The fundamental differential equation of wave motion and the mathematical linearity that allows waves to exist independently.
The wave equation is a second-order linear partial differential equation that describes the propagation of oscillations through a continuous medium. It links the spatial curvature of the wave to its temporal acceleration.
Here, $v$ is the wave propagation speed. This equation arises from applying Newton's laws to a continuous medium, like a stretched string or fluid.
Think of a wave propagating through a crowd in a stadium. Each person only moves up and down vertically, responding to the height of their immediate neighbors. Yet, this localized up-and-down motion creates a macro "wave" that travels horizontally around the stadium at a constant speed.
In 1747, Jean le Rond d'Alembert discovered the general solution to the 1D wave equation. He showed that the general solution consists of two arbitrary wave shapes, one traveling to the right and one to the left.
The function $f(x - vt)$ represents a wave traveling in the positive $x$ direction with speed $v$, while $g(x + vt)$ travels in the negative $x$ direction. Their shapes remain constant as they propagate.
Because the wave equation is linear (it contains no terms like $y^2$ or $y \cdot \frac{\partial y}{\partial x}$), the sum of any valid solutions is also a valid solution.
Physically, this means that two waves can pass through the same medium simultaneously without destroying or altering one another. When they overlap, the total displacement is merely the algebraic sum of their individual displacements.
If you throw two stones into a calm pond, circular ripples expand from both splash points. When the two sets of ripples meet, they don't bounce off each other. They smoothly pass right through one another, momentarily adding their heights together, and then continue on their original paths undisturbed.