Parametrising the initial curve; what is the meaning of this geometrically?

Question

In the book of PDE by Kumar, at page 21, it is given that

However, I cannot understand how and why does the author parametrises the curves as $$x = s, \quad y = 0, \quad z = f(x),$$ i.e what does this mean geometrically.

Answer 1

The pde alone has infinite number of solutions. We can visualize them as surfases, but when we want it that goes through a specific curve like the one you have here, there is only one solution surface.

Answer 2

$x=s$, $y=0$ is just the $X$-axis. On that axis you have an initial datum $f(x)$. Thus your characteristics will be issued from each point of the form $(s,0,f(s))$ in the $XYZ$ space. Their union forms a surface $z=z(x,y)$ which is the desired solution.