Through the eighties the theory for second order irregular wave generation was developed within the framework of Stokes wave theory This pioneering work, however, is not fully consistent Furthermore, due to the extensive algebra involved, the derived transfer functions appear in an unnecessarily complicated form The present paper develops the full second order wavemaker theory (including superharmonics as well as subharmonics) valid for a variety of different types of wave board motion In addition to the well known transfer functions some new terms evolve These axe related to the first order local disturbances (evanescent modes and accordingly they are significant when the wave board motion makes a poor fit to the velocity profile of the desired progressive wave component This is typically the case for the high-frequency part of a primary wave spectrum when using a piston type wavemaker The transfer functions are given in a relatively simple form by which the computational effort is reduced substantially This enhances the practical computation of second order wavemaker control signals for irregular waves, and no narrow band assumption is needed The software is conveniently included in a PC-based wave generation system - the DHI Wave Synthesizer The validity of the theory is analysed in a number of laboratory wave tests, covering the superharmonic generation for regular waves


First order wavemaker theory corresponding to linearized Stokes theory has long been well established (Havelock, 1929, Biesel, 1951, Ursell et al, 1960, and others, cf the review by Svendsen, 1985) and we shall devote this introduction to second order theories of wave generation]

The first step towards the development of wavemaker theory is of cause the knowledge of the underlying wave theory Already in 1847 Stokes gave results for regular waves in terms of a perturbation series using the wave steepness as the small ordering parameter For regular waves only the sum frequencies appear (since the difference frequencies vanish) and Stokes found the resulting superharmonics

Presumably the first approach to second order wavemaker theory was given by Fontanet (1961) for regular waves Using a Lagrangian description he found the spurious superharmonics generated by a purely sinusoidal oscillation of the wave board and gave directions as how to suppress these by adding a superharmonic component to the wavemaker control signal

Recently Hudspeth and Sulisz (1991) derived the complete second order lagrangian theory for regular waves with special emphasis on Stokes drift and return flow in wave flumes The theories of Fontanet (1961) and Hudspeth and Sulisz (1991) appear to be the most complete theoretical developments yet

Madsen (1971) developed an approximate theory for the suppression of spurious superharmomcs in regular waves generated in fairly shallow water

Buhr Hansen et al b975) chose an empirical approach to pursue the second order control signal for regular waves The second order regular wave field generated by a first order control signal has further been studied by Flick and Guza (1980)

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