# Thread: wave equation

1. ## wave equation

Let f and g be differentiable functions of one variable

set $\displaystyle \phi = f(x-t)+g(x+t)$

a) prove that $\displaystyle \phi$ satisfies the wave equation : $\displaystyle \frac{\partial^2 \phi}{\partial t^2} = \frac{\partial^2 \phi}{\partial x^2}$

b) sketch the graph of $\displaystyle \phi$ against $\displaystyle t$ and $\displaystyle x$ if $\displaystyle f(x)=x^2$ and $\displaystyle g(x)=0$

is part a) as simple as ( I can't see it being) $\displaystyle f''(x-t)+g''(x+t)=f''(x-t)+g''(x+t)$

2. ## Re: wave equation

Sort of. You need to evaluate \displaystyle \begin{align*} \frac{\partial ^2 \phi}{\partial t^2} \end{align*} and \displaystyle \begin{align*} \frac{\partial ^2 \phi}{\partial x^2} \end{align*} and show they're equal, which is what you have tried to do. But you just need to set it out better. Also you can't use Newton's notation " ' " for a derivative here, because these are partial derivatives...

3. ## Re: wave equation

Originally Posted by Prove It
Sort of. You need to evaluate \displaystyle \begin{align*} \frac{\partial ^2 \phi}{\partial t^2} \end{align*} and \displaystyle \begin{align*} \frac{\partial ^2 \phi}{\partial x^2} \end{align*} and show they're equal, which is what you have tried to do. But you just need to set it out better. Also you can't use Newton's notation " ' " for a derivative here, because these are partial derivatives...
$\displaystyle \frac{ \partial^2 f}{\partial t^2} + \frac{\partial^2 g}{\partial t^2} = \frac{ \partial^2 f}{\partial x^2} + \frac{\partial^2 g}{\partial x^2}$

like that??

4. ## Re: wave equation

And now you actually have to show that these things are equal...

5. ## Re: wave equation

Originally Posted by Prove It
And now you actually have to show that these things are equal...
is that

$\displaystyle f_{xx} = 0; f_{tt} = 0; g_{xx} = 0; g_{tt} = 0;$

so $\displaystyle 0+0=0+0$ ??