The wave equation: a derivation. We consider a string with mass per unit length ρ kg m-1.

We consider transverse vibrations (diplacement y of each particle is perpendicular to the direction of propagation). We assume a constant tension F in the string (although in practice, the tension will change if the string extends).

We consider the motion of a small element of the string , between P and Q. When in equilibrium the string lies along the x axis and PQ is at P0Q0. We let the displacement of PQ from the x axis be y.

The tension of the string acts along the tangent at any point. Therefore the resultant force parallel to the y axis is:

If P and Q are very close together then sss is very small:

Hence resultant force is:

The mass of PQ is and its acceleration is :

This could be written as:

We have .

Differentiating:

So we have:

Rearranging:

Given that:

we have:

We know that:

Therefore:

and

So we can write:

If the amplitude of the oscillations is small enough:

We put:

So:

The wave equation is usually written: