%!PS-Adobe-2.0 %%Creator: dvips 5.54 Copyright 1986, 1994 Radical Eye Software %%Title: TTDERSEG.dvi %%CreationDate: Thu Apr 01 10:26:08 1999 %%Pages: 4 %%PageOrder: Ascend %%BoundingBox: 0 0 612 792 %%EndComments %DVIPSCommandLine: C:\EMTEX\DVIPS16.EXE TTDERSEG %DVIPSParameters: dpi=300, comments removed %DVIPSSource: TeX output 1999.04.01:1026 %%BeginProcSet: tex.pro /TeXDict 250 dict def TeXDict begin /N{def}def /B{bind def}N /S{exch}N /X{S N}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 dup dup 4 get round 4 exch put dup dup 5 get round 5 exch put setmatrix}N /@landscape{/isls true N}B /@manualfeed{ statusdict /manualfeed true put}B /@copies{/#copies X}B /FMat[1 0 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58 2 v 1202 2509 a(@)r(x)1246 2513 y Fg(i)1289 2459 y Fh(@)p 1270 2475 61 2 v 1270 2509 a(@)r(x)1314 2513 y Fg(j)1335 2483 y Fh(H)m Fi(\()p Fh(x)1401 2487 y Fg(q)1417 2483 y Fh(;)g(p)1453 2487 y Fg(r)1471 2483 y Fi(\))p Fh(Q)1516 2487 y Fg(j)q(a)1551 2483 y Ff(\000)1606 2459 y Fh(@)p 1588 2475 58 2 v 1588 2509 a(@)r(x)1632 2513 y Fg(i)1673 2459 y Fh(@)p 1656 2475 V 1656 2509 a(@)r(p)1697 2513 y Fg(j)1719 2483 y Fh(H)m Fi(\()p Fh(x)1785 2487 y Fg(q)1801 2483 y Fh(;)g(p)1837 2487 y Fg(r)1854 2483 y Fi(\))p Fh(P)1894 2487 y Fg(j)q(a)1882 2550 y Fi(\(11\))p eop %%Page: 2 2 2 1 bop 550 29 a Fl(Third)14 b(and)h(higher)f(tra)o(v)o(el{ti)o(me)f (deriv)m(ativ)o(es)0 106 y Fi(can)h(b)q(e)f(obtained)i(b)o(y)f (di\013eren)o(tiating)i(the)e(ra)o(y)f(tracing)h(system)0 143 y(\(5\))i(with)h(resp)q(ect)f(to)h Fh(\015)361 147 y Fg(a)379 143 y Fi(.)27 b(After)15 b(insertion)k(of)c(\(10\),)i(the)g (\014rst)0 180 y(equation)d(of)e(the)g(dynamic)i(ra)o(y)f(tracing)g (system)g(\(11\))f(b)q(ecomes)170 242 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y(need)12 b(not)f(ha)o(v)o(e)h(an)o (y)g(sup)q(erscript.)18 b(Eac)o(h)12 b(p)q(ossible)i(summand)e(is)990 1740 y(con)o(tained)i(within)g(the)e(last)h(term)f(in)h(equation)g (\(17\))g(just)f(once.)990 1787 y(The)18 b(terms)g(of)g(forms)g Fh(H)1381 1771 y Fg(;a)1409 1787 y Fh(\034)1426 1791 y Fg(;a:::)1482 1787 y Fi(,)h Fh(H)1547 1769 y Fg(;a)1543 1792 y(;i)1575 1787 y Fh(\034)1592 1791 y Fg(;a:::)1667 1787 y Fi(and)g Fh(H)1782 1771 y Fg(;ab)1823 1787 y Fh(\034)1840 1791 y Fg(;ai)1880 1787 y Fh(\034)1897 1791 y Fg(;b:::)990 1837 y Fi(in)g(\(17\))g(cancel)g(out)g(for)f(the)h(third)g(and)g (higher)h(tra)o(v)o(el{time)990 1875 y(deriv)n(ativ)o(es,)15 b(the)e(other)g(terms)g(are)g(retained.)990 1946 y(Equations)f(\(14\),) e(\(15\))h(and)f(\(17\))g(ma)o(y)h(b)q(e)f(used)h(to)f(compute)h(the) 990 1983 y(third)19 b(and)f(higher)h(partial)h(tra)o(v)o(el{time)f (deriv)n(ativ)o(es)h(in)e(b)q(oth)990 2020 y(isotropic)13 b(and)f(anisotropic)i(mo)q(dels)e(b)o(y)g(n)o(umerical)h(quadrature)990 2058 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(for)f(the)g(\014rst{order)h(deriv)n(ativ)o(es,)184 157 y Fh(K)217 161 y Fg(i)240 157 y Fi(=)d Ff(\000)p Fh(H)342 162 y Fg(;i)379 157 y Fi(+)d Fh(H)452 