Laboratory-generated, shallow-water surface waves: Analysis using the periodic, inverse scattering transform

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Abstract
The space‐time evolution of laboratory‐generated, shallow‐water wave trains is considered in one‐space and one‐time dimensions. An electronic control‐and‐feedback system is used to generate periodic/quasiperiodic wave trains in a laboratory flume (0.76×0.8×46 m3) in which five spatially distributed, resistance wave gauges simultaneously record time series of the surface wave field. A concrete ramp with a slope of 0.02 minimizes reflections, and thus ensures near‐unidirectional wave motion. The data are analyzed using both (1) linear Fourier analysis and (2) a relatively new kind of nonlinear Fourier analysis based upon the inverse scattering transform (IST) for the periodic Korteweg–de Vries (KdV) equation. The periodic IST formalism consists of a linear superposition of the ‘‘hyperelliptic‐function oscillation modes,’’ which are intrinsically nonlinear, while simultaneously undergoing nonlinear interactions with each other. The KdV oscillation modes may be viewed as the ‘‘sine waves’’ of the periodic scattering transform, although they are generally nonsinusoidal in shape. The amplitudes of the oscillation modes are constants of the motion for wave motion governed purely by KdV. For a series of experiments in the Ursell number range 0.30<Ulinear Fourier modes have amplitudes that vary substantially in space and time, while the inverse scattering modes are found to be nearly constant. It is therefore suggested that the inverse scattering formulation may be more appropriate than linear Fourier analysis for describing the nonlinear dynamical motions studied experimentally herein. Introducing the concept of ‘‘phase locking’’ among the IST modes, features in the data that are referred to as ‘‘coherent structures’’ are identified. Such structures are found not to be strictly solitons (they constitute multiple cnoidal wave interactions), although they have many of the properties normally identified with solitons, including preservation of their amplitudes after collisions with other waves.

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