Third-order Asymptotic Solution Research Articles

INTERACTION OF NONLINEAR INTERFACIAL WAVE AND SURFACE GRAVITY WAVE IN TWO-LAYER FLUIDS

This paper describes the third-order solution for the nonlinear internal and surface progressive water wave in a two-layer fluid. We use the perturbation method to develop a mathematical derivation. In the previous studies, the second- and third-order solutions derived by Umeyama (2002) and Song (2004) can not satisfy the boundary conditions. The present third-order asymptotic solution which satisfies the governing equation and boundary conditions is obtained. The numerical results demonstrate the influence of the ratio density and thickness of the two fluids on the interfacial and surface profiles as well as the wave frequency. The wave elevations at the free surface and the interface are calculated in different wave conditions under the thicknesses and densities ratio between the upper and lower layers in two-layer fluids. This simple theoretical solution can be used to analyze the different dynamic mechanisms between a interfacial wave induced by the given surface wave (IWSW) and a surface wave induced by the given interfacial wave (SWIW). The effect of on the wave height of SWIW is shown in Fig1. The height of SWIW is directly proportional to . However, the effect of on the wave height of IWSW is opposite. The height of SWIW and are in inverse proportion relation (see Fig. 2).

Lagrangian modelling of fluid sloshing in moving tanks

The paper considers the problem of sloshing of incompressible fluid in a moving 2-D rectangular tank under horizontal and vertical excitation. The problem is solved in Lagrangian variables by applying two approaches. First, a third-order asymptotic solution for resonant sloshing with a dominant mode is derived using a recursive technique. Then, fully nonlinear set of equations in the material coordinates is solved numerically by employing a finite difference method. Both methods are applied to a problem of high amplitude resonant Faraday waves and the obtained results are compared with experimental data known from the literature and a good agreement between the results of the two methods and the empirical data is demonstrated.

Theoretical and experimental study of particle trajectories for nonlinear water waves propagating on a sloping bottom

A third-order asymptotic solution in Lagrangian description for nonlinear water waves propagating over a sloping beach is derived. The particle trajectories are obtained as a function of the nonlinear ordering parameter ε and the bottom slope α to the third order of perturbation. A new relationship between the wave velocity and the motions of particles at the free surface profile in the waves propagating on the sloping bottom is also determined directly in the complete Lagrangian framework. This solution enables the description of wave shoaling in the direction of wave propagation from deep to shallow water, as well as the successive deformation of wave profiles and water particle trajectories prior to breaking. A series of experiments are conducted to investigate the particle trajectories of nonlinear water waves propagating over a sloping bottom. It is shown that the present third-order asymptotic solution agrees very well with the experiments.

A Lagrangian asymptotic solution for finite-amplitude standing waves

A third-order asymptotic solution in a Lagrangian description is presented for the finite-amplitude standing waves in water of uniform depth. Regarding the frequency of particle motion to be a function of the wave steepness and using a successive Taylor series expansion to the path and the period of particle motion, the explicit parametric equation of water particles and the Lagrangian wave frequency up to third-order could be obtained. In particular, the Lagrangian mean level which differs from that in the Eulerian approach is also found as a part of the solutions. The variations in the wave profile and the water particle orbits for the nonlinear standing waves are also investigated. Comparison on the third-order wave profiles given by the Eulerian and Lagrangian solution with the experiments reveals that the latter is more accurate than the former in describing the shape of the wave profile.

A modified Euler–Lagrange transformation for particle orbits in nonlinear progressive waves

This paper presents a modified Euler–Lagrange transformation method to obtain the third-order trajectory solution in a Lagrangian form for the water particles in nonlinear water waves. We impose the assumption that the Lagrangian wave frequency is a function of wave steepness and an arbitrary vertical position for each water particle. Expanding the unknown function in a small perturbation parameter and using a successive expansion in a Taylor series for the water particle path and the period of a particle motion, the third-order asymptotic expressions for the Lagrangian particle trajectories, the mass transport velocity and the period of particle motion can be derived directly in Lagrangian form. The wave frequency and mean level of the particle motion in Lagrangian form differ from those of the Eulerian. Finally, the third-order asymptotic solution obtained is uniformly valid in contrast with early works containing resonant terms presented by Wiegel [1964. Oceanographical Engineering. Prentice-Hall, New Jersey, pp. 37–40] (Eqs. (B.1) and (B.2) in Appendix B) or Chen et al.[2006. Theoretical analysis of surface waves shoaling and breaking on a sloping bottom. Part 2 nonlinear waves. Wave motion, 43, 356–369] based on a straightforward expansion for two-dimensional progressive waves.