139 y Fg(;r)448 162 y(;i)479 157 y Fh(\034)491 161 y Fg(;r)526 157 y Fi(+)g Fh(H)599 139 y Fg(;r)q(s)640 157 y Fh(\034)652 161 y Fg(;r)678 157 y Fh(\034)690 161 y Fg(;si)766 157 y Fh(;)115 b Fi(\(22\))0 214 y(cannot)10 b(b)q(e)f(used)h(for)f(the)g (calculation)k(of)c(the)g(tra)o(v)o(el{time)i(gradi-)0 251 y(en)o(t,)f(but)g(will)g(turn)g(out)g(to)f(b)q(e)h(v)n(aluable)h (for)f(the)f(transformation)0 289 y(of)k(matrix)h Fh(Q)199 293 y Fg(ia)242 289 y Fi(to)f(the)g(one)g(de\014ned)h(for)f(di\013eren) o(t)i Fh(\015)788 293 y Fe(3)805 289 y Fi(.)0 360 y(On)9 b(the)g(other)g(hand,)h(equation)h(\(17\),)e(sp)q(eci\014ed)i(for)e (the)g(second{)0 397 y(order)k(tra)o(v)o(el{time)i(deriv)n(ativ)o(es,) 235 452 y Fh(K)268 456 y Fg(ij)306 452 y Fi(=)10 b Ff(\000)p Fh(H)407 457 y Fg(;ij)459 452 y Fi(+)e Fh(H)532 434 y Fg(;ab)574 452 y Fh(\034)586 456 y 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Ff(\000)r Fh(H)1718 833 y Fg(;abc)1774 851 y Fh(\034)1786 855 y Fg(;ai)1825 851 y Fh(\034)1837 855 y Fg(;bj)1876 851 y Fh(\034)1888 855 y Fg(;ck)1939 851 y Fh(;)1882 954 y Fi(\(37\))990 992 y(the)j(fourth)g(tra)o(v)o(el{time)i(partial)g (deriv)n(ativ)o(es,)984 1047 y Fh(K)1017 1051 y Fg(ij)q(k)q(l)1081 1047 y Fi(=)9 b Ff(\000)f Fh(H)1189 1052 y Fg(;ij)q(k)q(l)1268 1047 y Ff(\000)g Fh(H)1341 1029 y Fg(;a)1337 1053 y(;ij)q(k)1337 1079 y(;ij)q(l)1337 1106 y(;ik)q(l)1337 1132 y(;j)q(k)q(l)1397 1047 y Fh(\034)1408 1052 y Fg(;al)1408 1079 y(;ak)1408 1105 y(;aj)1408 1132 y(;ai)1468 1047 y Ff(\000)h Fh(H)1542 1029 y Fg(;a)1538 1053 y(;ij)1538 1079 y(;ik)1538 1106 y(;j)q(k)1538 1132 y(;il)1538 1159 y(;j)q(l)1538 1185 y(;k)q(l)1586 1047 y Fh(\034)1597 1052 y Fg(;ak)q(l)1597 1079 y(;aj)q(l)1597 1105 y(;ail)1597 1132 y(;aj)q(k)1597 1158 y(;aik)1597 1185 y(;aij)1672 1047 y Ff(\000)f Fh(H)1745 1029 y Fg(;ab)1741 1053 y(;ij)1741 1079 y(;ik)1741 1106 y(;j)q(k)1741 1132 y(;il)1741 1159 y(;j)q(l)1741 1185 y(;k)q(l)1790 1047 y Fh(\034)1801 1052 y Fg(;ak)1801 1079 y(;aj)1801 1105 y(;ai)1801 1132 y(;aj)1801 1158 y(;ai)1801 1185 y(;ai)1852 1047 y Fh(\034)1863 1052 y Fg(;bl)1863 1079 y(;bl)1863 1105 y(;bl)1863 1132 y(;bk)1863 1158 y(;bk)1863 1185 y(;bj)1120 1250 y Ff(\000)g Fh(H)1193 1232 y Fg(;ab)1189 1255 y(;i)1189 1282 y(;j)1189 1308 y(;i)1189 1335 y(;k)1189 1361 y(;j)1189 1388 y(;k)1189 1414 y(;i)1189 1441 y(;l)1189 1467 y(;j)1189 1494 y(;l)1189 1520 y(;k)1189 1547 y(;l)1235 1250 y Fh(\034)1246 1255 y Fg(;aj)1246 1281 y(;ai)1246 1308 y(;ak)1246 1334 y(;ai)1246 1361 y(;ak)1246 1387 y(;aj)1246 1414 y(;al)1246 1440 y(;ai)1246 1467 y(;al)1246 1493 y(;aj)1246 1520 y(;al)1246 1546 y(;ak)1297 1250 y Fh(\034)1308 1255 y Fg(;bk)q(l)1308 1281 y(;bk)q(l)1308 1308 y(;bj)q(l)1308 1334 y(;bj)q(l)1308 1361 y(;bil)1308 1387 y(;bil)1308 1414 y(;bj)q(k)1308 1440 y(;bj)q(k)1308 1467 y(;bik)1308 1493 y(;bik)1308 1520 y(;bij)1308 1546 y(;bij)1379 1250 y Ff(\000)h Fh(H)1453 1232 y Fg(;ab)1494 1250 y Fh(\034)1505 1255 y Fg(;aij)1505 1281 y(;aik)1505 1308 y(;aj)q(k)1571 1250 y Fh(\034)1582 1255 