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(Hamiltonian)e(\(17\).)1817 5824 y(230)p eop %%Page: 231 9 231 8 bop 0 100 a Fk(3)161 b(P)l(erturbation)53 b(expansions)0 390 y Fp(The)33 b(tra)m(v)m(el)g(time)e Fi(\034)43 b Fp(in)32 b(the)h(p)s(erturb)s(ed)g(medium)e(is)g(a)h(function)g(of)g (spatial)f(co)s(ordinates)h Fi(x)3451 405 y Fh(i)3512 390 y Fp(and)g(of)0 510 y(p)s(erturbation)g(parameter)g Fi(f)1086 525 y Fh(\013)1135 510 y Fp(,)1571 630 y Fi(\034)39 b Fp(=)28 b Fi(\034)11 b Fp(\()p Fi(x)1902 645 y Fh(m)1969 630 y Fi(;)17 b(f)2061 645 y Fh(\013)2111 630 y Fp(\))32 b Fi(:)1398 b Fp(\(23\))0 785 y(Similar)34 b(expression)k(can)g(b)s(e)f (written)g(for)g(partial)d(deriv)-5 b(ativ)m(e)37 b(of)g Fi(\034)48 b Fp(of)37 b(an)m(y)h(order.)58 b(F)-8 b(or)36 b(example,)0 906 y(for)c(tra)m(v)m(el)h(time)e(gradien)m(t)h Fi(\034)1069 921 y Fh(;i)1145 906 y Fp(=)27 b Fi(@)5 b(\034)11 b(=@)5 b(x)1517 921 y Fh(i)1548 906 y Fp(,)33 b(w)m(e)g(ha)m(v)m(e)1535 1081 y Fi(\034)1577 1096 y Fh(;i)1652 1081 y Fp(=)28 b Fi(\034)1798 1096 y Fh(;i)1846 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Fh(;\013)2175 4142 y Fp(\()p Fi(x)2268 4157 y Fh(m)2335 4142 y Fp(\))33 b Fi(;)1173 b Fp(\(34\))1164 4288 y(Im)o Fi(\034)1322 4303 y Fh(;i)1371 4288 y Fp(\()p Fi(x)1464 4303 y Fh(m)1531 4288 y Fi(;)17 b(f)1623 4303 y Fh(\013)1672 4288 y Fp(\))27 b Fj(\031)i Fi(f)1891 4303 y Fh(\013)1940 4288 y Fp(Im)p Fi(\034)2099 4303 y Fh(;i\013)2192 4288 y Fp(\()p Fi(x)2285 4303 y Fh(m)2352 4288 y Fp(\))j Fi(:)1157 b Fp(\(35\))0 4499 y(Note)39 b(that)g(the)g(\014rst)g(terms)g(of)f(RHS's)i(of)e (\(25\))g(and)h(\(26\),)h Fi(\034)11 b Fp(\()p Fi(x)2452 4514 y Fh(m)2519 4499 y Fp(\))39 b(and)g Fi(\034)2834 4514 y Fh(;i)2882 4499 y Fp(\()p Fi(x)2975 4514 y Fh(m)3042 4499 y Fp(\),)h(are)f(real-v)-5 b(alued)0 4620 y(and)33 b(th)m(us)g(do)g(not)f(app)s(ear)h(in)e(\(34\))h(and)h(\(35\).)146 4811 y(W)-8 b(e)36 b(use)h(the)e(follo)m(wing)e(notation.)51 b(The)36 b(tilde's)f(ab)s(o)m(v)m(e)h(sym)m(b)s(ols)f(denote)h(that)g (only)f(the)g(\014rst-)0 4931 y(order)g(p)s(erturbation)f(is)h 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233 11 233 10 bop 0 100 a Fp(where)30 b(the)g(p)s(erturbation)e(deriv)-5 b(ativ)m(e)29 b Fi(\034)1497 115 y Fh(;i\013)1590 100 y Fp(\()p Fi(\015)1679 115 y Fo(3)1718 100 y Fp(\))g(is)f(giv)m(en)i(b) m(y)g(\(29\).)41 b(The)30 b(atten)m(uation)f(v)m(ector)3529 74 y(~)3511 100 y Fl(A)g Fp(w)m(as)0 220 y(in)m(tro)s(duced)i(in)e(the) i(ab)s(o)m(v)m(e)h(w)m(a)m(y)g(b)m(y)1378 195 y(\024)1367 220 y(Cerv)m(en)q(\023)-50 b(y,)33 b(Klime)-5 