A wave is not an object sliding along a string. It is
a pattern of motion, carried by many tiny pieces passing a disturbance
from neighbor to neighbor. The material mostly stays put; the message moves.
Watch one bead. It mostly moves up and down, while
the shape itself slides sideways through the string. Like a
stadium wave, motion travels without every piece traveling with it.
The secret is local. Each small piece only feels its nearby pieces.
Where the string bends, the middle is pulled back toward the shape
around it. No piece needs to know the whole string.
That rule becomes the wave equation: acceleration in time equals wave speed
squared times curvature in space. The left side measures how the height
speeds up; the right side measures how sharply it bends.
The constant c is the wave speed. Change it, and the rule does
not change; only how quickly the disturbance hands itself down the line.
Now turn time into a second direction. The
moving string becomes a surface, where every cross-section
is the string at one particular instant.
Slide a bright slice through that surface. The slice is not a
different object; it is the same string, seen a little later.
The same idea scales up. On a drumhead, every point talks
to nearby points, and the equation becomes a living two-dimensional surface.
So the wave equation is a recipe for propagation:
curvature creates acceleration, and acceleration carries the disturbance forward.