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WuChong-shiWuChong-shi 1 !#$%& ’()*+,-. /01. 23456789!:;=?@:A6B9!/89B9!C$89!DEF•GHICJKILM6Laplace!NPoisson!•O6PQ3LM6OR!•SPTUVCWXUV26SPT!•YZ[\K 26Navier–Stockes!]CEuler!]•^_H‘abRcd6Maxwell!]•ef8g \bRhijk6Schr¨odinger!CDirac!•lmK 26Saint-Venant!]nnDop!(])qrstum789!(])D3viw!xy2z{|}~ 6stum789!Di & p UVT p~ 6stum789!Dv z{op!6 m\’ DWuChong-shi§13.1 2 §13.1 ¡¢£⁄¥ƒ§¤'“«‹›fi fl –†‡· ¶D•‚ „ ¶“”D»…‰ ¿ `´xˆ˜¯˘˙¨ ´x=0˚x=lD¸u(x,t) ¨ ´x‰…˝˛˙ˇt— ‰( ) Dˇ…˝ ´dx‰˛ (… )D… ‰… Æ ª ˙D Ł… ØŒºªˇ ˘˙x x+dx Ø æŒ‰ D 13.1ı łøtanθ1=∂u∂xx,tanθ2=∂u∂xx+dxœfßr 6 K KT6e : KD ; K6e D (Tsinθ)x+dx−(Tsinθ)x=dm∂2u∂t2,(Tcosθ)x+dx−(Tcosθ)x=0.· ¶ ºx+dx?x ; u(x+dx,t)−u(x,t)?dx r |∂u/∂x| 1.ˇ !#$%sinθ≈tanθ=∂u∂x&’(∂u∂x‰)*+,cosθ≈1&’(∂u∂x‰,*+.WuChong-shi 3 -./ (T)x+dx−(T)x=00(T)x+dx=(T)x,12T34x56 789 :;=D ρdx∂2u∂t2=T∂u∂xx+dx−∂u∂xx#=T∂2u∂x2dx,0ρ∂2u∂t2−T∂2u∂x2=0,?ρ …‰@AB(C B‰ D)DEFa=sTρ,GHIÆ J ∂2u∂t2−a2∂2u∂x2=0.a/ …‰!KLMBDNOvPQºR SRTUV:;TWtXYDorœfßZ6[\ds−dx=pdu2+dx2−dx=s1+∂u∂x2−1dx=O∂u∂x2,3vR]^ ∂u/∂x6 _‘( SRTU)6abVßZ6\cde; cdDœ@fgHookehkTde; cdDijklmPQnTdexcd3vTo pqDrs…ˇ (0 u‰t )˝uØ vŒ‰ ¸C Bw؉vŒ´fGxyz{‰|} ρdx∂2u∂t2=T∂2u∂x2dx+fdx. ∂2u∂t2−a2∂2u∂x2=fρ,?‰~ +f/ρ C Dw؉vŒDWuChong-shi§13.2 4 §13.2 “ ‡· ¶D fl ¡ “ –†89 ¶ ( ) DFr 13.2» H ´xˆH H ‰ { ª‰ ¿ `x DFˇ ˛— t { ¿ `‰ ´u(x,t)DFˇ ? ˛ (x,x+dx) ŁØŒº• ¡ {xØ ¢£ŒP(x,t)S‰ • ¡ {x+dxØ ¢£ŒP(x+dx,t)S‰ P(x,t)´C {⁄w؉¢£Œ(¥Œ)ƒxH ´tD 13.2§ ¤łø'“«'‹ ›fiNewtonfl,E /– dm∂2u∂t2=[P(x+dx,t)−P(x,t)]S.† ‰AB´ρGdm=ρdx·Sρ∂2u∂t2=∂P∂x.rs&’ H ‰‡·›fiHookeE ¥ŒP˚¥·∂u/∂x t P=E∂u∂x, ¶•‚E„´ ‰Young”Dª ˛ » …‚D-./– ( ‰‰!HI∂2u∂t2−a2∂2u∂x2=0,?a=sEρ. ‰‰!˚…‰ ! ¿ `´ˆ ˜¯ª˘˙ ‰¨ HI‰‡˚¸´ˆ˛.D-˛ HI˝„´˛¶“”Dˇ˛— ˇ) ?‰ HI ∂2u∂t2−a2∇2u=0,?∇2≡∂2∂x2+∂2∂y2+∂2∂z2„´Laplace ∇2=∇·∇0∇2u=∇·(∇u).WuChong-shi 5 §13.3 TSPT!3 6 C j6 Dd R Æ 6 jkd Doª 6rS j6 hijk ŁØCŒº FourierŁØDŒº FourierŁØ¸ ˛ D»E˛E¨ • u(x,y,z,t)æ ¨ ´(x,y,z)‰˛˙ˇt— ‰ BD†ƒxH ˛E‰ BıˇxH /˛E D‰KłDøœß˝ ¡x“ – Œ qW 56 0q=−k∂u∂x,q„´ ABk„´} Dk?[\6\ :A ? cu !#$%&’()*k+,-u. /0123456789:;=?@AB;=CDEF#G9HIJ?:=K? LMNOPQRSTU34VWX#YTU3NZ[Q/\]Y^_‘#abqx=−k∂u∂x,qy=−k∂u∂y,qz=−k∂u∂z,cq=−k∇u,dVef_ghqi^_j_∇uklm noFourierpqrshtupqvXwxPQRSTU34VWX[y YTUz{|}~ Z 0 ( 13.3)# Z0\r 0 WuChong-shi§13.3 6 13.3VWX[y (x,y,z) 4 0 FΔt z x[Qe 0 4Vh(qx)x−(qx)x+dxΔyΔzΔt=hk∂u∂xx+dx−k∂u∂xxiΔyΔzΔt=k∂2u∂x2ΔxΔyΔzΔt.FΔt z y[Qe 0 4Vhh(qy)y−(qy)y+dyiΔxΔzΔt=k∂2u∂y2ΔxΔyΔzΔt,FYΔt z z[Qe 0 4Vh(qz)z−(qz)z+dzΔxΔyΔt=k∂2u∂z2ΔxΔyΔzΔt. 0 z b Vh c ¡#anoshtupq#¢:£;9H⁄¥ƒ§¤'“«‹›fi?@flJ –†;9H#k∂2u∂x2+∂2u∂y2+∂2u∂z2ΔxΔyΔzΔt=ρΔxΔyΔz·c·Δu.‡·∂u∂t−kρc∇2u=0, 3ρ TU4f_#c mV¶ WuChong-shi•‚„”»…‰ ¿»… ` 7 ´κ=k/ρc#ab∂u∂t−κ∇2u=0, 3κˆ˜¯˘˙#c^_WX˙ YTUzbVh¨ (˚ #b¸ ˝˛ˇ #c—b e#······)# z TU3¨ 4Vh˜F(x,y,z,t)#abk∇2uΔxΔyΔzΔt+F(x,y,z,t)ΔxΔyΔzΔt=ρΔxΔyΔz·c·Δu,∂u∂t−κ∇2u=1ρcF(x,y,z,t)=f(x,y,z,t). TU wx#aXV˙ki (x,y,z)b #VWX[y ˜ρc∂u∂t−∇·(k∇u)=F(x,y,z,t).9 ; ´j=ρcu#ˆ˜Ve(Æ_)#a∂j∂t+∇·q=F(x,y,z,t). Z[yªˆ˜ PQ STU#aFourierpq˛ŁØkq=−K·∇u.Œº K 3×3 #æ ∇u ı $ ł˛ø#VWX[y ˜ρc∂u∂t−∇·(K·∇u)=F(x,y,z,t).œß 4 _ #^_4 ß V y_4˝ ß V 4 #— sh#Y / ˜Vh4W · # TUz]Y wx #˚ U _4 wx#— ß 4 ˇ U4 # Y / ˜ß 4¯˘ Y /4ł S# p!¯˘[yrVWX[ybłR42#∂u∂t−D∇2u=f(x,y,z,t), 34u(x,y,z,t)# ß _#D ¯˘˙#f(x,y,z,t)a zY 3$ ß 4¨˙ WuChong-shi§13.4%&’( 8 §13.4)*+,-.?@/0Y p123# 4^_456p7d 8 ¸ #a^_ß9:;Poisson[y∇2u=−fκ. # f=0#abLaplace[y#∇2u=0. = [yØ4 456u4 ? @AB;AC D [y3u 8 ¸#˚ E F4 Gu(x,y,z)# :;Poisson[y∇2u=−ρε0, 3ρ Hf_#ε0ˆ˜IJ ¶˙(IJT ªK) Hf_ρ≡0#aE F4 G:;Laplace[y∇2u(x,y,z)=0.LMN D [y∂2u∂t2−a2∇2u=03#u(x,y,z,t)8 OPø ¸#Q˙˜ω#u(x,y,z,t)=v(x,y,z)e−iωt,av(x,y,z):;Helmholtz[y∇2v(x,y,z)+k2v(x,y,z)=0, 3k=ω/aˆ˜DK ·/4R ST4U ß[y#œ / #bFVWXYZ[ XY\[FVW]^Z[ _‘a\[FVWbcde Poisson\[ Laplace\[œK / # lfł˛øߘNghFXY\[#ijklmnopq\[F_‘a\[#ijklmnrsq\[FPoisson\[ Laplace\[#ijklmntuq\[ Ng[y4vwxy#z T{y43|}~