So suppose I have 2 sound waves given by a function $a_n(t)$ which represents the amplitude $a$ of the sound $n$ at time $t$. And I want to compare the two waves. How much effective is this approach:
1. Imagine a vector, $\vec{r}$, fixed at (0,0) on the $a-t$ plane.
2. The vector is given by $\vec{r} = <a_n(t), t>$.
3. I calculate the average rate of change of the vector divided by the total time length $T$ of the wave $n$ and call this the score $S(n) = \dfrac{d\vec{r}}{dt\cdot T}$.
4. I compare $S(n)$ to see how much the sounds match.