北大数学物理方法(A)-数学物理方程教案03分离变量法2

Outline1nù©lCþ{()ÔnÆ2007cSC.S.Wu1nù©lCþ{()OutlineùLJ:1Ý/«S­½¯K2üà½uɽЧ9.^‡ÓžàgzUƒAàg¯K¼êÐmC.S.Wu1nù©lCþ{()OutlineùLJ:1Ý/«S­½¯K2üà½uɽЧ9.^‡ÓžàgzUƒAàg¯K¼êÐmC.S.Wu1nù©lCþ{()SteadyStateProblemsForcedVibrationinaStringFixedatBothEndsReferencesÇÂÁ§5êÆÔn{6§§14.3,14.5ù&œ§5êÆÔn{6§§8.2nÎ!X1Á§5êÆÔn{6§§10.5C.S.Wu1nù©lCþ{()SteadyStateProblemsForcedVibrationinaStringFixedatBothEndsReferencesÇÂÁ§5êÆÔn{6§§14.3,14.5ù&œ§5êÆÔn{6§§8.2nÎ!X1Á§5êÆÔn{6§§10.5C.S.Wu1nù©lCþ{()SteadyStateProblemsForcedVibrationinaStringFixedatBothEndsReferencesÇÂÁ§5êÆÔn{6§§14.3,14.5ù&œ§5êÆÔn{6§§8.2nÎ!X1Á§5êÆÔn{6§§10.5C.S.Wu1nù©lCþ{()SteadyStateProblemsForcedVibrationinaStringFixedatBothEndsÝ/«S­½¯KC.S.Wu1nù©lCþ{()SteadyStateProblemsForcedVibrationinaStringFixedatBothEndsÝ/«S­½¯K©lCþ{·^u9D§Ú­½¯K(~X§Laplace§)½)¯K∂2u∂x2+∂2u∂y2=00xa,0ybux=0=0∂u∂xx=a=00≤y≤buy=0=f(x)∂u∂yy=b=00≤x≤aEŒ^©lCþ{¦)E,Uìþ¡o(o‡IOÚ½¦)C.S.Wu1nù©lCþ{()SteadyStateProblemsForcedVibrationinaStringFixedatBothEndsÝ/«S­½¯K©lCþ{·^u9D§Ú­½¯K(~X§Laplace§)½)¯K∂2u∂x2+∂2u∂y2=00xa,0ybux=0=0∂u∂xx=a=00≤y≤buy=0=f(x)∂u∂yy=b=00≤x≤aEŒ^©lCþ{¦)E,Uìþ¡o(o‡IOÚ½¦)C.S.Wu1nù©lCþ{()SteadyStateProblemsForcedVibrationinaStringFixedatBothEndsÝ/«S­½¯K©lCþ{·^u9D§Ú­½¯K(~X§Laplace§)½)¯K∂2u∂x2+∂2u∂y2=00xa,0ybux=0=0∂u∂xx=a=00≤y≤buy=0=f(x)∂u∂yy=b=00≤x≤aEŒ^©lCþ{¦)E,Uìþ¡o(o‡IOÚ½¦)C.S.Wu1nù©lCþ{()SteadyStateProblemsForcedVibrationinaStringFixedatBothEndsÝ/«S­½¯K©lCþ{·^u9D§Ú­½¯K(~X§Laplace§)½)¯K∂2u∂x2+∂2u∂y2=00xa,0ybux=0=0∂u∂xx=a=00≤y≤buy=0=f(x)∂u∂yy=b=00≤x≤aEŒ^©lCþ{¦)E,Uìþ¡o(o‡IOÚ½¦)C.S.Wu1nù©lCþ{()SteadyStateProblemsForcedVibrationinaStringFixedatBothEnds©lCþ∂2u∂x2+∂2u∂y2=00xa,0ybux=0=0∂u∂xx=a=00≤y≤buy=0=f(x)∂u∂yy=b=00≤x≤aC.S.Wu1nù©lCþ{()SteadyStateProblemsForcedVibrationinaStringFixedatBothEnds©lCþ∂2u∂x2+∂2u∂y2=00xa,0ybux=0=0∂u∂xx=a=00≤y≤buy=0=f(x)∂u∂yy=b=00≤x≤aE^©lCþ{¦)©-u(x,y)=X(x)Y(y)C.S.Wu1nù©lCþ{()SteadyStateProblemsForcedVibrationinaStringFixedatBothEnds©lCþ∂2u∂x2+∂2u∂y2=00xa,0ybux=0=0∂u∂xx=a=00≤y≤buy=0=f(x)∂u∂yy=b=00≤x≤a“\§§©lCþ§=C.S.Wu1nù©lCþ{()SteadyStateProblemsForcedVibrationinaStringFixedatBothEnds©lCþ∂2u∂x2+∂2u∂y2=00xa,0ybux=0=0∂u∂xx=a=00≤y≤buy=0=f(x)∂u∂yy=b=00≤x≤aX00(x)X(x)=−Y00(y)Y(y)=−λ=⇒X00(x)+λX(x)=0Y00(y)−λY(y)=0C.S.Wu1nù©lCþ{()SteadyStateProblemsForcedVibrationinaStringFixedatBothEnds©lCþ∂2u∂x2+∂2u∂y2=00xa,0ybux=0=0∂u∂xx=a=00≤y≤buy=0=f(x)∂u∂yy=b=00≤x≤aX00(x)X(x)=−Y00(y)Y(y)=−λ=⇒X00(x)+λX(x)=0Y00(y)−λY(y)=0“\'ux˜éàg.