij)31186 42694 y Fg(E)32227 42212 y Fd(\003)32149 43022 y Ff(k)g(l)33443 42694 y Fg(;)15061 44284 y Fn(\026)14769 44620 y Fg(P)15607 44819 y Ff(s)16467 44620 y Fn(=)17980 44097 y Fm(1)p 17980 44310 V 17980 45078 a(2)18583 44620 y Fg(V)19631 44138 y Fd(\003)19341 44948 y Ff(i)20156 44620 y Fe(F)21111 44819 y Ff(i)21920 44620 y Fg(;)24029 44284 y Fn(\026)23736 44620 y Fg(P)24574 44819 y Ff(d)25483 44620 y 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y(represen)-36 b(tation)486 b(of)h(the)g(w)-36 b(a)g(v)g(e)487 b(\014eld,)500 b(ma)-36 b(y)488 b(b)36 b(e)486 b(used)g(as)i(a)f(generalization)h(of)f (the)g(approac)-36 b(h)486 b(based)0 72652 y(on)433 b(the)h(real-v)-72 b(alued)433 b(represen)-36 b(tation)433 b(of)h(the)f(w)-36 b(a)g(v)g(e)435 b(\014eld.)24221 77652 y(189)p eop end %%Page: 190 12 TeXDict begin 190 11 bop 1951 1328 a Fn(In)502 b(the)g(follo)-36 b(wing,)522 b(w)-36 b(e)502 b(consider)h(v)-36 b(ectors)502 b Fg(v)24123 1527 y Ff(i)24499 1328 y Fn(,)520 b Fg(f)26021 1527 y Ff(i)26899 1328 y Fn(and)502 b(tensors)g Fg(\034)34553 1527 y Ff(ij)35360 1328 y Fn(,)519 b Fg(e)36843 1527 y Ff(ij)38153 1328 y Fn(to)502 b(b)36 b(e)502 b(complex-v)-72 b(alued)0 2878 y(time-harmonic)433 b(quan)-36 b(tities:)17162 5924 y Fg(v)17791 6123 y Ff(i)18535 5924 y Fn(=)369 b Fg(V)20674 6123 y Ff(i)21050 5924 y Fg(e)21653 5442 y Fm(i)p Ff(!)32 b(t)23360 5924 y Fg(;)2149 b(\034)26435 6123 y Ff(ij)27611 5924 y Fn(=)368 b Fg(T)29753 6123 y 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Ff(i)22053 70555 y Fg(e)22656 70006 y Fm(2i)p Ff(!)g(t)24833 70555 y Fg(:)0 72781 y Fn(Inserting)550 b(\(49\))g(in)-36 b(to)550 b(\(56\),)580 b(w)-36 b(e)551 b(can)f(see)g(that)g(the)g(resulting)g(expressions)g (are)h(the)e(same)i(as)f(the)0 74331 y(expressions)434 b(\(13\))g(and)f(\(18\))h(for)g(time-dep)36 b(enden)-36 b(t)431 b(energy-related)i(quan)-36 b(tities.)24221 77652 y(191)p eop end %%Page: 192 14 TeXDict begin 192 13 bop 0 1328 a Fl(5.4)1793 b(Complete)599 b(energy-balance)g(equation)0 4681 y Fn(In)451 b(Sections)g(5.2)h(and)e (5.3,)457 b(w)-36 b(e)451 b(deriv)-36 b(ed)451 b(t)-36 b(w)g(o)452 b(energy-balance)f(equations:)614 b(one)451 b(for)g(time-a)-36 b(v)g(eraged)0 6231 y(energy-related)313 b(quan)-36 b(tities,)338 b(\(52\),)g(and)313 b(the)g(other)g(for)h (time-dep)36 b(enden)-36 b(t)311 b(energy-related)j(quan)-36 b(tities)0 7780 y(\(55\).)868 b(It)530 b(is)g(natural)g(to)g(com)-36 b(bine)529 b(b)36 b(oth)530 b(these)f(equations)i(in)f(suc)-36 b(h)529 b(a)h(w)-36 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30100 a(2)15486 29642 y Fg(v)16115 29841 y Ff(i)16491 29642 y Fg(f)17132 29841 y Ff(i)17876 29642 y Fn(=)369 b(Re\()p Fg(v)21927 29841 y Ff(i)22303 29642 y Fn(\))p Fg(f)23450 29841 y Ff(i)24259 29642 y Fg(:)0 32038 y Fn(W)-108 b(e)317 b(call)h Fk(F)5330 31556 y Fm(+)6434 32038 y Fn(the)e(complete)h(energy-\015ux)g(v)-36 b(ector,)341 b(and)316 b Fg(K)28320 31556 y Fm(+)29107 32038 y Fg(;)221 b(U)30711 31556 y Fm(+)31499 32038 y Fg(;)g(P)33100 31488 y Fm(+)32919 32372 y Ff(d)34205 32038 y Fn(and)316 b Fg(P)37636 31556 y Fm(+)37455 32366 y Ff(s)38740 32038 y Fn(the)h(complete)g(energy-)0 33587 y(related)370 b(quan)-36 b(tities.)557 b(It)370 b(is)g(in)-36 b(teresting)369 b(to)h(note)f(a)h(simplicit)-36 b(y)371 b(of)f(the)f(expressions)h(for)g(the)g(complete)0 35137 y(energy-related)433 b(quan)-36 b(tities,)434 b(particularly)h(of)f Fk(F)24322 34655 y Fm(+)25110 35137 y Fg(;)221 b(K)26894 34655 y Fm(+)28115 35137 y Fn(and)433 b Fg(P)31663 34655 y Fm(+)31482 35466 y Ff(s)32450 35137 y 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b(ophys.)598 b(J.)464 b(Int.)p Fn(,)432 b(in)i(press.)24221 77652 y(194)p eop end %%Page: 195 17 TeXDict begin 195 16 bop 1951 1328 a Fn(Desc)-36 b(hamps,)506 b(M.)491 b(\(1990\).)752 b(R)-36 b(\023)-614 b(e\015exion-r)-36 b(\023)-614 b(efraction)492 b(de)f(l'onde)g(plane)g(h)-36 b(\023)-614 b(et)-36 b(\023)-614 b(erog)-36 b(\022)-614 b(ene:)694 b(Repartition)0 2878 y(de)433 b(l')-36 b(\023)-614 b(energie.)579 b Fh(J.)465 b(A)-66 b(c)g(oust.)p Fn(,)432 b Fk(3)p Fn(,)i(251{261.)1951 5373 y(Ha)-36 b(y)g(es,)572 b(M.)544 b(\(1980\).)910 b(Energy)544 b(\015ux)f(for)h(trains)g(of)g (inhomogeneous)g(plane)g(w)-36 b(a)g(v)g(es.)909 b Fh(Pr)-66 b(o)g(c.)901 b(R.)0 6922 y(So)-66 b(c.)596 b(L)-66 b(ondon)465 b(A)p Fn(,)433 b Fk(370)p Fn(,)h(417{429.)1951 9417 y(Mann,)576 b(J.A.,)i(Tic)-36 b(h)g(y)-108 b(,)577 b(I.,)h(and)547 b(Romano,)578 b(A.J.)549 b(\(1987\).)922 b(Instan)-36 b(taneous)548 b(and)f(time-a)-36 b(v)g(eraged)0 10967 y(energy)434 b(transfer)f(in)h(acoustic)g(\014elds.)578 b Fh(J.)464 b(A)-66 b(c)g(oust.)597 b(So)-66 b(c.)596 b(A)-33 b(m.)p Fn(,)432 b Fk(82)p Fn(,)j(17{29.)1951 13461 y(Scott,)493 b(N.H.)482 b(\(1997\).)723 b(Energy)482 b(\015ux)f(and)g(dissipation)g(of)h(inhomogeneous)g(plane)f(w)-36 b(a)g(v)g(e)482 b(forms)0 15011 y(in)433 b(Kelvin-V)-108 b(oigt)435 b(visco)36 b(elasticit)-36 b(y)-108 b(.)581 b Fh(Int.)596 b(J.)464 b(Engng.)596 b(Sci.)p Fn(,)433 b Fk(35)p Fn(,)h(561{572.)1951 17506 y(Sh)-36 b(uv)-72 b(alo)-36 b(v,)352 b(A.L.,)h(and)331 b(Scott,)352 b(N.H.)333 b(\(1999\).)545 b(On)331 b(the)g(prop)36 b(erties)331 b(of)i(homogeneous)e(visco)36 b(elastic)0 19056 y(w)-36 b(a)g(v)g(es.)579 b Fh(Q.J.)464 b(Me)-66 b(ch.)597 b(Appl.)g(Math.)p Fn(,)433 b Fk(52)p Fn(,)i(405{417.)24221 77652 y(195)p eop end %%Trailer userdict /end-hook known{end-hook}if %%EOF