y Fg(;bk)q(l)1582 1281 y(;bj)q(l)1582 1308 y(;bil)1649 1250 y Ff(\000)f Fh(H)1722 1232 y Fg(;abc)1718 1255 y(;i)1718 1282 y(;j)1718 1308 y(;k)1718 1335 y(;l)1778 1250 y Fh(\034)1789 1255 y Fg(;aj)1789 1281 y(;ai)1789 1308 y(;ai)1789 1334 y(;ai)1837 1250 y Fh(\034)1848 1255 y Fg(;bk)1848 1281 y(;bk)1848 1308 y(;bj)1848 1334 y(;bj)1897 1250 y Fh(\034)1908 1255 y Fg(;cl)1908 1281 y(;cl)1908 1308 y(;cl)1908 1334 y(;ck)1120 1611 y Ff(\000)g Fh(H)1193 1593 y Fg(;abc)1249 1611 y Fh(\034)1260 1616 y Fg(;aij)1260 1643 y(;aik)1260 1669 y(;aj)q(k)1260 1696 y(;ail)1260 1722 y(;aj)q(l)1260 1749 y(;ak)q(l)1325 1611 y Fh(\034)1336 1616 y Fg(;bk)1336 1643 y(;bj)1336 1669 y(;bi)1336 1696 y(;bj)1336 1722 y(;bi)1336 1749 y(;bi)1385 1611 y Fh(\034)1396 1616 y Fg(;cl)1396 1643 y(;cl)1396 1669 y(;cl)1396 1696 y(;ck)1396 1722 y(;ck)1396 1749 y(;cj)1453 1611 y Ff(\000)g Fh(H)1526 1593 y Fg(;abcd)1598 1611 y Fh(\034)1609 1616 y Fg(;ai)1654 1611 y Fh(\034)1665 1616 y Fg(;bj)1711 1611 y Fh(\034)1722 1616 y Fg(;ck)1770 1611 y Fh(\034)1781 1616 y Fg(;dl)1838 1611 y Fh(;)33 b Fi(\(38\))990 1801 y(and)14 b(so)f(on.)990 1872 y Fj(4)30 b(Examples)15 b(of)f(the)h(in)o(tegrands)f(in)f (isotropic)h(media)990 1942 y Fi(In)c(isotropic)j(mo)q(dels,)f(the)e (Hamiltonian)j(is)e(a)g(function)h(of)e Fh(p)1882 1946 y Fg(k)1901 1942 y Fh(p)1920 1946 y Fg(k)1939 1942 y Fi(,)990 1980 y(but)j(is)h(indep)q(enden)o(t)h(of)d(the)h(direction)i (of)e(the)g(slo)o(wness)h(v)o(ector)990 2017 y Fh(p)1009 2021 y Fg(i)1022 2017 y Fi(.)j(If)c(the)g(Hamiltonian)i(is)f(c)o(hosen) g(in)g(the)f(form)g(of)1189 2071 y Fh(H)s Fi(\()p Fh(x)1261 2075 y Fg(q)1278 2071 y Fh(;)6 b(p)1314 2075 y Fg(r)1332 2071 y Fi(\))k(=)p 1398 2041 V 11 w Fh(H)r Fi(\()p Fh(x)1469 2075 y Fg(q)1486 2071 y Fh(;)c(p)1522 2075 y Fg(r)1540 2071 y Fi(\))i Ff(\000)h Fh(V)g Fi(\()p Fh(x)1670 2075 y Fg(q)1687 2071 y Fi(\))38 b Fh(;)131 b Fi(\(39\))990 2128 y(where)p 1100 2097 V 12 w Fh(H)r Fi(\()p Fh(x)1171 2132 y Fg(q)1188 2128 y Fh(;)6 b(p)1224 2132 y Fg(r)1242 2128 y Fi(\))11 b(is)i(a)e Fa(homo)n(gene)n(ous)f(function)f(of)j(the)g (\014rst)g(or-)990 2173 y(der)18 b Fi(with)i(resp)q(ect)f(to)g Ff(j)p Fh(p)p Ff(j)1387 2157 y Fe(2)1424 2173 y Fi(=)i Fh(p)1494 2177 y Fg(k)1513 2173 y Fh(p)1532 2177 y Fg(k)1552 2173 y Fi(,)e(all)i(partial)g(deriv)n(ativ)o(es)990 2210 y(of)e(the)g(Hamiltonian)j(con)o(taining)f(the)e(third)h(or)f(higher)i (par-)990 2248 y(tial)16 b(deriv)n(ativ)o(es)h(with)e(resp)q(ect)h(to)e (the)h(slo)o(wness{v)o(ector)i(com-)990 2285 y(p)q(onen)o(ts)d(are)f (zero,)1351 2341 y Fh(H)1386 2323 y Fg(;abc:::)1382 2346 y(;:::)1480 2341 y Fi(=)e(0)39 b Fh(;)292 b Fi(\(40\))990 2400 y(whic)o(h)16 b(considerably)i(simpli\014es)g(equations)f(suc)o(h) e(as)g(\(37\))g(and)990 2438 y(\(38\),)1071 2495 y Fh(K)1104 2499 y Fg(ij)q(k)1158 2495 y Fi(=)c Ff(\000)p Fh(H)1260 2500 y Fg(;ij)q(k)1328 2495 y Ff(\000)e Fh(H)1402 2477 y Fg(;a)1398 2501 y(;ij)1398 2527 y(;ik)1398 2554 y(;j)q(k)1447 2495 y Fh(\034)1458 2500 y Fg(;ak)1458 2527 