On April 12, 2000, at 1900 central daylight time (cdt), a Cessna 172N, N5909E, operated by a private pilot porpoised while landing on runway 08L (5,000' x 75', dry asphalt) at the Spirit of St. Louis Airport, Chesterfield, Missouri. The pilot was not injured. The airplane was substantially damaged. The 14 CFR Part 91 personal flight was operating in visual meteorological conditions without a flight plan. The local flight originated at the Spirit of St. Louis Airport, at 1830 cdt. The pilot reported the accident occurred on the fourth touch and go of the flight. She stated that while she was on downwind, she requested a wind check and was informed the winds were from 120 degrees at 12 knots. She reported she decided to use 20 degrees of flaps for the landing. The pilot reported that she pulled the power to idle when the airplane was over the runway. The airplane touched down and porpoised. She reported the airplane porpoised three times prior to coming to rest on the runway. The pilot stated that she was unaware that the nose tire was flat and she initiated another takeoff. She stated that after becoming airborne she noticed a shimmer and the airplane had a tendency to veer to the left. She aborted the takeoff and stopped the airplane on the runway. Post accident inspection of the airplane revealed the nose wheel was flat, the firewall had sustained damage, and one propeller blade tip was missing.