Maximum temporal amplitude and designs of experiments for generation of extreme waves

This paper aims to describe a deterministic generation of extreme waves in a typical towing tank. Such a generation involves an input signal to be provided at the wavemaker in such a way that at a certain position in the wave tank, say at a position of a tested object, a large amplitude wave emerges. For the purpose, we consider a model called a spatial nonlinear Schr\"odinger equation describing the spatial propagation of a slowly varying envelope of a signal. Such a model has an exact solution known as (spatial) Soliton on a Finite Background (SFB) that is a nonlinear extension of Benjamin-Feir instability. This spatial-SFB is characterized by wave focusing leading to almost time-periodic extreme waves that appear in between phase singularities. Although phase singularities and wave focusing have been subject to a number of studies, this spatial-SFB written in the field variables has many interesting properties among which are the existence of many critical values related to the modulation length of the monochromatic signal in the far fields. These properties will be used in choosing parameters for designing experiments on extreme wave generation. In doing so, a quantity called maximum temporal amplitude (MTA) is used. This quantity measures at each location the maximum over time of the wave elevation. For a given modulation length of SFB and desired maximum amplitude at a position in a towing tank, the MTA readily shows the maximum signal that is required at the wavemaker and the amplitude amplification factor of the requested signal. Some examples of such a generation in realistic laboratory variables will be displayed.