b(\024)-44 b(s)29 b(&)h(P)-5 b(\024)-44 b(sen)m(\024)e(c)-11 b(\023)-38 b(\020k)32 b(\(2008\),)e(who)h(used)h(the)f(p)s(ertur-)0 340 y(bation)38 b(Hamiltonian)e(\(17\).)64 b(The)40 b(prop)s(erties)g (of)f(the)h(atten)m(uation)e(v)m(ector)2956 315 y(~)2938 340 y Fl(A)h Fp(in)g(heterogeneous)0 461 y(anisotropic)31 b(w)m(eakly)j(dissipativ)m(e)e(medium)g(w)m(ere)i(thoroughfully)d(in)m (v)m(estigated)i(there.)45 b(The)34 b(ab)s(o)m(v)m(e)0 581 y(authors)41 b(sp)s(eci\014ed)g(their)f(results)h(also)e(for)h 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b(not)g(in\015uenced)h(b)m(y)g(the)g(eigen)m(v)m(ectors)g(of)f (the)h(Christo\013el)e(matrices.)86 b(Consequen)m(tly)-8 b(,)53 b(the)0 5475 y(di\013erences)34 b(b)s(et)m(w)m(een)856 5474 y(~)851 5475 y Fl(g)h Fp(and)d Fl(g)i Fp(do)f(not)f(pla)m(y)h(an)m (y)g(role)f(in)f(isotropic)g(media.)1817 5824 y(238)p eop %%Page: 239 17 239 16 bop 146 100 a Fp(Let)45 b(us)g(no)m(w)g(discuss,)k(in)44 b(a)g(greater)h(detail,)h(the)f(most)f(imp)s(ortan)m(t)e(expression)k (for)e Fi(\034)3517 115 y Fh(;\013)3587 100 y Fp(\()p Fi(\015)3676 115 y Fo(3)3714 100 y Fp(\).)0 220 y(F)-8 b(rom)31 b(\(28\))h(with)g(\(45\),)g(w)m(e)i(obtain)d(a)i(simple)e (expression)j(for)e Fi(\034)2411 235 y Fh(;\013)2480 220 y Fp(\()p Fi(\015)2569 235 y Fo(3)2608 220 y Fp(\),)h(v)-5 b(alid)30 b(for)i(an)m(y)i Fi(N)10 b Fp(:)646 480 y Fi(\034)688 495 y Fh(;\013)757 480 y Fp(\()p Fi(\015)846 495 y Fo(3)885 480 y Fp(\))28 b(=)f Fj(\000)1156 441 y Fo(1)p 1141 457 64 4 v 1141 515 a Fh(N)1232 363 y Fc(Z)1315 389 y Fh(\015)1351 398 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y(Klime)-5 b(\024)-44 b(s)29 b(\(2007\))g(that)h(the)h (most)f(accurate)h(results)g(for)f(real-v)-5 b(alued)29 b(tra)m(v)m(el)h(times)g(are)g(obtained)g(for)0 2715 y Fi(N)38 b Fp(=)28 b Fj(\000)p Fp(1.)146 2906 y(T)-8 b(o)43 b(mak)m(e)g(the)g(expressions)i(for)d Fi(\034)1463 2921 y Fh(;\013)1532 2906 y Fp(\()p Fi(\015)1621 2921 y Fo(3)1660 2906 y Fp(\))h(more)f(transparen)m(t,)k(w)m(e)e(express)h (the)e(p)s(erturbation)0 3027 y(term)31 b Fi(b)i Fp(in)e(terms)h(of)f (Lam)m(\023)-46 b(e's)32 b(visco)s(elastic)f(complex-v)-5 b(alued)30 b(mo)s(duli)f Fi(\025)j Fp(and)g Fi(\026)p Fp(.)43 b(According)32 b(to)f(the)0 3147 y(de\014nition,)h Fi(b)i Fp(is)f(the)h(p)s(erturbation)e(of)h(the)g(densit)m (y-normalized)f(elastic)g(mo)s(dulus)h(corresp)s(onding)0 3267 y(to)40 b(the)g(w)m(a)m(v)m(e)i(under)f(consideration.)66 b(It)40 b(is)g(p)s(ossible)f(to)h(discuss)h Fi(b)g Fp(indep)s(enden)m (tly)g(for)f(P)g(and)g(S)0 3388 y(w)m(a)m(v)m(es.)k(F)-8 