Origin of the wave equation: vibrating springs

The wave equation is a partial differential equation that describes the propagation of waves through a medium, such as mechanical waves (such as sound) through air or water, or electromagnetic waves (such as visible light or radio waves) through vacuum or glass.

For a function $f(t)$ with a single variable $t$ representing time, the zero-dimensional wave equation has the form

This equation describes the horizontal position of a point mass attached to a (idealized) spring, where the force (second-derivative) is proportional and opposite to the displacement from equilibrium. For a spring with spring constant $k$ (representing the stiffness), the equation becomes (we have normalized mass to 1) $$f''(t) = -k f(t)$$ In the applet below, the point mass is represented by a black square and can be dragged horizontally to change its position. Its equilibrium position is depicted by a dashed line, and the force acting on the mass is shown as a red arrow. By clicking the play button, you can observe how the mass moves over time. To its right, the position $f(t)$ is graphed through time. This forward-evolution through time is analogous to the forward-evolution of the wave equation.

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