Background
I am self-studying Introduction to PDEs by Walter Strauss. In chapter 1, Strauss describes that the characteristic lines of the PDE
$$
au_x+bu_t=0
$$
are given by $bx-at=C$, and the function $u(x,t)$ is constant along these lines in xt space.
In chapter 2, Strauss shows that the wave equation, $u_{tt}-c^2u_{xx}=0$ has two families of characteristic lines
$$
x+ct=C
$$
$$
x-ct=C
$$
and the general solution to the wave equation is
$$
u(x,t)=f(x+ct)+g(x-ct)
$$
The families of characteristic lines are sketched here:
Question
What is an intuitive meaning of the two "families" $x\pm ct=C$ for the wave equation? If we consider $u$ on the line $x+ct=C$, is $u$ strictly constant?
$$\begin{align}
u(x,t)&=u\left(x,\frac{1}{c}(C-x)\right)=f(C)+g(2x-C)
\end{align}
$$
So it seems that $u(x,t)$ is not totally constant on the line $x+ct=C$, because the other wave can "pass over" and change the value. Am I understanding this correctly?
Best Answer
Mariano's comment pointed me in the right direction, I believe.
The characteristic lines are "lines on which information can move". So, when $x+ct=C$ is constant, $f(x+ct)$ is constant, but $g(x-ct)$ is not. For a given time $t$, the x-value $x=C-ct$ is a particular point on the left-running wave $f(x+ct)$ with constant contribution to $u$.
Below, I visualize two Gaussian pulse solutions to the wave equation. One moves to the left, and one to the right. The black point on the left-running wave corresponds to $x+ct=1$. Note that $u(x,t)$ is not constant for $(1-ct,t)$.
I had some more time, so I decided to plot more points and visualize their path along the characteristic lines.