搜索
WuChong-shi ( )§15.1 ( ) l !#$%&’()*+!,-. ∂2u∂t2−a2∂2u∂x2=0,0xl,t0,u x=0=0,u x=l=0,t≥0,u t=0=φ(x),∂u∂t t=0=ψ(x),0≤x≤l./0123456F789:;=?@,u(x,t)=X(x)T(t)FABCDX(x)EFGH9)*IJK-.L+T(t)EFGH9)*FMNOH9)*IJK-.PQRSTUVTWXYX(x)WZ[\]^_‘ab(cde2[\]^fghi_ZYλ(_‘jkelmnopqjkrstW_‘abuvwZ[\]^Wxyabrz{mw|}λy(~h noZ[\]^ nopqjkW{ ‘r h λ c_y ( h noZ[\]^ nopqjkW{ ‘X(x)rλWs c_y ( W{ ‘ rXYX(x)WZ[\]^_‘ab( r/ 12 λn=nπl2,n=1,2,3,··· CDXn(x)=sinnπlx. ¡(¢£⁄L¥ƒ§Dn¤'(“«(‹›fifl –(† I‡· CDP' λnIXn(x)rWuChong-shi§15.1 ¶•‚„”»…‰ (¿) 2`/´12 ˆ (˜¯˘˙¨ ‹ ,˚ ¸ ˝(˛ˇ—0¡ λn(%)*T00(t)+λa2T(t)=0⁄L ‡·Tn(t)(Tn(t)=Cnsinnπlat+Dncosnπlat.“«( ˚EFOH9)*IJK-. ,un(x,t)=Cnsinnπlat+Dncosnπlatsinnπlx(n=1,2,3,···).F , ¡F—0¡ ,PEFRSOH9)*IRSJK-.F0 ( 0¡ , ⁄ EF!,¸ –Æ -.(ª0 GDCnIDn(EFDnsinnπlx=φ(x),Cnnπalsinnπlx=ψ(x).FOH9)*IJK-.PQRS(†¢£( ŁØŒ) ,º ( QEFRS)*IRSJK-.,rQ ⁄ EFÆ -.æF† ¡ ,º u(x,t)=∞Xn=1Cnsinnπlat+Dncosnπlatsinnπlx, ı D Fł øœß( (⁄L OH )( ( u(x,t) QRSOH9)*‹RSJK-. ,r =, ¨ r¢ ˇOH9)* ,(“ 0 , QEFOH9)*( EFRSJK-. } l ‘fW YCn Dnæ∞Xn=1Dnsinnπlx=φ(x),(z)∞Xn=1Cnnπalsinnπlx=ψ(x)()/ 12 ¯˘ Zl0Xn(x)Xm(x)dx=0,n6=m.WuChong-shi !#$%&’(() 3`‹(z) )Lsinmπlx( *9( Zl0φ(x)sinmπlxdx=Zl0∞Xn=1Dnsinnπlxsinmπlxdx=∞Xn=1DnZl0sinnπlxsinmπlxdx=Dm·l2.+LDn=2lZl0φ(x)sinnπlxdx. (%()(⁄L Cn=2nπaZl0ψ(x)sinnπlxdx. ˚§¡!,¸ ,rF ,-.Xn(x)=sinnπlxIXm(x)=sinmπlxQ9/˛·ˇ λnIλm ¡ CD(λn6=λm(ªn6=m)r¢£9/EFX00n(x)+λnXn(x)=0,Xn(0)=0,Xn(l)=0,IX00m(x)+λmXm(x)=0,Xm(0)=0,Xm(l)=0.¥Xm(x))LXn(x))*(¥Xn(x))LXm(x))*(‡0((Xm(x)X00n(x)−Xn(x)X00m(x))+(λn−λm)Xm(x)Xn(x)=0,‹12[0,l]›*9(ª (λn−λm)Zl0Xn(x)Xm(x)dx=Zl0[Xn(x)X00m(x)−Xm(x)X00n(x)]dx=[Xn(x)X0m(x)−Xm(x)X0n(x)] l0=0.