Do you mean the ratio of the averagespeedto the maximum speed?

No, you can't possibly mean that since an object in harmonic motion returns to its starting point after one oscillation and so its "average speed" is 0. What average is it that you are talking about?

Harmonic motion can be modeled as . Its speed, then, is . The maximum speed occurs when takes its maximum value which is 1- the maximum speed is .

The average of a continuous function over an interval is its integral over that interval divided by the length of the interval. Here, an anti-derivative of v= x' is x itself so we just need to determine the interval. sin(T) has period so completes one cycle when or .

But, the integral of v= x' is which is 0 at both ends so, like I said before, its average is 0.