A third-order asymptotic solution of nonlinear standing water waves in Lagrangian coordinates

Asymptotic solutions up to third-order which describe irrotational finite amplitude standing waves are derived in Lagrangian coordinates. The analytical Lagrangian solution that is uniformly valid for large times satisfies the irrotational condition and the pressure p = 0 at the free surface, which is in contrast with the Eulerian solution existing under a residual pressure at the free surface due to Taylor's series expansion. In the third-order Lagrangian approximation, the explicit parametric equation and the Lagrangian wave frequency of water particles could be obtained. In particular, the Lagrangian mean level of a particle motion that is a function of vertical label is found as a part of the solution which is different from that in an Eulerian description. The dynamic properties of nonlinear standing waves in water of a finite depth, including particle trajectory, surface profile and wave pressure are investigated. It is also shown that the Lagrangian solution is superior to an Eulerian solution of the same order for describing the wave shape and the kinematics above the mean water level.

Head-on collision between two solitary waves in a compressible Mooney–Rivlin elastic rod

The interaction between two solitary waves in hyperelastic rods had been studied in literature by numerical methods, and here we study the head-on collision between two solitary waves in a circular cylindrical rod composed of a compressible Mooney–Rivlin material by a perturbation approach which combines the reductive perturbation method with the technique of strained coordinates. The third-order asymptotic solution which describes the evolution process of interaction is derived. It is found that the head-on collision does have imprints on the colliding waves with nonuniform phase shifts at O( ε 2), which cause the tilting of the wave profiles. Our analytical results provide a formula for the maximum amplitude during the collision and also successfully explain the phenomenon that the smaller solitary wave has a larger distortion while the larger solitary wave has a smaller distortion.

A Two-Dimensional Dirichlet Problem with an Exponential Nonlinearity

We consider the two-dimensional nonlinear Dirichlet problem \[ \begin{gathered} - \Delta u = \lambda \left( {e^u + \psi e^{ - u} } \right),\quad y \in \Omega , \hfill \\ u = \phi ,\quad y \in \partial \Omega , \hfill \\ \end{gathered} \] where $y = (y_1 ,y_2 )$, $\Delta $ is the Laplacian operator, $\Omega $ is a simply connected region bounded by a smooth closed Jordan curve, the boundary data is continuous, $\psi $ is analytic in $\Omega $ and continuous on $\bar \Omega $, and $\lambda $ is positive. Our concern is with obtaining a large norm (second) solution for $\lambda $ tending to $0_ + $. This is accomplished by obtaining a third-order asymptotic solution which is used as a first approximation for a modified Newton’s method. Additionally, we obtain three solutions as $\lambda \to 0^ + $ if $\psi $ is a negative constant.

Uniform Asymptotic Technique for Analyzing Wave Propagation in Inhomogeneous Slab Waveguides

The guided modes of inhomogeneous dielectric slab waveguides are analyzed by a uniform asymptotic technique based on the related equation method. This technique gives highly accurate solutions in the sense of asymptotic expansion. The algorithm for calculating the guided modes of slab waveguides with an even polynomial refractive-index medium is presented. As an example, we calculate the third-order approximate solutions for the guided modes in an analytic form. The results show that the WKB solutions for higher order modes are more accurate than for the lower order modes and the correction to the WKB solutions is significant for the lower order modes. The numerical result for eigenvalues and modal fields confirms that the third-order asymptotic solution is accurate for all the guided modes of the near-parabolic profile waveguides and for higher order modes in the case of the quasi-Gaussian profile.

Wave Propagation in Inhomogenous Slab Waveguides Embedded in Homogeneous Media

Wave propagation in inhomogeneous slab waveguides embedded in homogeneous media is analyzed by rising the uniform asymptotic technique. This technique accurately evaluates the effect of the refractive-index profiles with various core and cladding structures on the guided modes. We calculate the guided modes of waveguides with asymmetric claddings in the eases of a near-parabolic profile core and a quasi-Gaussian profile core. The results show that the third-order asymptotic solution is accurate for all the guided modes in the case of the near-parabolic profile core and for modes far from cutoff in the quasi-Gaussian core case. The dispersion relation indicates that modes guided in strongly asymmetric profiles have almost the same propagation constants as odd-order modes of propagation in the symmetric structure.