파동 방정식(wave equation) #5

Pixabay

$$\psi (x,t)=f(x^{\prime })=f(x\mp vt)$$

 

First, differentiate with $x$

$$\frac{\partial \psi }{\partial x}=\frac{\partial f}{\partial x}=\frac{\partial f}{\partial x^{\prime }}\frac{\partial x^{\prime }}{\partial x}=\frac{\partial f}{\partial x^{\prime }}$$

$$x^{\prime }=x\mp vt$$

$$\frac{\partial x^{\prime }}{\partial x}=1$$

 

$$\frac{\partial \psi }{\partial x}=\frac{\partial f}{\partial x}=\frac{\partial f}{\partial x^{\prime }}$$

$◼$

 

Then, differentiate with $t$

$$\frac{\partial \psi }{\partial t}=\frac{\partial \psi }{\partial x^{\prime }}\frac{\partial x^{\prime }}{\partial t}=\mp v\frac{\partial \psi }{\partial x^{\prime }}$$

$$x^{\prime }=x\mp vt$$

$$\frac{\partial x^{\prime }}{\partial t}=\mp v$$

 

$$\frac{\partial \psi }{\partial t}=\frac{\partial f}{\partial t}=\frac{\partial f}{\partial x^{\prime }}\frac{\partial x^{\prime }}{\partial t}=\mp v\frac{\partial f}{\partial x^{\prime }}$$

$◼$

 

Now, bring the wave equation below

$$\frac{{\partial }^{2}f}{\partial t^2}=v^2\frac{{\partial }^{2}f}{\partial x^2}$$

 

Let's make the simple first term with the equation we derived above

$$\frac{\partial }{\partial t}\left(\frac{\partial \psi }{\partial t}\right)=\frac{\partial }{\partial t}\left(\frac{\partial f}{\partial t}\right)=\frac{{\partial }^{2}f}{\partial t^2}(\psi =f)$$

 

Then rearrange the second term with the equation we derived above

$$v^2\frac{\partial }{\partial x}\left(\frac{\partial \psi }{\partial x}\right)=v^2\frac{\partial }{\partial x}\left(\frac{\partial f}{\partial x^{\prime }}\right)=v^2\frac{\partial }{\partial x^{\prime }}\frac{\partial x^{\prime }}{\partial x}\left(\frac{\partial f}{\partial x^{\prime }}\right)=v^2\frac{{\partial }^{2}f}{\partial x^2}$$

$◼$

 

다 작성하고 보니 이걸 내가 전에 왜 했을까 생각이 드네요. (그냥 자명한거 아닌가)