b(or)25 b(simplicit)m(y)-8 b(,)24 b(ho)m(w)m(ev)m(er,)30 b(w)m(e)d(consider)f(only)f(P)i(w)m(a)m(v)m(es;)j(for)c(S)g(w)m(a)m(v)m (es)i(the)e(results)g(w)m(ould)g(b)s(e)0 3508 y(quite)35 b(analogous.)51 b(The)36 b(exact)h(expressions)g(for)e(the)g(densit)m (y)i(normalized)c(visco)s(elastic)h(complex-)0 3628 y(v)-5 b(alued)32 b(mo)s(dulus)g(of)g(P)h(w)m(a)m(v)m(es)i(read)1199 3805 y Fi(\025)22 b Fp(+)g(2)p Fi(\026)p 1199 3850 285 4 v 1316 3941 a(\032)1521 3873 y Fp(=)27 b Fi(\013)1687 3832 y Fo(2)1686 3898 y Fh(P)1773 3873 y Fp(=)g(Re)17 b Fi(\013)2071 3832 y Fo(2)2070 3898 y Fh(P)2151 3873 y Fj(\000)22 b Fp(iIm)16 b Fi(\013)2474 3832 y Fo(2)2473 3898 y Fh(P)2564 3873 y Fi(:)1015 b Fp(\(64\))0 4147 y(In)41 b(the)g(reference,)j(p)s(erfectly)d(elastic)e(medium,)i Fi(\013)1944 4111 y Fo(2)1943 4172 y Fh(P)2043 4147 y Fp(=)h(Re\()p Fi(\025)27 b Fp(+)h(2)p Fi(\026)p Fp(\))p Fi(=\032)40 b Fp(=)i Fi(V)2983 4111 y Fo(2)2962 4172 y Fh(P)3023 4147 y Fp(,)g(where)g Fi(V)3439 4162 y Fh(P)3538 4147 y Fp(is)e(the)0 4268 y(real-v)-5 b(alued)31 b(v)m(elo)s(cit)m(y)h (of)g(P)h(w)m(a)m(v)m(es.)46 b(In)33 b(the)g(p)s(erturb)s(ed)g(medium,) e(ho)m(w)m(ev)m(er,)k(w)m(e)f(ha)m(v)m(e)964 4555 y(~)-59 b Fi(\013)1017 4514 y Fo(2)1016 4579 y Fh(P)1103 4555 y Fp(=)27 b(Re)1336 4461 y(~)1332 4487 y Fi(\025)22 b Fp(+)g(2)8 b(~)-57 b Fi(\026)p 1332 4531 V 1458 4623 a Fp(~)f Fi(\032)1648 4555 y Fj(\000)23 b Fp(iIm)1905 4461 y(~)1901 4487 y Fi(\025)f Fp(+)g(2)8 b(~)-57 b Fi(\026)p 1901 4531 V 2027 4623 a Fp(~)f Fi(\032)2223 4555 y Fp(=)2342 4530 y(~)2327 4555 y Fi(V)2406 4514 y Fo(2)2384 4579 y Fh(P)2445 4555 y Fp(\(1)22 b Fj(\000)g Fp(i)2686 4528 y(~)2681 4555 y Fi(\016)t Fp(\))32 b Fi(;)781 b Fp(\(65\))0 4819 y(where)1275 4914 y(~)1270 4940 y Fi(\016)32 b Fp(=)27 b(Im\()1607 4914 y(~)1603 4940 y Fi(\025)22 b Fp(+)g(2)8 b(~)-57 b Fi(\026)o Fp(\))p Fi(=)p Fp(Re\()2131 4914 y(~)2127 4940 y Fi(\025)22 b Fp(+)g(2)8 b(~)-57 b Fi(\026)p Fp(\))32 b Fi(:)1097 b Fp(\(66\))0 5114 y(The)43 b(signs)g(\\{")f(in)g 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y Fb(3)1452 1648 y Fh(\015)1492 1626 y Fb(0)1488 1671 y(3)1581 1406 y Fc(0)1581 1555 y(@)1860 1510 y Fp(1)p 1663 1554 444 4 v 1678 1662 a(~)1663 1687 y Fi(V)1720 1702 y Fh(P)1779 1570 y Fc(q)p 1862 1570 245 4 v 117 x Fp(1)22 b Fj(\000)g Fp(i)2065 1661 y(~)2060 1687 y Fi(\016)2138 1577 y Fj(\000)2281 1510 y Fp(1)p 2248 1554 116 4 v 2248 1645 a Fi(V)2305 1660 y Fh(P)2374 1406 y Fc(1)2374 1555 y(A)2463 1577 y Fi(V)2520 1592 y Fh(P)2578 1577 y Fp(d)p Fi(\015)2688 1512 y Fa(0)2683 1602 y Fo(3)2755 1577 y Fi(:)824 