›fi¥ ˚Xn(x)IXm(x)EFJK-.r λn6=λm( 3 Zl0Xn(x)Xm(x)dx=0,n6=m.4‹›fi34– ¥ ˚21. CDEFH9)*2. CDEFJK-.5 ¥ CD 6CD=4“«(78 9: ;3=X00(x)+λX(x)=0,?fl (λn−λm)Zl0Xn(x)Xm(x)dx=[Xn(x)X0m(x)−Xm(x)X0n(x)] l0 @ArWuChong-shi§15.1 ¶•‚„”»…‰ (¿) 4`4 B CDEFJK-.C α1X(0)+β1X0(0)=0,α2X(l)+β2X0(l)=0,D–α1Iβ1 α2Iβ2E F 0(? α1Xn(0)+β1X0n(0)=0,α1Xm(0)+β1X0m(0)=0Iα2Xn(l)+β2X0n(l)=0,α2Xm(l)+β2X0m(l)=0.“ α1Iβ1 F 0(+L Xn(0)X0n(0)Xm(0)X0m(0) =0.G“ α2Iβ2 F 0(+LG Xn(l)X0n(l)Xm(l)X0m(l) =0.FflH2IJ X00(x)+λX(x)=0,α1X(0)+β1X0(0)=0,α2X(l)+β2X0(l)=0 Zl0Xn(x)Xm(x)dx=0,n6=mKLMNr4›fiJK-.OP˚0 ´Q´ QRJK-.rF STkXnk2≡Zl0X2n(x)dx=l2.TkXnkUVWXYZ[\]^_‘abcdefgZ1kXnk2Zl0X2n(x)dx=1hijkWXn(x)/kXnkUlZ1dmnopqrstuvZl0Xn(x)Xm(x)dx=l2δnm.YZ[\]^_wx‘aydWuChong-shi !#$%&’(() 5`Fz{|}~ = ˚ ( L %Æ ’ r t0F(Æ ‹ K#›9/ ( Q x=0 x=lF( (˜ ‡ ¡π(ª‹ x=0Ix=l ¢£⁄¥( Q% ! JK-.ƒ!)r #› §0 ‹ §0¡F¤ '(¢ QÆ ‹ ¡ 2 S “ º fl r˛ˇÆ«‹›fi ’( ⁄LQfl –HrF †‡6‹ 0F¤t(#’ I 9/Q12Zl0ρ∂u∂t2dxI12Zl0T∂u∂x2dx,· E(t)=12Zl0ρ∂u∂t2dx+12Zl0T∂u∂x2dx.B, ¶(•¥ CDƒ‚„0ß( ”» E(t)=mπ2a24l2∞Xn=1n2|Cn|2+|Dn|2.… ‰ QGD( t ¿(ª#· `TrF ´¨ «!,¸ ¡,(u1(x,t)Iu2(x,t)( (v(x,t)≡u1(x,t)−u2(x,t) 0!EF!,¸ ∂2v∂t2−a2∂2v∂x2=0,0xl,t0,v x=0=0,v x=l=0,t≥0,v t=0=0,∂v∂t t=0=0,0≤x≤l. ı ł34v(x,t)=0ª⁄rˆ˜¯›⁄L˘˙( ¨!Qƒ rˆ `ı ˚( t=0F#· 0(“«L˝ 0F¤t(E(t)E 0r §¸ 0! ∂v∂x=0,∂v∂t=0,ªv(x,t) GDr%Æ -. JK-.(P ! «GD 0rT˝˛ˇU— f 13.6 U o dE/dt=0o UÆ ª ( o ŁØ )dWuChong-shi§15.1 ¶•‚„”»…‰ (¿) 6`F 3456Œ º;3= 1./01(9:;rsl æ ıT( jkeł[\]^ pqjk~enoWrø\œß W (e (l )ghi_ZYWnoZ[\]^ nopqjk( (l ) yabr2./ 1( , ¸ r3./´1( ,(˜ 01º 0 ,r h|} fW|}l c‘r4./ 1(•¥ CDƒ‚ß!