Dali Xu

A HAM-based analytic approach for physical models with an infinite number of singularities

Based on the Homotopy Analysis Method (HAM), an analytic approach is proposed to solve physical models with an infinite number of “singularities”. The nonlinear interaction of double cnoidal waves governed by the Korteweg-de Vries (KdV) equation is used to illustrate its validity. The HAM is an analytic technique for highly nonlinear problems, which is based on the homotopy in topology and thus has nothing to do with small physical parameters. Besides, the HAM provides us great freedom to choose proper equation-type and solution-expression for high-order approximation equations. Especially, unlike other methods, the HAM can guarantee the convergence of solution series. Using the HAM, an infinite number of zero denominators of the considered problem are avoided once for all by properly choosing an auxiliary linear operator, as illustrated in this paper. This HAM-based approach has general meanings and can be used to solve many physical problems with lots of “singularities”. It also suggests that the so-called “singularity” might not exist physically, but only due to the imperfection of used mathematical methods, because the nature should not contain any singularities at all.

On the Discovery of the Steady-State Resonant Water Waves

In 1960 Phillips gave the criterion of wave resonance and showed that the amplitude of a resonant wave component, if it is zero initially, grows linearly with time. In 1962 Benney derived evolution equations of wave-mode amplitudes and demonstrated periodic exchange of wave energy for resonant waves. However, in the past half century, the so-called steady-state resonant waves with time-independent spectrum have never been found for order higher than three, because perturbation results contain secular terms when Phillips’ criterion is satisfied so that “the perturbation theory breaks down due to singularities in the transfer functions”, as pointed out by Madsen and Fuhrman in 2012.

WEC Design Based on Refined Mean Annual Energy Production for the Israeli Mediterranean Coast

Using the Israeli Mediterranean as an example, we address the impact of resource variability and device survivability on the design of floating-body wave-energy converters (WECs). Employing a simplified heaving-cylinder as a prototypical WEC, several device sizes, corresponding to the most frequently encountered and most energetic sea-states in the Israeli Mediterranean, are investigated. Mean-annual energy production is calculated based on the scatter-diagram/power-matrix approach. Subsequently, a measure for significant device motions under irregular sea-states akin to the spectral significant wave-height is developed, and cut-offs to regular operation are explored from the perspective of these significant displacements. The impact of this WEC down-time is captured in a refinement of mean-annual energy production, which consists of supplementing the scatter-diagram/power-matrix calculations by a Boolean displacement matrix. In the Israeli Mediterranean, where most of the annual incident wave power comes in infrequent winter storms, larger WECs outperform smaller WECs by a greater margin when down-time is taken into account. Analogous displacement cut-offs for refining calculations of mean-annual energy production may inform WEC design for other sites.

Mass, momentum, and energy flux conservation between linear and nonlinear steady-state wave groups

This paper provides a mass, momentum, and energy flux conservation analysis between the linear and nonlinear steady-state wave groups. Convergent high-order solutions for nonlinear wave groups with multiple steady-state near resonances in deep water have been obtained by means of the homotopy analysis method. The small divisors associated with nearly resonant components are transformed to singularities that are originally caused by exact resonances by a piecewise auxiliary linear operator. Both two primary components and other nearly resonant ones are considered in the initial guess to search for finite amplitude wave groups. It is found that as nonlinearity of wave groups increases, more wave components appear in the spectrum due to the nearly resonant interactions. The nonlinear wave fields change from the initial bi-chromatic waves that contain only two nontrivial primary components into the steady-state resonant waves that contain both two primary components and other nearly resonant ones. The conserv...

The experimental study on the standing solitary waves

The experimental study on the characteristic of solitons observed by Rajchenbach[1] have been done. The driving frequency, nondimensional acceleration and depth of the fluid are chosen to analyze the characteristics of the wave. The wave height changes with the first two mentioned as the same way as faraday waves. But the depth of fluid has no effect. The heights of the stable, localized wave are similar to each other while the length of these waves seems no discipline with the same parameters.

On the quartet resonance of gravity waves in water of finite depth

The interaction among a quartet of resonant progressive waves in water of finite depth is considered by the Homotopy Analysis Method (HAM). The problem is governed by a linear PDE with a group of nonlinear boundary conditions on the unknown free surface. Convergent multiple steady-state solutions have been gained by means of the HAM. Its worth noting that, in the most cases of the quartet resonance, wave energy is often exchanged periodically among different wave modes. However, our computations indicate for the first time that there exist some cases in which energy exchange disappears and besides the resonant wave component may contain much small percentage of the total wave energy. This work verifies that the HAM is valid even for some rather complicated nonlinear PDEs, and can be used as a powerful tool to understand some nonlinear phenomena.