y(;aj)1458 2553 y(;ai)1518 2495 y Ff(\000)f Fh(H)1591 2477 y Fg(;ab)1587 2501 y(;i)1587 2527 y(;j)1587 2554 y(;k)1633 2495 y Fh(\034)1644 2500 y Fg(;aj)1644 2527 y(;ai)1644 2553 y(;ai)1692 2495 y Fh(\034)1703 2500 y Fg(;bk)1703 2527 y(;bk)1703 2553 y(;bj)1790 2495 y Fh(;)81 b Fi(\(41\))p eop %%Page: 4 4 4 3 bop 550 29 a Fl(Third)14 b(and)h(higher)f(tra)o(v)o(el{ti)o(me)f (deriv)m(ativ)o(es)16 110 y Fh(K)49 114 y Fg(ij)q(k)q(l)114 110 y Fi(=)8 b Ff(\000)g Fh(H)221 115 y Fg(;ij)q(k)q(l)300 110 y Ff(\000)g Fh(H)373 92 y Fg(;a)369 116 y(;ij)q(k)369 142 y(;ij)q(l)369 169 y(;ik)q(l)369 195 y(;j)q(k)q(l)429 110 y Fh(\034)440 115 y Fg(;al)440 142 y(;ak)440 168 y(;aj)440 195 y(;ai)500 110 y Ff(\000)h Fh(H)574 92 y Fg(;a)570 116 y(;ij)570 142 y(;ik)570 169 y(;j)q(k)570 195 y(;il)570 222 y(;j)q(l)570 248 y(;k)q(l)619 110 y Fh(\034)630 115 y Fg(;ak)q(l)630 142 y(;aj)q(l)630 168 y(;ail)630 195 y(;aj)q(k)630 221 y(;aik)630 248 y(;aij)704 110 y Ff(\000)f Fh(H)777 92 y Fg(;ab)773 116 y(;ij)773 142 y(;ik)773 169 y(;j)q(k)773 195 y(;il)773 222 y(;j)q(l)773 248 y(;k)q(l)822 110 y Fh(\034)833 115 y Fg(;ak)833 142 y(;aj)833 168 y(;ai)833 195 y(;aj)833 221 y(;ai)833 248 y(;ai)885 110 y Fh(\034)896 115 y Fg(;bl)896 142 y(;bl)896 168 y(;bl)896 195 y(;bk)896 221 y(;bk)896 248 y(;bj)152 313 y Ff(\000)g Fh(H)225 295 y Fg(;ab)221 318 y(;i)221 345 y(;j)221 371 y(;i)221 398 y(;k)221 424 y(;j)221 451 y(;k)221 477 y(;i)221 504 y(;l)221 530 y(;j)221 557 y(;l)221 583 y(;k)221 610 y(;l)267 313 y Fh(\034)278 318 y Fg(;aj)278 344 y(;ai)278 371 y(;ak)278 397 y(;ai)278 424 y(;ak)278 450 y(;aj)278 477 y(;al)278 503 y(;ai)278 530 y(;al)278 556 y(;aj)278 583 y(;al)278 609 y(;ak)330 313 y Fh(\034)341 318 y Fg(;bk)q(l)341 344 y(;bk)q(l)341 371 y(;bj)q(l)341 397 y(;bj)q(l)341 424 y(;bil)341 450 y(;bil)341 477 y(;bj)q(k)341 503 y(;bj)q(k)341 530 y(;bik)341 556 y(;bik)341 583 y(;bij)341 609 y(;bij)411 313 y Ff(\000)h Fh(H)485 295 y Fg(;ab)527 313 y Fh(\034)538 318 y Fg(;aij)538 344 y(;aik)538 371 y(;aj)q(k)603 313 y Fh(\034)614 318 y Fg(;bk)q(l)614 344 y(;bj)q(l)614 371 y(;bil)685 313 y Fh(;)196 b Fi(\(42\))0 664 y(and)14 b(so)f(on.)0 735 y Fj(4.1)30 b(In)o(tegration)14 b(with)g(resp)q(ect)h(to)g(tra)o(v)o(el)h(time)0 806 y Fi(F)m(or)g(example,)i(if)e(w)o(e)f(c)o(ho)q(ose)i(the)f(Hamiltonian) j(in)e(Cartesian)0 843 y(co)q(ordinates)e(in)f(the)f(sp)q(ecial)i(form) e(of)246 899 y Fh(H)s Fi(\()p Fh(x)318 903 y Fg(q)334 899 y Fh(;)6 b(p)370 903 y Fg(r)388 899 y Fi(\))11 b(=)459 884 y Fe(1)p 459 891 16 2 v 459 912 a(2)479 899 y Fh(v)499 881 y Fe(2)517 899 y Fh(p)536 903 y Fg(k)555 899 y Fh(p)574 903 y Fg(k)602 899 y Ff(\000)645 884 y Fe(1)p 645 891 V 645 912 a(2)704 899 y Fh(;)177 b Fi(\(43\))0 955 y Fh(v)15 b Fi(=)f Fh(v)q Fi(\()p Fh(x)135 959 y Fg(i)148 955 y Fi(\))h(b)q(eing)i(the)e(phase)h(v)o(elo)q(cit)o(y)m(,)h(then)f Fh(\015)723 959 y Fe(3)754 955 y Fi(=)e Fh(\034)19 b Fi(in)d(\(14\),)0 993 y(and)e(only)g(partial)h(deriv)n(ativ)o(es)110 1050 y Fh(H)141 1054 y Fg(;:::)191 1050 y