The numerical wave flume using first order wavemaker theory has been well established and widely used for a long time. But the existing numerical models based on the first order wave-maker theory will lose accuracy as the nonlinear effects enhance. Because of the different propagation velocities of the spurious harmonic waves and the primary waves, the simulated waves with the first order wave-maker theory have an unstable wave profile. In this paper, a numerical wave flume with piston-type wave maker is established. The comparison of the surface elevation using first order and second order wave-maker theory proves that second order wave-maker theory can make stable wave profile in both temporally and spatially. Harmonic analysis is applied to prove the superiority of second order wave-maker theory.
Piston-type wave-maker has been widely used to generate waves in laboratory flume or basin. Havelock (1929), Svendsen (1985) and Dean & Dalrymple (1991) had well established first order wave-maker theory. Flick & Guza (1980) and Ursell et al. (1960) had verified the first order wave-maker theory by experiments in the laoratory (see also Galvin, 1964; Keating & Webber, 1977). Small-amplitude assumption is the basic assumption of first order wave-maker theory. The small-amplitude waves will decompose into a primary wave and spurious superharmonic wave, which will affect the stability of the wave profile (see Gōda and Kikuya, 1964; Multer and Galvin, 1967; Iwagaki and Sakai, 1970), when the motion of the wave-maker is sinusoidal. In early 1847, Stokes found the superharmonic wave by regular wave in terms of a perturbation series using the wave steepness as the small ordering parameter. But the problem of generated nonlinear wave was gave a solution by Fontanet (1961). He found the spurious superharmonic wave by piston-type wave-maker with sinusoidal motion in Lagrangian coordinates and suggested that it can be restrained using wave paddle control signal with an addition component.
Further, Moubayed & Williams (1994) extended second order wave-maker theory from the regular wave to the bichromatic wave. For the irregular waves, second order wave-maker theory has both sum and difference frequencies in the interaction terms. Longuet-Higgins & Stewart (1962, 1964) deduced the subharmonics generated by wave components interaction under the narrow band assumption. Flick & Guza (1980) pointed out that spurious long wave will be generated by a first order bichromatic control signal. Barthel et al. (1983) used the second order difference frequencies wave paddle control signal to restrain spurious long wave and expended the theory to the flap-type wave-maker. Schaffer (1996) derived second order wave-maker theory including sum frequencies and difference frequencies components without the narrow band assumption. The theory was applied to the piston-type and flap-type wave-makers and was verified by experiments. Schaffer & Steenberg (2003) extended the second order wave-maker theory to multidirectional waves.