º Dr (›fi Q=,r˛ˇ 6¸ ( 321.st Wu(x,t)e ł[\]^( ( Y‘e !ł[#$2.st Wu(x,t)e pqjk( ( Y‘W XYe %&$3.U_ Y ( ’\e ()*¿+,-./0123456789:;*=+7Cn?Dn@=φ(x)?ψ(x)AB91CDφ(x)?ψ(x)9EFABGH,-./09IJ*KLMNO1PQRST9UV1WXA+YZ[.\]^1._‘abcdef2.gebcheigjklm_‘nopq1rstu1_‘nohvwxyzhf3._‘noig{d|}~* IJ,[./0*WuChong-shi ( ) 7 8un(x,t)=Cnsinnπlat+Dncosnπlatsinnπlx=Ansin(ωnt+δn)sinknx, ωn=nπla,kn=nπl,Ancosδn=Cn,Ansinδn=Dn.Fun(x,t) .¡¢FAnsinknx £⁄N¥ƒ9§¤P'Fsin ωnt+δn £“«C‹Fωn@¡¢9›fifl1 –†‡·B⁄9· fifl¶•‚fifl1„”»\]…‰Fkn –¢71@ «¿ N¢9`´7Fδn@”“«1=”»\]ABF knx=mπ1ˆx=mπ/kn=(m/n)l,m=0,1,2,3,···,n9¥ƒN1§˜9§¤¯–01 –¢˘*˙¨⁄9†.‡ƒ 1¢˘ƒ˚ n+1.*F knx=(m+1/2)π1ˆx=(2m+1)π/2kn=(2m+1)l/2n,m=0,1,2,3,···,n−19¥ƒN1§˜§¤9¸H ¯–˝˛1 –¢ˇ*¢ˇƒ˚ n.*F—./098 @, ¡¢9 *|x 1 e) *F†‡·B9⁄ O1· fifl .˝Æ 1ˆω1=πla, – ª1 · fiflωn2@ fiω19—7 1ωn=nω1,n=2,3,···, – ª*F⁄9 fiFABGŁØŒº9 * ⁄ 1 ⁄9E B(ˆρ B) 1æ R⁄9 ı (ˆ R łT9˛Æ)1Fø œ˘ fiω19˛Æ*F8ß fi? fi9 7{Cn}?{Dn}9“H˛ÆABGŒº9fi P'1ˆABGŒº9 *WuChong-shi§15.1 ( ) 8 F?7∞Xn=1n2|Cn|2+|Dn|2„⁄9 SU 1Ł FABGŒº9 *F ”»\]φ(x)?ψ(x) Φ(x)=−φ(−x),−l≤x≤0,φ(x),0≤x≤l,Ψ(x)=−ψ(−x),−l≤x≤0,ψ(x),0≤x≤l, –`´–2l9`´ 7( –Φ(x)?Ψ(x))*, 9!#$G ‡ƒx=l%@ * Φ(x)?Ψ(x)&’–Fourier67Φ(x)=∞Xn=1αnsinnπlx,Ψ(x)=∞Xn=1βnsinnπlx, αn=1lZl−lΦ(x)sinnπlxdx=2lZl0φ(x)sinnπlxdx,βn=1lZl−lΨ(x)sinnπlxdx=2lZl0ψ(x)sinnπlxdx.„()B*9Cn?Dn“ +1Fø *αn=Dn,βn=nπalCn.Ł u(x,t)=∞Xn=1Cnsinnπlat+Dncosnπlatsinnπlx=12∞Xn=1Dnhsinnπl(x−at)+sinnπl(x+at)i+12∞Xn=1Cnhcosnπl(x−at)−cosnπl(x+at)i=12∞Xn=1αnhsinnπl(x−at)+sinnπl(x+at)