The long wavelength limit of periodic solutions of water wave models

The present essay is concerned with providing rigorous justification of a long-standing practice in numerical simulation of partial differential equations. Theory often sets initial-value problems on all of $R$ or $R^d$. If the initial data are localized in space, it has been usual practice to approximate the problem by an associated periodic problem or a homogeneous Dirichlet problem set on a finite interval. While these strategies are commonplace, rigorous justification of the practice is sparse. It is our purpose here to indicate justification of this practice in the concrete context of a surface water wave model. While the theory worked out here is specific to the particular partial differential equation, it will be apparent to the reader that more general results may be derived using the same approach.