The optimum shape of a two-dimensional undulatory plate in its motion in fluid flow is analyzed and the physical parameters (the thrust and the power) are calculated by the boundary element method. With the commonly used Lagrange multiplier method, only a saddle point of the optimum solution can be obtained, but not the solution itself due to the singularity of the problem. To eliminate the singularity, a method is proposed by adding an amplitude constraint. The new method is a generalization of that proposed by Wu, and it can be applied to more complex cases. The optimum shape of the linearly varying amplitude motion is studied in detail. It is shown that both the maximum and the minimum solutions exist when the frequency is higher than a critical value. For a high frequency, the maximum efficient motion with a large amplitude at the leading edge and a small amplitude at the trailing edge induces a large leading edge suction force. As the frequency decreases, the leading edge suction force decreases to a minimum and then increases. For high wavenumbers, there exists an optimum frequency with the maximum efficiency. High efficiency holds over a large range of frequency. The optimum efficiency increases as the wavenumber increases. The increase of the wavenumber can also reduce the leading edge suction force. The optimization method can also be applied to a quadratically varying amplitude motion. It is found that the optimum efficiency is larger than that for the linearly varying amplitude motion. However, the additional efficiency is relatively small, especially as the original efficiency is already high enough.