We develop modulation theory for undular bores (dispersive shock waves) in the framework of the Gardner, or extended Korteweg--de Vries, equation, which is a generic mathematical model for weakly nonlinear and weakly dispersive wave propagation, when effects of higher order nonlinearity become important. Using a reduced version of the finite-gap integration method we derive the Gardner-Whitham modulation system in a Riemann invariant form and show that it can be mapped onto the well-known modulation system for the Korteweg--de Vries equation. The transformation between the two counterpart modulation systems is, however, not invertible. As a result, the study of the resolution of an initial discontinuity for the Gardner equation reveals a rich phenomenology of solutions which, along with the KdV type simple undular bores, include nonlinear trigonometric bores, solibores, rarefaction waves and composite solutions representing various combinations of the above structures. We construct full parametric maps of such solutions for both signs of the cubic nonlinear term in the Gardner equation. Our classification is supported by numerical simulations.Comment: 25 pages, 24 figures, slightly revised and corrected versio

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Transcritical flow of a stratified fluid past a broad localised topographic obstacle is studied analytically in the framework of the forced extended Korteweg-de Vries (eKdV), or Gardner, equation. We consider both possible signs for the cubic nonlinear term in the Gardner equation corresponding to different fluid density stratification profiles. We identify the range of the input parameters: the oncoming flow speed (the Froude number) and the topographic amplitude, for which the obstacle supports a stationary localised hydraulic transition from the subcritical flow upstream to the supercritical flow downstream. Such a localised transcritical flow is resolved back into the equilibrium flow state away from the obstacle with the aid of unsteady coherent nonlinear wave structures propagating upstream and downstream. Along with the regular, cnoidal undular bores occurring in the analogous problem for the single-layer flow modeled by the forced KdV equation, the transcritical internal wave flows support a diverse family of upstream and downstream wave structures, including solibores, rarefaction waves, reversed and trigonometric undular bores, which we describe using the recent development of the nonlinear modulation theory for the (unforced) Gardner equation. The predictions of the developed analytic construction are confirmed by direct numerical simulations of the forced Gardner equation for a broad range of input parameters.

Unsaturated polyester filled with coloring agent is commonly used as the surface material of a GFRP yacht and is called a gel-coating layer. The reflection on the gel-coating layer surface will be imperfect if twists and wrinkles exist on the gel-coating surface. This phenomenon is called print-through phenomenon (PTP) in this investigation. The PTP seriously reduces the beauty of a yacht and therefore limits the application of GFRP to the yacht. Therefore, it is urgent to solve the PTP problem. The first goal of this study is to objectively define and quantitatively measure the existence of PTP and its level. The surface of the gel-coating layer of GFRP is scanned by using the portable high resolution surface roughness and form measurement instrument. Based on a large number of experiments, the average parameter, arithmetic mean deviation (R a ), and the peak parameter, the altitude of the surface profile (R t ), are proposed to determine the existence of PTP and its level. The second goal of this study is to consider the causes of PTP happening and find out the factors that can influence it. Through experimental observations and qualitative analysis, it is believed that PTP is related to the non-uniform residual stress in the gel-coating layer of GFRP. So, any factors which can produce the non-uniform residual stress in the gel-coating layer can influence PTP, such as the shrinkage of the resin during its hardening chemical reaction and the atmospheric pressure during the Seemann composite resin infusion molding process (SCRIMP). After ascertaining what causes PTP, research methods that can be used to reduce PTP will then be studied.

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