Understanding the wave equation

[Eclair_de_XII]

Homework Statement

"The solution $u(x,t)=f(x+ct)+g(x-ct)$ solves the PDE, $u_{tt}=c^2u_{xx}$. By graphing the solution $u(x,t)=f(x+ct)$ on the $ux$-plane, please show that as $t$ increases, the graph shifts to the left at a velocity $c$. Conversely, show that for $u(x,t)=g(x-ct)$, the graph shifts to the right as $t$ increases."

Homework Equations

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The Attempt at a Solution

The first thing I was going to do was to take the partial derivative of $u$ with respect to $t$. Then I can observe that $u_t(x,t)=c⋅f'(x+ct)$ and that $u_t=c⋅u_x$. I honestly don't understand how I would apply this derivative to the concept of shifting the graph of $u(x,t)$ to the left or to the right. I have a factor of $c$ there. So I guess the instantaneous rate at which $u(x,t)$ changes with respect to $t$ is $c$ times greater than the instantaneous rate at which $u(x,t)$ changes with respect to $x$? I don't know how to start this problem, honestly.

I mean I could try setting $u(x,t)=f(x+ct)=α$ and graphing it. From there I could then set $u(x,0)=α$. Then I guess I could set $u(x,1)=α$ and by changing the boundary conditions each time, I could perhaps show that as $t$ increases, this arbitrary point of $u(x,t)$ moves to the left for each increment of $t$. And I guess I could use the partial derivative of $u(x,t)$, $c⋅f'(x+ct)$ to show that $f(x+ct)$ changes at an instantaneous rate of $c$. The problem is, I don't exactly understand how $f'(x+ct)$ would correspond to a shift in the graph of $u$ in the $ux$-plane...

 

[vela]

You're not supposed to differentiate anything here. It's just asking you to plot $u(x,t_0)$ and $u(x,t_1)$ for some values of $t_0$ and $t_1$ and show that the wave is moving to the left or right with speed $c$. Just set $c=1$.

 

[Eclair_de_XII]

Thanks.