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              <p>This paper investigates the convergence properties of Luenberger observers applied to a linearized water wave model. The study is motivated by the challenge of estimating wave dynamics when only partial free surface measurements are available. We identify fundamental obstructions to convergence, showing that the classical Luenberger observer fails to achieve full-state reconstruction due to challenges associated with mean-value modes and high-frequency components.&lt;p&gt;To overcome these limitations, we introduce modified observer schemes that incorporate frequency filtering and projection techniques. Our theoretical results are reinforced by numerical experiments that demonstrate the practical effectiveness of these observer-based estimation methods for water waves.</p>
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