left- and right-moving waves
Here is the single most important picture of the one-dimensional wave equation: every solution is just a shape sliding to the right plus a shape sliding to the left, both at speed c. Nothing else ever happens. If you understand this you understand waves on a line.
Why is it true? A function of the form F(x - ct) is a fixed shape F that moves to the right at speed c (as t grows, you must increase x to keep x - ct the same, so the bump travels right). Likewise G(x + ct) moves to the left. The general solution of u_tt = c^2 u_xx is exactly u(x,t) = F(x - ct) + G(x + ct) — a right-mover plus a left-mover, and that is all there is. You can see it cleanly by factoring the wave operator: u_tt - c^2 u_xx = (d/dt - c d/dx)(d/dt + c d/dx) u, a product of two transport operators. The first factor kills right-movers, the second kills left-movers; setting the product to zero leaves exactly their sum. This is why d'Alembert's formula has the form it does: the two halves of phi are precisely the launched F and G.
This is a defining feature of the hyperbolic world. The arbitrary FUNCTIONS F and G (not mere constants, as in an ODE) are pinned down by the initial displacement and velocity. The decomposition also explains reflection: when a right-mover hits a wall it is reborn as a left-mover, and the method of images is just a bookkeeping trick for which left-mover to add. By contrast, the heat equation has no such travelling-wave structure at all — it diffuses rather than propagates.
Two equal pulses approach each other, one a right-mover, one a left-mover. They overlap and momentarily add (or cancel, if one is inverted) — but because each obeys its own transport equation independently, they pass straight through and re-emerge unchanged. Superposition, made visible.
Right-movers and left-movers pass through each other untouched — pure superposition.
This clean splitting into two travelling waves is special to one space dimension. In two or three dimensions waves spread on expanding circles or spheres and there is no simple left/right split — that is where Huygens' principle and spherical means take over.