Fi(=)230 1034 y Fe(1)p 230 1041 V 230 1062 a(2)257 1017 y Fd(\000)276 1050 y Fh(v)296 1032 y Fe(2)313 1017 y Fd(\001)327 1067 y Fg(;:::)373 1050 y Fh(p)392 1054 y Fg(k)412 1050 y Fh(p)431 1054 y Fg(k)463 1050 y Fh(;)44 b(H)553 1032 y Fg(;a)549 1055 y(;:::)600 1050 y Fi(=)634 1017 y Fd(\000)653 1050 y Fh(v)673 1032 y Fe(2)690 1017 y Fd(\001)704 1067 y Fg(;:::)750 1050 y Fh(p)769 1054 y Fg(a)801 1050 y Fh(;)333 1122 y(H)368 1105 y Fg(;ab)364 1128 y(;:::)414 1122 y Fi(=)448 1090 y Fd(\000)467 1122 y Fh(v)487 1105 y Fe(2)505 1090 y Fd(\001)518 1140 y Fg(;:::)565 1122 y Fh(\016)582 1126 y Fg(ab)892 1122 y Fi(\(44\))0 1183 y(of)14 b(the)h(Hamiltonian)i(are)d(nonzero.)22 b(The)15 b(tra)o(v)o(el{time)h(deriv)n(a-)0 1220 y(tiv)o(es)i(on)g(the)g(righ)o (t{hand)i(side)e(of)g(equation)h(\(16\))f(with)g(\(44\))0 1257 y(ma)o(y)f(then)g(b)q(e)g(cancelled,)j(for)c(the)h(third)h(and)g (higher)g(tra)o(v)o(el{)0 1295 y(time)h(deriv)n(ativ)o(es,)j(b)o(y)d (subtracting)h(equation)g(\(7\))f(with)g(\(43\))0 1332 y(from)13 b(equation)i(\(16\),)p 0 1358 960 2 v 0 1719 2 361 v 109 1408 a Fh(K)142 1412 y Fg(ij)q(k)q(:::mn)265 1408 y Fi(=)t Fh(v)319 1390 y Fe(2)330 1408 y Fh(\034)342 1412 y Fg(;r)369 1408 y Fh(\034)381 1412 y Fg(;r)q(ij)q(k)q(:::mn)526 1408 y Fi(+)r Fh(v)578 1390 y Fe(2)596 1408 y Fh(\034)607 1412 y Fg(;r)q(i)607 1439 y(;r)q(j)616 1454 y Fi(.)616 1470 y(.)616 1487 y(.)607 1513 y Fg(;r)q(n)659 1408 y Fh(\034)670 1412 y Fg(;r)q(j)q(k)q(:::mn)670 1439 y(;ir)q(k)q(:::mn)679 1454 y Fi(.)679 1470 y(.)679 1487 y(.)670 1513 y Fg(;ij)q(k)q(:::mr)40 1588 y Fi(+\()p Fh(v)105 1570 y Fe(2)122 1588 y Fi(\))131 1593 y Fg(;i)131 1620 y(;j)140 1634 y Fi(.)140 1651 y(.)140 1667 y(.)131 1694 y Fg(;n)168 1588 y Fh(\034)179 1593 y Fg(;r)179 1620 y(;r)188 1634 y Fi(.)188 1651 y(.)188 1667 y(.)179 1694 y Fg(;r)212 1588 y Fh(\034)223 1593 y Fg(;r)q(j)q(k)q(:::mn)223 1620 y(;ir)q(k)q(:::mn)232 1634 y Fi(.)232 1651 y(.)232 1667 y(.)223 1694 y Fg(;ij)q(k)q(:::mr)351 1588 y Ff(\000)383 1556 y Fd(\000)407 1573 y Fe(1)p 407 1580 16 2 v 407 1601 a(2)427 1588 y Fh(v)447 1570 y Fe(2)465 1588 y Fh(\034)477 1592 y Fg(;r)503 1588 y Fh(\034)515 1592 y Fg(;r)537 1588 y Ff(\000)574 1573 y Fe(1)p 574 1580 V 574 1601 a(2)594 1556 y Fd(\001)603 1606 y Fg(;ij)q(k)q(:::mn) 750 1588 y Fh(:)125 b Fi(\(45\))p 958 1719 2 361 v 0 1721 960 2 v 0 1769 a(Equations)19 b(suc)o(h)e(as)g(\(41\))f(or)h (\(42\))f(for)h(the)f(third)i(and)f(higher)0 1806 y(tra)o(v)o(el{time)e (deriv)n(ativ)o(es)g(then)e(tak)o(e)h(the)f(form)25 1855 y Fh(K)58 1859 y Fg(ij)q(k)106 1855 y Fi(=)n Ff(\000)164 1822 y Fd(\000)188 1839 y Fe(1)p 188 1846 16 2 v 188 1867 a(2)208 1855 y Fh(v)228 1837 y Fe(2)245 1822 y Fd(\001)254 1872 y Fg(;ij)q(k)308 1855 y Fh(\034)320 1859 y Fg(;r)346 1855 y Fh(\034)358 1859 y Fg(;r)386 1855 y Ff(\000)418 1822 y Fd(\000)438 1855 y Fh(v)458 1837 y Fe(2)475 1822 y Fd(\001)488 1860 y Fg(;ij)488 1886 y(;ik)488 1912 y(;j)q(k)530 1855 y Fh(\034)541 1860 y Fg(;r)541 1886 y(;r)541 1912 y(;r)568 1855 y Fh(\034)579 