二维波动方程的导出

数学物理方法

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October 13, 2014

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数学物理方法

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Gauss-Greenúª: R n´±v 1w -¡ >. k.« ( ±´õëÏ ), u 34« ∪ þëY, 3 SkëYê. n ü{ . K uxi dx1··· dxn=

u cos(xi, n)ds .

(1)

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数学物理方法

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Newton-Leibnizúª

= (a, b ) R 1, K (2)

u (x )dx= u (b ) cos 0+ u (a) cosπ= u (b ) u (a).

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数学物理方法

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Newton-Leibnizúª

= (a, b ) R 1, K (2)

u (x )dx= u (b ) cos 0+ u (a) cosπ= u (b ) u (a).

Greenúª:

= D R 2, D=Γ, K (D

P Q )dxdy= x y

Pdx+ Qdy .Γ

(3)

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数学物理方法

Greenúª y²y²: dGauss-Greenúªk Q dxdy= x P dxdy= y Q cos(n, x )ds,Γ

D

P cos(n, y )dsΓ

D

Ïd (D

Q P )dxdy= x y

Q cos(n, x )ds ΓΓ

P cos(n, y )ds

Γ

τ, K cos(n, y )ds= cos(τ, x )ds= dx .

cos(n, x )ds= sin(τ, x )ds= dy,= Greenúª.m

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数学物理方法

È©úªGaussúª: R 3, u= P (x, y, z )i+ Q (x, y, z )j+ R (x, y, z )k, K div u dxdydz=

(

Q R P++ )dxdydz x y z P cos(n, x )+ Q cos(n, y )+ R cos(n, z )ds

=

=

Pdydz+

Qdxdz+

Rdxdy (4)

=

u· nds .

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数学物理方法

È©úªGaussúª: R 3, u= P (x, y, z )i+ Q (x, y, z )j+ R (x, y, z )k, K div u dxdydz=

(

Q R P++ )dxdydz x y z P cos(n, x )+ Q cos(n, y )+ R cos(n, z )ds

=

=

Pdydz+

Qdxdz+

Rdxdy (4)

=

u· nds .

Ñݽn:

R n, ¤áe¡'X div u dx=

u· nds .

(5)

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数学物理方法

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数学物理方法

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©ÙÈ©úª: uxi vdx=

u v cos(n, xi )ds

u vxi dx .

(6)

y²: dGauss-Greenúªk (uv )xi dx=

u v cos(n, xi )ds .

5¿ (uv )xi dx=

u vxi dx+

uxi vdx .

= ©ÙÈ©úª.

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数学物理方法

È©úªGreen1 úª: u, v 9§ ¤k ê34« ∪ þëY.§ ¤kê3 SëY. K u vdx=

u

v ds n

u· vdx .

(7)

y²: 3Ñݽn¥^u v O u div (u v )dx=

u v· nds .

(8)

5¿ div (u v )= · (u v )= u· v+ u v,òþª“<(8)= .m‘ÅÄ §Ñ

v· n=

v . n

数学物理方法

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Green1úª: (u v v u )dx=

(u

u v v )ds . n n

(9)

y²:dGreen1 úª v udx=

v u· nds

v· udx .

(10)

(7) (10) = Green1úª.

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数学物理方法

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åF ²ï ¤3²¡R ; )?Û-|å.

´R^,§Ï

- u)/Cج

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Γ ;

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Ñ£^u (x, y, t ) L«,

ÜåT τ×ν —. § u= u (x, y, t ). -¡{ ν ±

( ux, uy, 1). sτ (cos(x, s), cos(y, s), 0). (cos(x, s), cos(y, s),¯¢þÏ τ s²1.τ u ). s

(cos(x, s), cos(y, s),η ). Kν⊥τ ν·τ= 0.=η= ux cos(x, s)+ uy cos(y, s)=m‘ÅÄ §Ñ

u . s

数学物理方法

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τ×ν

(α1,α2,α3 ),Ù¥α1= cos(y, s)+ u uy . s u ux . s

α2= cos(x, s)

α3= ux cos(y, s) uy cos(x, s)= ux cos(x, n)+ uy cos(y, n) u= . n

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数学物理方法

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¯¢þÏ τ×ν=Ú (y, s )= (x, n), (x, s )=π (y, n). dd ,Üå3R ©þ´ T|u= Tα32+α2+α2α1 2 3

i j cos(x, s ) cos(y, s ) ux uy

k u s

1

.

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