Accuracy Examination of the Fourier Series Approximation for Almost Limiting Gravity Waves on Deep Water

A permanent gravity wave propagating on deep water is a classic mathematical problem. However, the Fourier series approximation (FSA) based on the physical plane was examined to be valid for almost waves at all depths. The accuracy of the FSA for almost-limiting gravity waves remains unevaluated, which is the purpose of this study. We calculate some physical properties of almost-limiting waves on deep water using the FSA and compare them with other studies on the complex plane. The comparison results show that the closer the wave is, the greater the difference. We find that the main reason for this difference is that the wave profile in the FSA retains an original implicit form and is not represented by Fourier series. Therefore, the kinematic and dynamic conditions of the free surface around the wave crest cannot be satisfied at the same time.