How does the restoring force connect to waves on a string?
When Newton’s second law is applied to a problem with a linear, restoring force, you typically get an equation of motion- a differential equation for the displacement as a function of space and time – that’s a wave equation. What it means to be a wave equation is that the solution gives some sort of wavy motion. The simplest example we saw is the case of a spring in one dimension: in that case, we got an equation of motion that led to a sinusoid.