b Fp(\(71\))0 1861 y(Note)43 b(that)g Fi(V)525 1876 y Fh(P)583 1861 y Fp(d)p Fi(\015)693 1802 y Fa(0)688 1886 y Fo(3)773 1861 y Fp(=)i(d)p Fi(s)p Fp(,)h(where)e Fi(s)f Fp(is)f(the)h(arclength)g(along)e(the)j(ra)m(y)-8 b(.)75 b(Th)m(us,)47 b(the)c(in)m(tegration)0 1982 y(v)-5 b(ariable)33 b(in)i(\(71\))f(ma)m(y)h(b)s(e)g(in)m(terpreted)h(as)g (the)f(arclength,)h(and)f(do)s(es)g(not)g(dep)s(end)i(on)e(the)g(tra)m (v)m(el)0 2102 y(time.)146 2293 y(W)-8 b(e)34 b(no)m(w)f(determine)g 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(follo)m(ws:)930 3317 y(1)p Fi(=)1050 3292 y Fp(~)1028 3317 y Fi(Q)27 b Fp(=)h Fj(\000)1337 3278 y Fo(2)p 1323 3294 V 1323 3351 a Fh(N)1396 3317 y Fp(Im)1530 3171 y Fc(")1578 3317 y Fp(\(1)22 b Fj(\000)g Fp(i)1819 3291 y(~)1814 3317 y Fi(\016)s Fp(\))1915 3171 y Fc( )1981 3317 y Fp(1)g(+)2175 3224 y(~)2160 3249 y Fi(V)2238 3213 y Fo(2)2217 3274 y Fh(P)2300 3249 y Fj(\000)g Fi(V)2478 3213 y Fo(2)2456 3274 y Fh(P)p 2160 3294 358 4 v 2279 3385 a Fi(V)2358 3351 y Fo(2)2336 3410 y Fh(P)2527 3171 y Fc(!#)2641 3184 y Fh(N)q(=)p Fo(2)2823 3317 y Fi(:)756 b Fp(\(73\))0 3583 y(This)26 b(is)f(a)g(v)m(ery)i(general)d (expression,)29 b(v)-5 b(alid)23 b(for)i(arbitrary)g Fi(N)10 b Fp(,)27 b(and)f(con)m(taining)e(b)s(oth)h(the)h(structural)0 3703 y(p)s(erturbations)32 b(\()663 3678 y(~)648 3703 y Fi(V)727 3667 y Fo(2)705 3728 y Fh(P)788 3703 y Fj(\000)23 b Fi(V)966 3667 y Fo(2)945 3728 y Fh(P)1006 3703 y Fp(\))p Fi(=V)1171 3667 y Fo(2)1150 3728 y Fh(P)1211 3703 y Fp(,)32 b(and)h(the)g(dissipativ)m(e)f(p)s(erturbations)2721 3677 y(~)2716 3703 y Fi(\016)t Fp(.)146 3895 y(If)40 b(b)s(oth)g(these)h(p)s(erturbations)e(are)h(small)d(enough,)42 b(w)m(e)f(can)f(neglect)g(the)g(quadratic)g(term)f(of)0 4015 y(their)32 b(pro)s(duct,)h(and)g(obtain,)973 4224 y(1)p Fi(=)1093 4199 y Fp(~)1071 4224 y Fi(Q)28 b Fj(\031)g(\000)1382 4185 y Fo(2)p 1368 4201 64 4 v 1368 4259 a Fh(N)1441 4224 y Fp(Im[\(1)22 b Fj(\000)g Fp(i)1826 4198 y(~)1821 4224 y Fi(\016)j Fp(+)d(\()2040 4199 y(~)2025 4224 y Fi(V)2104 4183 y Fo(2)2082 4249 y Fh(P)2166 4224 y Fj(\000)g Fi(V)2344 4183 y Fo(2)2322 4249 y Fh(P)2383 4224 y Fp(\))p Fi(=V)2548 4183 y Fo(2)2527 4249 y Fh(P)2588 4224 y Fp(])2615 4183 y Fh(N)q(=)p Fo(2)2779 4224 y Fi(:)800 b Fp(\(74\))0 4434 y(Linear)32 b(expansion)h(of)f(\(74\))g(then)h(yields:)1223 4643 y(1)p Fi(=)1343 4618 y Fp(~)1321 4643 y Fi(Q)27 b Fj(\031)h Fp(Im[i)1707 4617 y(~)1702 4643 y Fi(\016)d Fj(\000)e Fp(\()1923 4618 y(~)1908 4643 y Fi(V)1986 4602 y Fo(2)1965 4668 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