1860 y Fg(;r)q(k)579 1886 y(;r)q(j)579 1912 y(;r)q(i)631 1855 y Ff(\000)663 1822 y Fd(\000)682 1855 y Fh(v)702 1837 y Fe(2)719 1822 y Fd(\001)732 1860 y Fg(;i)732 1886 y(;j)732 1912 y(;k)760 1855 y Fh(\034)771 1860 y Fg(;r)q(j)771 1886 y(;r)q(i)771 1912 y(;r)q(i)812 1855 y Fh(\034)823 1860 y Fg(;r)q(k)823 1886 y(;r)q(k)823 1912 y(;r)q(j)886 1855 y Fh(;)892 1958 y Fi(\(46\))33 2006 y Fh(K)66 2010 y Fg(ij)q(k)q(l)125 2006 y Fi(=)-5 b Ff(\000)182 1974 y Fd(\000)206 1991 y Fe(1)p 206 1998 V 206 2019 a(2)226 2006 y Fh(v)246 1988 y Fe(2)264 1974 y Fd(\001)272 2024 y Fg(;ij)q(k)q(l)336 2006 y Fh(\034)348 2010 y Fg(;r)374 2006 y Fh(\034)386 2010 y Fg(;r)415 2006 y Ff(\000)447 1974 y Fd(\000)466 2006 y Fh(v)486 1988 y Fe(2)503 1974 y Fd(\001)516 2011 y Fg(;ij)q(k)516 2038 y(;ij)q(l)516 2064 y(;ik)q(l)516 2091 y(;j)q(k)q(l)570 2006 y Fh(\034)581 2011 y Fg(;r)581 2038 y(;r)581 2064 y(;r)581 2091 y(;r)607 2006 y Fh(\034)618 2011 y Fg(;r)q(l)618 2038 y(;r)q(k)618 2064 y(;r)q(j)618 2091 y(;r)q(i)670 2006 y Ff(\000)702 1974 y Fd(\000)721 2006 y Fh(v)741 1988 y Fe(2)758 1974 y Fd(\001)771 2011 y Fg(;ij)771 2038 y(;ik)771 2064 y(;j)q(k)771 2091 y(;il)771 2117 y(;j)q(l)771 2144 y(;k)q(l)814 2006 y Fh(\034)825 2011 y Fg(;r)825 2038 y(;r)825 2064 y(;r)825 2091 y(;r)825 2117 y(;r)825 2144 y(;r)851 2006 y Fh(\034)862 2011 y Fg(;r)q(k)q(l)862 2038 y(;r)q(j)q(l)862 2064 y(;r)q(il)862 2091 y(;r)q(j)q(k)862 2117 y(;r)q(ik)862 2144 y(;r)q(ij)150 2209 y Ff(\000)182 2176 y Fd(\000)201 2209 y Fh(v)221 2191 y Fe(2)238 2176 y Fd(\001)251 2213 y Fg(;ij)251 2240 y(;ik)251 2266 y(;j)q(k)251 2293 y(;il)251 2319 y(;j)q(l)251 2346 y(;k)q(l)294 2209 y Fh(\034)305 2213 y Fg(;r)q(k)305 2240 y(;r)q(j)305 2266 y(;r)q(k)305 2293 y(;r)q(j)305 2319 y(;r)q(i)305 2346 y(;r)q(i)348 2209 y Fh(\034)359 2213 y Fg(;r)q(l)359 2240 y(;r)q(l)359 2266 y(;r)q(l)359 2293 y(;r)q(k)359 2319 y(;r)q(k)359 2346 y(;r)q(j)411 2209 y Ff(\000)443 2176 y Fd(\000)462 2209 y Fh(v)482 2191 y Fe(2)500 2176 y Fd(\001)513 2213 y Fg(;i)513 2240 y(;j)513 2266 y(;i)513 2293 y(;k)513 2319 y(;j)513 2346 y(;k)513 2372 y(;i)513 2399 y(;l)513 2425 y(;j)513 2452 y(;l)513 2478 y(;k)513 2505 y(;l)541 2209 y Fh(\034)552 2213 y Fg(;r)q(j)552 2240 y(;r)q(i)552 2266 y(;r)q(k)552 2293 y(;r)q(i)552 2319 y(;r)q(k)552 2346 y(;r)q(j)552 2372 y(;r)q(l)552 2399 y(;r)q(i)552 2425 y(;r)q(l)552 2452 y(;r)q(j)552 2478 y(;r)q(l)552 2505 y(;r)q(k)596 2209 y Fh(\034)607 2213 y Fg(;r)q(k)q(l)607 2240 y(;r)q(k)q(l)607 2266 y(;r)q(j)q(l)607 2293 y(;r)q(j)q(l)607 2319 y(;r)q(il)607 2346 y(;r)q(il)607 2372 y(;r)q(j)q(k)607 2399 y(;r)q(j)q(k)607 2425 y(;r)q(ik)607 2452 y(;r)q(ik)607 2478 y(;r)q(ij)607 2505 y(;r)q(ij)673 2209 y Ff(\000)r Fh(v)725 2191 y Fe(2)742 2209 y Fh(\034)753 2213 y Fg(;r)q(ij)753 2240 y(;r)q(ik)753 2266 y(;r)q(j)q(k)811 2209 y Fh(\034)822 2213 y Fg(;r)q(k)q(l)822 2240 y(;r)q(j)q(l)822 2266 y(;r)q(il)894 2209 y Fh(;)892 2550 y Fi(\(47\))990 106 y(and)14 b(so)f(on.)990 177 y Fj(4.2)30 b(In)o(tegration)14 b(with)g(resp)q(ect)h(to)g