According to the student pilot, he was performing his second solo flight in the pattern. During the flare on the first landing, the airplane porpoised on the runway, resulting in the nose wheel impacting the runway three times, oscillating harder each time. After the first landing, an air traffic controller contacted the flight school to inform them of the porpoised landing. The student pilot completed three more landings, taxied back to the ramp, and exited the airplane. The first assessment of the airplane revealed no obvious damage, and it was flown two more times before another examination of the airplane by the operator revealed substantial damage to the firewall. According to a Federal Aviation Administration inspector, there were no preexisting mechanical anomalies with the airplane.