In this paper we extend the theory of synchronously pumped mode-locking and obtain the precise solution, i.e. supermodes solution. The physical meaning of the supermodes solution is given. Under the condition of the lowest supermodes, the analytic expressions for pulsewidth, intensity and peak position are deduced. The effect of noise on supermodes solution of the synchronously pumped mode-locking dye laser is studied. The expressions for the spectrum of noise, the method of evaluating the noise source and the ratio of noise to signal are also obtained.