Fh(\015)1634 181 y Fe(3)1662 177 y Fi(=)1703 144 y Fd(R)1737 177 y Fh(v)1757 161 y Fe(2)1774 177 y Fi(d)p Fh(\034)990 254 y Fi(Moreo)o(v)o(er,)g(if)g(w)o(e)f(c)o(ho)q(ose)i(the)e (Hamiltonian)k(in)d(Cartesian)h(co-)990 291 y(ordinates)f(in)f(the)f (sp)q(ecial)i(form)d(of)1224 347 y Fh(H)s Fi(\()p Fh(x)1296 351 y Fg(q)1312 347 y Fh(;)6 b(p)1348 351 y Fg(r)1366 347 y Fi(\))11 b(=)1437 331 y Fe(1)p 1437 338 V 1437 359 a(2)1457 347 y Fh(p)1476 351 y Fg(k)1496 347 y Fh(p)1515 351 y Fg(k)1542 347 y Ff(\000)1586 331 y Fe(1)p 1586 338 V 1586 359 a(2)1606 347 y Fh(v)1626 329 y Fb(\000)p Fe(2)1706 347 y Fh(;)165 b Fi(\(48\))990 405 y(then)13 b(d)p Fh(\015)1118 409 y Fe(3)1147 405 y Fi(=)d Fh(v)1207 389 y Fe(2)1224 405 y Fi(d)p Fh(\034)18 b Fi(in)c(\(14\),)e(and)i(only) g(partial)h(deriv)n(ativ)o(es)1031 464 y Fh(H)1062 469 y Fg(;:::)1119 464 y Fi(=)c Ff(\000)1196 431 y Fd(\000)1220 448 y Fe(1)p 1220 455 V 1220 476 a(2)1240 464 y Fh(v)1260 446 y Fb(\000)p Fe(2)1301 431 y Fd(\001)1315 481 y Fg(;:::)1374 464 y Fh(;)45 b(H)1465 446 y Fg(;a)1503 464 y Fi(=)11 b Fh(p)1563 468 y Fg(a)1595 464 y Fh(;)44 b(H)1685 446 y Fg(;ab)1738 464 y Fi(=)10 b Fh(\016)1795 468 y Fg(ab)1882 464 y Fi(\(49\))990 528 y(of)j(the)g(Hamiltonian)j(are)e(nonzero.)k (Equation)d(\(16\))f(then)f(sim-)990 566 y(pli\014es)i(to)1083 613 y Fh(K)1116 617 y Fg(ij)q(k)q(:::mn)1245 613 y Fi(=)c Fh(\034)1298 617 y Fg(;r)1324 613 y Fh(\034)1336 617 y Fg(;r)q(ij)q(k)q(:::mn)1488 613 y Fi(+)e Fh(\034)1538 618 y Fg(;r)q(i)1538 644 y(;r)q(j)1547 659 y Fi(.)1547 675 y(.)1547 692 y(.)1538 719 y Fg(;r)q(n)1590 613 y Fh(\034)1601 618 y Fg(;r)q(j)q(k)q(:::mn)1601 644 y(;ir)q(k)q(:::mn) 1610 659 y Fi(.)1610 675 y(.)1610 692 y(.)1601 719 y Fg(;ij)q(k)q(:::mr)1778 613 y Fh(:)93 b Fi(\(50\))990 774 y(The)11 b(tra)o(v)o(el{time)i(deriv)n(ativ)o(es)g(on)e(the)h(righ) o(t{hand)h(side)f(of)f(\(50\))990 811 y(ma)o(y)f(then)g(b)q(e)g (cancelled)i(for)e(the)g(third)g(and)h(higher)g(tra)o(v)o(el{time)990 848 y(deriv)n(ativ)o(es)k(b)o(y)f(subtracting)h(\(7\))d(with)i(\(48\))f (from)g(\(50\),)p 990 873 960 2 v 990 1137 2 265 v 1122 922 a Fh(K)1155 926 y Fg(ij)q(k)q(:::mn)1285 922 y Fi(=)1325 890 y Fd(\000)1349 907 y Fe(1)p 1349 914 16 2 v 1349 935 a(2)1369 922 y Fh(v)1389 905 y Fb(\000)p Fe(2)1431 890 y Fd(\001)1444 940 y Fg(;ij)q(k)q(:::mn)1582 922 y Fi(+)8 b Fh(\034)1632 926 y Fg(;r)1658 922 y Fh(\034)1670 926 y Fg(;r)q(ij)q(k)q(:::mn)1174 1007 y Fi(+)p Fh(\034)1215 1012 y Fg(;r)q(i)1215 1038 y(;r)q(j)1224 1053 y Fi(.)1224 1070 y(.)1224 1086 y(.)1215 1113 y Fg(;r)q(n)1267 1007 y Fh(\034)1278 1012 y Fg(;r)q(j)q(k)q(:::mn)1278 1038 y(;ir)q(k)q(:::mn)1287 1053 y Fi(.)1287 1070 y(.)1287 1086 y(.)1278 1113 y Fg(;ij)q(k)q(:::mr)1425 1007 y Ff(\000)1464 974 y Fd(\000)1488 992 y Fe(1)p 1488 999 V 1488 1020 a(2)1508 1007 y Fh(\034)1520 1011 y Fg(;r)1546 1007 y Fh(\034)1558 1011 y Fg(;r)1585 974 y Fd(\001)1604 1025 y Fg(;ij)q(k)q(:::mn)1752 1007 y Fh(:)113 b Fi(\(51\))p 1948 1137 2 265 v 990 1139 960 2 v 990 1186 