파동 방정식(wave equation) #5

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

 

First, differentiate with \(x\)

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

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

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

 

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

\(\blacksquare\)

 

Then, differentiate with \(t\)

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

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

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

 

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

\(\blacksquare\)

 

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

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

 

Then rearrange the second term with the equation we derived above

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

\(\blacksquare\)

 

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