Phase velocity for waves on a slinky

The phase velocity v was introduced in describing traveling waves. It satisfies v = &lambda&nu. We also know what &lambda and &nu mean for standing waves; therefore, we can find v by studying standing waves instead of traveling waves. Stretch your slinky to a length of 10 feet or so and hold the two ends firm. Pluck one end to send a short pulse along the slinky to measure v. (Note: the velocity of such a pulse is not strictly the phase velocity v, but for the slinky the two are equivalent.) Compare it to value you determine from the standing wave frequency. Specifically, the "down and back" time for the pulse should equal the period of the lowest-frequency transverse mode. Show that this is true and explain why. Also, if you double the length of the slinky by stretching it farther then the phase velocity should double. Confirm this and explain why.