Interesting. I've never heard of this before, but according to

wikipedia, simple harmonic motion is given by:

where

is displacement,

is time,

is amplitude,

is frequency, and

is phase.

We also note that period

is given by:

This means that

. So:

To find velocity, we take the derivative:

To find velocity extrema, we take the derivative of

and set it equal to zero:

Let's plug that into our velocity function:

Since the amplitude must be a positive value, we can just say:

Now, the center of oscillation is another way of saying

. So:

Let's let

and

:

And we plug that into our velocity function:

Then plug in

:

But of course we know that speed is relative, and so the value is actually: