The fun thing is that a PDE’s solution is emergent by definition: it is an interplay of the dynamics (specified by the the PDE), the boundary conditions, initia

The fun thing is that a PDE’s solution is emergent by definition: it is an interplay of the dynamics (specified by the the PDE), the boundary conditions, initial conditions, and maybe a forcing function (energy or information pump). Change one of those things and you’ll get a different solution. You might be reminded of fractals, which are a special case of iterated function systems, roughly a discrete version of iterated function systems. Or maybe you might think of cellular automata, all based on local update rules and some initial conditions. The answer to your actual question is literally called the wave equation (first in the list on the linked webpage). The left side talks about some variable u and how it changes over time. The right side is how u changes over space. And the two sides are linked via a constant c. By observing the solution or by working out the units, we can understand c to be the phase velocity, or roughly, the wave speed. So the way that u evolves over space is limited by how it evolves over time (and vice versa)! u cannot react to all things in space instantaneously. Therefore, a wave emerges, carrying updates from one part of space to another.