User:EGM6321.f09.team1.Zhichao Gong/Mtg24

Mtg 24: Thu, 15 Oct 09

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Chladni :


 * PEA1.F09.Mtg24.pg1.fig1.new.svg

p=mass / area T=uniform tension in membr.

$$ \underset{({\color{black} \underset{ \overset{\uparrow}{spatial \ coord.(1-D) \ \psi=ps;}}},{\color{black} \underset{ \overset{\uparrow}{time}}})} \ {\color{blue} \underset{ \overset{\uparrow}{generic \ dep. \ var.}}} \equiv {\color{blue} \underset{ \overset{\uparrow}{trans. \ disp \ membr.}}}$$

$${\color{red}(2)} \ \frac{{\partial}^{2}\psi}{\partial {x}^{2}}=\frac{1}{{c}^{2}}\frac{{\partial}^{2}\psi}{\partial {t}^{2}}$$

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$$Sep. \ of \ var: \ {\color{red} \underset{ \overset{\uparrow}{{\color{blue}Ansatz}(guess) \ (trial \ soln)}}}=X(x){\color{blue} \underset{T(t)}{ \underbrace}} \ {\color{red}(3)}$$

$$i=\sqrt{-1}$$

(3) in (2): $${\color{blue} \underset{ \overset{||}{0}}}{\color{blue} \underset{\neq 0}{ \underbrace}}=0$$

1.D Helmholtz eq.

$${\color{red}\underset{separeted \ ed. \ for \ X}{\underline{{X}^{''}+{k}^{2}X=0, \ {k}^{2}=\frac{{\omega}^{2}}{{c}^{2}}}}}$$

Trial solns$$\Rightarrow$$ coskx, sinkx

Note:

$$\ast \ \frac{{\partial}^{2}\psi}{\partial {t}^{2}}=X{{\color{red} \underset{-}{+}}i\omega}^{2}{e}^{{\color{red} \underset{-}{+}}i\omega t}$$

$$\ast \ Add \ \underset{-}{+}2\pi \ in \ exp({\color{red} \underset{-}{+}}i\omega t \underset{-}{+}{\color{blue} \underset{\psi}{ \underbrace}})$$

$$\psi(x, \ t)=X(x)exp({\color{red} \underset{-}{+}}i\omega t \ + \ \psi)$$

$$X(x)={e}^{ikx}$$

$$\psi(x, \ t)={\color{red} \underset{ \overset{\uparrow}{Amplitude(Const)}}}{e}^{ikx}{e}^{\underset{-}{+}i\omega t}$$

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$$\psi(x, \ t)=A{e}^{i(kx \underset{-}{+}\omega t)}$$

$$=A{e}^{ik(x \underset{-}{+}ct)}$$

$$\omega \ = \ freq. \ in \ time$$

$$k \ = \ freq. \ in \ space{\color{blue}(ware \ number)}$$

More gen. soln:

$$\psi(x, \ t)=G(x{\color{red}\underset{-}{+}}ct) \ {\color{blue}K. \ p. \ 82}$$


 * PEA1.F09.Mtg24.pg3.fig1.new.svg

Wave eq. in 1-D is shape preserving. c&gt;0 speed. + out going wave (+ x direction) - incoming wave (- x direction) Wave eq in 2-D more complex <P>                    3-D even more complex</P> <P>      curv. coordinate (even more complex)<SUP>2</SUP></P>