We propose a Luenberger-like observer for a water wave system. The water wave dynamics are assumed to be governed by the linearized water wave equation (LWWE). Our proposed observer estimates the velocity potential of water waves by measuring only the surface wave elevation. The information of the water-wave velocity field is then extracted from the gradient of the velocity potential. It is shown that the estimation error of the observer converges exponentially to zero as time tends to infinity. Furthermore, we numerically solve the LWWE and observer equation by the finite difference method. The efficiency of our proposed numerical scheme and the performance of the observer are validated by numerical simulation results.

我们建议一个类似于Luenberger的水波观测器。假定水波动力学受线性化水波方程（LWWE）控制。我们建议的观察者仅通过测量表面波高程来估计水波的速度势。然后从速度势的梯度中提取水波速度场的信息。结果表明，随着时间趋于无穷大，观察者的估计误差呈指数收敛至零。此外，我们通过有限差分法对LWWE和观测器方程进行数值求解。数值仿真结果验证了我们提出的数值方案的效率和观测器的性能。