DA40 pilot experienced a bounced landing in strong wind conditions that turned into porpoising and a prop strike.

Narrative

A normal approach to the runway; at air speeds recommended by the POH; was performed; and a normal landing flare was executed. Stall warning horn went off shortly before touchdown; as is normal in that aircraft. After the main gear touched down; the aircraft bounced back into the air; likely due to a wind gust. A small amount of power was added and another touchdown was attempted; which also resulted in a bounce. At that top of the second bounce; the decision to go around was made and full power was applied. Some forward elevator pressure was applied in an effort to keep the aircraft in ground effect and build up airspeed. It became immediately apparent that the aircraft was headed for a nose strike and back pressure was rapidly applied; but that was not sufficient to prevent a nose wheel touchdown and possibly a prop strike. This was followed by porpoising of increasing severity; and likely further prop strikes. When it became clear that the go-around attempt was not succeeding; likely due to prop damage; power was reduced to idle and the porpoising oscillations immediately decreased in severity and soon stopped altogether. At the time of landing winds as reported by AWOS were 15 KTS; gusting 25; out of 260; and the aircraft had a total of three occupants. The incident was likely caused by the pilot-in-command's relative inexperience in type; particularly in high-wind conditions with passengers on board. Remedial action will consist of further training in type with emphasis on high-wind landings; go-around procedures; and porpoising recovery.