a(Equations)19 b(suc)o(h)e(as)g(\(41\))f(or)h(\(42\))f(for)h(the)f(third)i(and)f (higher)990 1224 y(tra)o(v)o(el{time)e(deriv)n(ativ)o(es)g(then)e(tak)o (e)h(the)f(form)1292 1271 y Fh(K)1325 1275 y Fg(ij)q(k)1379 1271 y Fi(=)1420 1238 y Fd(\000)1444 1256 y Fe(1)p 1444 1263 16 2 v 1444 1284 a(2)1464 1271 y Fh(v)1484 1253 y Fb(\000)p Fe(2)1525 1238 y Fd(\001)1539 1289 y Fg(;ij)q(k)1638 1271 y Fh(;)233 b Fi(\(52\))1189 1342 y Fh(K)1222 1346 y Fg(ij)q(k)q(l)1286 1342 y Fi(=)1327 1309 y Fd(\000)1351 1326 y Fe(1)p 1351 1333 V 1351 1354 a(2)1371 1342 y Fh(v)1391 1324 y Fb(\000)p Fe(2)1432 1309 y Fd(\001)1446 1359 y Fg(;ij)q(k)q(l)1518 1342 y Ff(\000)9 b Fh(\034)1568 1347 y Fg(;r)q(ij)1568 1373 y(;r)q(ik)1568 1400 y(;r)q(j)q(k)1632 1342 y Fh(\034)1643 1347 y Fg(;r)q(k)q(l)1643 1373 y(;r)q(j)q(l)1643 1400 y(;r)q(il)1741 1342 y Fh(;)130 b Fi(\(53\))1142 1458 y Fh(K)1175 1462 y Fg(ij)q(k)q(lm)1267 1458 y Fi(=)1308 1425 y Fd(\000)1332 1443 y Fe(1)p 1332 1450 V 1332 1471 a(2)1352 1458 y Fh(v)1372 1440 y Fb(\000)p Fe(2)1413 1425 y Fd(\001)1427 1476 y Fg(;ij)q(k)q(lm)1526 1458 y Ff(\000)9 b Fh(\034)1576 1463 y Fg(;r)q(ij)1576 1489 y(;r)q(ik)1576 1516 y(;r)q(j)q(k)1576 1542 y(;r)q(il)1576 1569 y(;r)q(j)q(l)1576 1595 y(;r)q(k)q(l)1576 1622 y(;r)q(im)1576 1648 y(;r)q(j)q(m)1576 1675 y(;r)q(k)q(m)1576 1701 y(;r)q(lm)1646 1458 y Fh(\034)1657 1463 y Fg(;r)q(k)q(lm)1657 1489 y(;r)q(j)q(lm)1657 1516 y(;r)q(ilm)1657 1542 y(;r)q(j)q(k)q(m)1657 1569 y(;r)q(ik)q(m)1657 1595 y(;r)q(ij)q(m)1657 1622 y(;r)q(j)q(k)q(l)1657 1648 y(;r)q(ik)q(l)1657 1675 y(;r)q(ij)q(l)1657 1701 y(;r)q(ij)q(k)1787 1458 y Fh(;)84 b Fi(\(54\))990 1751 y(and)14 b(so)f(on.)990 1822 y Fj(5)30 b(Conclusions)990 1893 y Fi(Equations)12 b(\(14\),)e(\(15\))f(and)i(\(17\))e(ma)o(y)h(b)q (e)g(used)g(to)g(calculate)i(the)990 1930 y(third)19 b(and)f(higher)h(partial)h(tra)o(v)o(el{time)f(deriv)n(ativ)o(es)h(in)e (b)q(oth)990 1968 y(isotropic)23 b(and)e(anisotropic)i(mo)q(dels)e(b)o (y)g(n)o(umerical)i(quadra-)990 2005 y(tures)14 b(along)h(ra)o(ys.)k(F) m(or)14 b(a)f(sp)q(ecial)j(c)o(hoice)f(of)e(the)h(Hamiltonian)990 2042 y(in)h(isotropic)h(mo)q(dels,)f(equation)h(\(17\))e(considerably)j (simpli\014es)990 2080 y(to)c(equations)i(suc)o(h)e(as)h(\(45\))f(or)g (\(51\).)990 2151 y Fj(Reference)990 2221 y Fi(Babic)o(h,)19 b(V.M.,)e(Buldyrev,)j(N.J.)d(&)g(Molotk)o(o)o(v,)j(I.A.)d(\(1985\):)990 2259 y(Space{time)e(ra)o(y)f(metho)q(d.)f(Linear)i(and)f(non{linear)i (w)o(a)o(v)o(es)d(\(in)990 2296 y(Russian\).)h(Leningrad,)h(Leningrad)g (Univ.)f(Press.)990 2367 y Fj(Ac)o(kno)o(wledgemen)o(ts)990 2438 y Fi(The)e(researc)o(h)h(has)g(b)q(een)g(supp)q(orted)h(b)o(y)e (the)h(Gran)o(t)f(Agency)h(of)990 2475 y(the)g(Czec)o(h)h(Republic)h (under)f(Con)o(tract)g(205/95/1465,)h(and)e(b)o(y)990 2513 y(consortium)i(\\Seismic)h(W)m(a)o(v)o(es)e(in)h(Complex)f(3-D)g (Structures",)990 2550 y(see)f(\\h)o(ttp:/seis.k)n(arlo)o(v.m\013.cuni) q(.cz/cons)q(ort/mai)q(n.h)o(tm".)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF