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We then introduced the wave equation in 1d and showed that the sinusoidal waves indeed satisfy this equation.

So to my current understanding a 1d wave is a phenomena described by a function which associates to each point in time a scalar displacement at each point on a line, that satisfies the wave equation. So far so good.

Now there is an example where a rod attached to a wall is pulled and we want to analyse the displacement at each point on the rod from its original position over time. Using hooks Law we are able to derive a relationship between the second partial of time and the second partial of space(of the displacement). We then plug this into the wave equation, which gives us an expression for the velocity at which this wave propagates through the rod.

My question is, why are we able to plug these values into the wave equation? What about this example tells us that these displacements will indeed satisfy the wave equation?