The generalized channel Boussinesq (gcB) two-equation model and the forced channel Kortewegae Vries (cKdV) one-equation model previously derived by the authors are further analysed and discussed in the present study. The gcB model describes the propagation and generation of weakly nonlinear, weakly dispersive and weakly forced long water waves in channels of arbitrary shape that may vary both in space and time, and the cKdV model is applicable to unidirectional motions of such waves, which may be sustained under forcing at resonance of the system. These two models are long-wave approximations of a hierarchy set of section-mean conservation equations of mass, momentum and energy, which are exact for inviscid fluids.Results of these models are demonstrated with four specific channel shapes, namely variable rectangular, triangular, parabolic and semicircular sections, in which case solutions are obtained in closed form. In particular, for uniform channels of equal mean water depth, different cross-sectional shapes have a leading-order effect only on the variations of a K-factor of the coefficient of the term bearing the dispersive effects in the model equations. For this case, the uniform-channel analogy theorem enunciated here shows that long waves of equal (mean) height in different uniform channels of equal mean depth but distinct K-shape factors will propagate with equal velocity and with their effective wavelengths appearing K times of that in the rectangular channel, for which K = 1. It also shows that the further channel shape departs from the rectangular, the greater the value of K . Based on this observation, the solitary and cnoidal waves in a K-shaped channel are compared with experiments on wave profiles and wave velocities. Finally, some three-dimensional features of these solitary waves are presented for a triangular channel.

Propagation of solitary waves in curved shallow water channels of constant depth and width is investigated by carrying out numerical simulations based on the generalized weakly nonlinear and weakly dispersive Boussinesq model. The objective is to investigate the effects of channel width and bending sharpness on the transmission and reflection of long waves propagating through significantly curved channels. Our numerical results show that, when travelling through narrow channel bends including both smooth and sharp-cornered 90°-bends, a solitary wave is transmitted almost completely with little reflection and scattering. For wide channel bends, we find that, if the bend is rounded and smooth, a solitary wave is still fully transmitted with little backward reflection, but the transmitted wave will no longer preserve the shape of the original solitary wave but will disintegrate into several smaller waves. For solitary waves travelling through wide sharp-cornered 90°-bends, wave reflection is seen to be very significant, and the wider the channel bend, the stronger the reflected wave amplitude. Our numerical results for waves in sharp-cornered 90°-bends revealed a similarity relationship which indicates that the ratios of the transmitted and reflected wave amplitude, excess mass and energy to the original wave amplitude, mass and energy all depend on one single dimensionless parameter, namely the ratio of the channel width b to the effective wavelength λe. Quantitative results for predicting wave transmission and reflection based on b/λe are presented.

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ObjectivePrevious genome-wide association studies have indicated an association between CDH13 genotypes and adiponectin levels. In this study, we used mediation analysis to assess the statistical association between CDH13 locus variants and adiponectin levels, metabolic syndrome, and related metabolic phenotypes.Methods and resultsA sample population of 530 Taiwanese participants was enrolled. Four CDH13 gene variants in the promoter and intron 1 regions were genotyped. After adjustment for clinical covariates, the CDH13 genotypes/haplotypes exhibited an association with the adiponectin levels (lowest P = 1.95 × 10−11 for rs4783244 and lowest P = 3.78 × 10−13 for haplotype ATTT). Significant correlations were observed between the adiponectin levels and the various metabolic syndrome-related phenotypes (all P ≤ 0.005). After further adjustment for the adiponectin levels, participants with a minor allele of rs12051272 revealed a considerable association with a more favorable metabolic profile, including higher insulin sensitivity, high-density lipoprotein cholesterol levels, lower diastolic blood pressure, circulating levels of fasting plasma glucose, and triglycerides, and as a lower risk of metabolic syndrome (all P < 0.05). The mediation analysis further revealed a suppression effect of the adiponectin levels on the association between CDH13 genotypes and metabolic syndrome and its related phenotypes (Sobel test; all P < 0.001).ConclusionThe genetic polymorphisms at the CDH13 locus independently affect the adiponectin levels, whereas the adiponectin levels exhibit a suppressive effect on the association between CDH13 locus variants and various metabolic phenotypes and metabolic syndrome. In addition, these results provide further evidence of the association between the CDH13 gene variants and the risks of metabolic syndrome and atherosclerotic cardiovascular disease.