Wave crest height and steepness are crucial parameters for the design of ships and offshore structures. For irregular sea states, they are commonly predicted by using linear wave theory and a Eulerian description of the fluid motion. This theory only applies when the wave steepness is small and it fails to capture extreme wave events. Such linear solutions can also be extended by including second-order terms in order to provide more realistic wave properties. The paper describes a model for irregular long-crested waves that is based on a modified linear solution derived from a Lagrangian description of the fluid, i.e. by considering the motion of individual fluid particles. Lagrangian solutions have the advantage of showing realistic wave profiles with sharp crests and broad troughs already at the first order, whereas these features only appear at the second order when using the Eulerian approach. Still, a severe drawback with the former is that the mass conservation is not fulfilled exactly. The aim of the modification in the present Lagrangian model is to ensure that the mass conservation is always fulfilled in the solution. This is done by using the second-order residual in the continuity equation to lift up the fluid particles vertically. Comparative investigations of wave properties such as the crest height and the wave steepness are further carried out by making use of both numerical case studies and wave tank recordings. The wave models used in the comparisons include linear and second-order Eulerian solutions as well as the modified linear Lagrangian one.

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