homework and exercises - Kinetic energy stored in a triangular wave

I usually used to solve questions based on plane progressive harmonic wave. Recently I encountered a triangular plane progressive wave. Here is a snapshot of the wave.

I am provided with mass/unit length of the string and the tension in the string. I had to find out the kinetic energy of the wave pulse travelling in the taut string as in the figure above.

Here is my attempt:

The triangular wave seems to me like $\sin^{-1}(\sin x)$ with some modifications. I am struggling finding out correct expression of $y$ as expression of $x$ and then integrate to find out the required energy. Any help is appreciated.

Answer

Outline of solution:

• Find the total amount of potential energy stored. This is simply $T \Delta \ell$ where $T$ is the tension and $\Delta \ell$ is the change in length of the rope.


• Don't bother doing integrals, just compute $\Delta \ell$ with the Pythagorean theorem.


• Note that for waves on a string, the kinetic and potential energies are equal.