^‡C.S.Wu1nù©lCþ{()SteadyStateProblemsForcedVibrationinaStringFixedatBothEnds©lCþ∂2u∂x2+∂2u∂y2=00xa,0ybux=0=0∂u∂xx=a=00≤y≤buy=0=f(x)∂u∂yy=b=00≤x≤aX00(x)X(x)=−Y00(y)Y(y)=−λ=⇒X00(x)+λX(x)=0Y00(y)−λY(y)=0X(0)Y(y)=0X0(a)Y(y)=0=⇒X(0)=0X0(a)=0C.S.Wu1nù©lCþ{()SteadyStateProblemsForcedVibrationinaStringFixedatBothEnds¦)Š¯KX00(x)+λX(x)=0X(0)=0X0(a)=0C.S.Wu1nù©lCþ{()SteadyStateProblemsForcedVibrationinaStringFixedatBothEnds¦)Š¯KX00(x)+λX(x)=0X(0)=0X0(a)=0eλ=0‡©§Ï)X(x)=A0x+B0C.S.Wu1nù©lCþ{()SteadyStateProblemsForcedVibrationinaStringFixedatBothEnds¦)Š¯KX00(x)+λX(x)=0X(0)=0X0(a)=0eλ=0‡©§Ï)X(x)=A0x+B0.^‡=⇒A0=0,B0=0C.S.Wu1nù©lCþ{()SteadyStateProblemsForcedVibrationinaStringFixedatBothEnds¦)Š¯KX00(x)+λX(x)=0X(0)=0X0(a)=0eλ=0‡©§Ï)X(x)=A0x+B0.^‡=⇒A0=0,B0=0`²λ=0žk)©=λ=0Ø´ŠC.S.Wu1nù©lCþ{()SteadyStateProblemsForcedVibrationinaStringFixedatBothEnds¦)Š¯KX00(x)+λX(x)=0X(0)=0X0(a)=0λ6=0ž‡©§Ï)X(x)=Asin√λx+Bcos√λxC.S.Wu1nù©lCþ{()SteadyStateProblemsForcedVibrationinaStringFixedatBothEnds¦)Š¯KX00(x)+λX(x)=0X(0)=0X0(a)=0λ6=0ž‡©§Ï)X(x)=Asin√λx+Bcos√λx.^‡=⇒B=0A6=0cos√λa=0C.S.Wu1nù©lCþ{()SteadyStateProblemsForcedVibrationinaStringFixedatBothEnds¦)Š¯KX00(x)+λX(x)=0X(0)=0X0(a)=0λ6=0ž§Ò¦Šλn=2n+12aπ2,n=0,1,2,3,···¼êXn(x)=sin2n+12aπx.C.S.Wu1nù©lCþ{()SteadyStateProblemsForcedVibrationinaStringFixedatBothEndsA)9˜„)§Y00n(y)−λnYn(y)=0λn=2n+12aπ2,n=0,1,2,3,···)Yn(y)=Cnsinh2n+12aπy+Dncosh2n+12aπyC.S.Wu1nù©lCþ{()SteadyStateProblemsForcedVibrationinaStringFixedatBothEndsA)9˜„)Ïd§Q÷vLaplace§!q÷vàg.^‡A)un(x,y)=Cnsinh2n+12aπy+Dncosh2n+12aπy×sin2n+12aπxC.S.Wu1nù©lCþ{()SteadyStateProblemsForcedVibrationinaStringFixedatBothEndsA)9˜„)Ïd§Q÷vLaplace§!q÷vàg.^‡A)un(x,y)=Cnsinh2n+12aπy+Dncosh2n+12aπy×sin2n+12aπxòùáõ‡A)U\å5§Ò˜„)C.S.Wu1nù©lCþ{()SteadyStateProblemsForcedVibrationinaStringFixedatBothEndsA)9˜„)u(x,y)=∞Xn=0Cnsinh2n+12aπy+Dncosh2n+12aπy×sin2n+12aπx#C.S.Wu1nù©lCþ{()SteadyStateProblemsForcedVibrationinaStringFixedatBothEnds|^¼ê5½U\Xê“\'uy˜é(šàg).^‡uy=0=∞Xn=0Dnsin2n+12aπx=f(x)∂u∂yy=b=∞Xn=02n+12aπCncosh2n+12aπb+Dnsinh2n+12aπbsin2n+12aπx=0Šâ¼ê8˜5Za0sin2n+12aπxsin2m+12aπxdx=a2δnmC.S.Wu1nù©lCþ{()SteadyStateProblemsForcedVibrationinaStringFixedatBothEnds|^¼ê5½U\Xê“\'uy˜é(šàg).^‡uy=0=∞Xn=0Dnsin2n+12aπx=f(x)∂u∂yy=b=∞Xn=02n+12aπCncosh2n+12aπb+Dnsinh2n+12aπbsin2n+12aπx=0Šâ¼ê8˜5Za0sin2n+12aπxsin2m+12aπxdx=a2δnmC.S