# Visualising traveling waves as solutions of $\partial_t ^{2} u - \partial_x ^{2} u=0$

I have seen that there are particular solutions for the wave equation called traveling waves. I have also seen stationary waves, but I would like to understand the physical meaning of the former. While I can imagine a stationary wave $u(t,x)$ in the space $O_{t,x,u}$ if the problem has some boundaries (let's say for example we have a string fixed at both extremes. Then I can see the wave in that space that oscillates for $x$ between those extremes), I am not sure I understand what happens with traveling waves.

We have seen that in general every function like $f(t,x)= \phi(x-t)+ \psi(x+t)$ with $\phi$ and $\psi$ regular enough is a solution (we have also seen the concept of weak solutions, but this is a more abstract concept that I don't need to discuss). Then we have seen that given some initial conditions, the d'Alembert formula gives us the solution. The problem is that I can't imagine that function propagating. I don't "see" what happens as time goes on, and how things go.

• Well, there's left- and right-movers, cf. the method of characteristics. – Qmechanic Feb 13 '18 at 16:10
• @Qmechanic yeah I kinda see that, but does that mean that the wave only propagates in two spcifical directions? I mean, I think it should follw the lines $t=x$ and $t=-x$, right? – tommy1996q Feb 13 '18 at 16:48
• You could go down to the seas again (to the lonely sea and the sky) and watch the waves. That will give you a pretty good idea. – John Rennie Feb 13 '18 at 17:39

$f(t,x)= \phi(x-t)+ \psi(x+t)$
$\phi(x-ct)$ and $\psi(x+ct)$
where c is the propagation speed. As t increases from t=0, x must increase in $\phi(x-ct)$ and decrease in $\psi(x+ct)$ to maintain the same numerical value of their arguments (to select a fixed point in $\phi$ or $\psi$ ). So then $\phi$ would be moving in the +x direction (say 'forward'or 'right') and $\psi$ would be moving in the -x direction (say 'backward' or 'left').
In general $g(x+a)$ is a translation of $g$ to the right or left depending on the sign of $a$. Here $a$ is the time dependent $ct$.