User:EGM6321.f09.team1.Zhichao Gong/Mtg28

Mtg 28: Tue, 27 Oct 09

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(Mtgs 26+27 : Exam 1, Thu, 22 Oct)


 * PEA1.F09.Mtg28.pg1.fig1.svg :PEA1.F09.Mtg28.pg1.fig2.svg
 * PEA1.F09.Mtg28.pg1.fig3.svg

Gen. to 3-D: polar (2D) $$\rightarrow$$ spherical (3D)

Elliptic (2D) $$\rightarrow$$ Elliseidal coord. (3D)

Ellipse $$\rightarrow$$ Ellipseisl (around x axis)

Hyperbole $$\rightarrow$$ Hyperboloid with one sheet(y axis)

" $$\rightarrow$$ " with 2 sheet along z axis

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3-D Wave equation(2-D, 1-D, ..., n-D): $$ \triangle \phi=\frac{1}{{c}^{2}}\frac{{\partial}^{2}\phi}{\partial {t}^{2}} \ {\color{red}(1)}$$

Plane wave: $$\phi(\underset{-}{x}, \ t)=Aexp[i(\underset{-}{k}\underset{-}{x}-\omega t)] \ {\color{red}(2)}$$

$$\underset{-}{x}=({x}_{1}, \ {x}_{2}, \ {x}_{3})={x}_{i}{\underset{-}{e}}_{i}$$

$$\underset{-}{k}={k}_{i}{\underset{-}{e}}_{i}$$


 * PEA1.F09.Mtg28.pg2.fig1.svg

Ansatz: $$\phi({\color{blue} \underset{ \overset{\uparrow}{x=({x}_{1}, \ {x}_{2}, \ {x}_{3})}}}, \ t)=X({\color{blue} \underset{ \overset{\uparrow}{x=({x}_{1}, \ {x}_{2}, \ {x}_{3})}}}){e}^{-i\omega t} \ {\color{red}(3)}$$

$${\color{red}(3) \ and \ (2)}\Rightarrow \ \triangle X+{k}^{2}X=0 \ {\color{red}(4)}$$

$${k}^{2}=\frac{{\omega}^{2}}{{c}^{2}} \ {\color{red}3-D \ Helmhottz \ eq.}$$

Sep. of var. : $$X(x) \underset{=}{X}_{1}({x}_{1}){X}_{2}({x}_{2}){X}_{3}({x}_{3})$$

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(5) into (4) : $$\frac{{X}^{}_{1}}{{X}_{1}}+\frac{{X}^{}_{2}}{{X}_{2}}+\frac{{X}^{''}_{3}}{{X}_{3}}+{k}^{2}=0 \ {\color{red}(1)}$$

Curv. lin. coord. $$ \xi=({\xi}_{1}, \ {\xi}_{2}, \ {\xi}_{3})$$

$$X(\xi)={X}_{1}({\xi}_{1}){X}_{2}({\xi}_{2}){X}_{3}({\xi}_{3}) \ {\color{red}(2)}$$

Separated eq. :

$${\color{red} \underset{\triangle X}{ \underbrace}}+{\color{blue} \underset{ \overset{\uparrow}{{X}_{i}}}{[{\color{black}\sum_{j=1}^{3}{\phi}_{ij}({\xi}_{i}){{k}_{j}}^{2}}]}} \ {\color{red}(3)}$$

$${\color{red} \underset{{k}^{2}x, \ {\color{black}=0, \ i=1, \ 2, \ 3}}{ \underbrace}}$$

Application Eng: Elliptic, Ellipsoidel coordinate

- Earth : spheroial (ellipsoial of revolution) - Flow with particles : serierse, eng. ash flow (e.g. volcanoes) liquid crystal


 * PEA1.F09.Mtg28.pg3.fig1.svg

Ellipsoial: diverse shapes  

disk $$\rightarrow$$ needle microfluidics