Of course, I could not leave my colleagues and the rest of the people in the Ships Division and. Papoutsellis, for his help in understanding the variational formulation of the water wave problem, and also N. Although there is a standard and well-understood procedure to achieve this by implementing MSM [(Mei, Stiassnie, and Yue 2005)] , in the case of AVP several issues occur that make its applicability and its relation to other established methods, such as MSM, unclear.
However, in both cases the vertical dependence of the velocity potential is included a priori in the corresponding anzatz and is inspired by the results of other formal perturbation methods. Namely, so far it seems that the AVP is not self-contained and, in order to ensure its consistency with other recognized results, an important part of the solution (i.e. the vertical structure of the potential) needs to be supplemented with “external” meanings. We first apply our modification of the method to the case of weakly nonlinear, uniform wave trains (Stokes waves), where we re-derive the results (Fenton 1985) variationally.
Interestingly, in the context of AVP both definitions (Stokes 1847) of wave velocity occur naturally. Therefore, our approach can be considered as a generalization of the works of Yuen, Lake, and Sedletsky, making Whitham's AVP an autonomous and consistent method for studying periodic or near-periodic waves.
Weakly nonlinear waves in water of intermediate depth
The Water-Wave Problem
- Introduction
- The classical problem of surface gravity waves
- The problem of steady periodic waves
- Some methods for the analysis of nonlinear water waves
The reason for this is its inherent nonlinearity, due to the nonlinear boundary conditions at the upper boundary of the fluid domain, i.e. Extensive information on that era of the WWP can be found in the historical review articles by (Darrigol 2003) and (Craik 2004). However, because of what follows later, we mainly refer to important contributions regarding the variational formulation of the classical WWP.
Based on the previous section, a special case of the classical gravitational WWP is that of stable progressive waves. Furthermore, due to the -periodicity of the waves, the domain is confined within their periodic cell. As a result, the action functional S is limited within the periodic cell of the problem and the change of variables.
Thus, reexploiting the arbitrariness of the admissible variations and the fundamental lemma results in. Because the latter then becomes a function of the (fast) phase and other, slowly varying functions, the average is taken over that phase, under the assumption that the other quantities remain approximately constant during that integration [(Whitham 1974; Jeffrey and Kawahara 1982 )].
A consistent, autonomous approach to Stokes waves, using a variational method
- Introduction
- Derivation of Stokes waves by means of the AVP
A requirement for the implementation of AVP is the existence of a variational formulation of the problem at hand [(Whitham 1974; Karpman 1975)]. Therefore, in that case, the introduction of proper trial functions into the variational principle of the problem is equivalent to the implementation of AVP. However, the presence of 1/2 is desired as it simplifies the form of the integrand, in the case where trigonometric representations are used for the unknown fields.
We consider the problem of determining the functions K{i j}( )z as independent of the total function. In this way the z-dependence of the functions K{ij}, contained in , becomes fully known. Thus, no variations with respect to the velocity can be assumed, as these already existed from the formulation of the stable periodic WWP.
Thus the velocity changes and the reason for that is the presence of the wave induced current. This time the results of the AVP correspond to the results of (Fenton 1985) when Stokes' second approximation for the celerity is used.
The Multiple-Scales Method in weakly nonlinear, narrow-banded wavetrains .… 39
- The effects of weak nonlinearity and narrowbandedness in the classical WWP
- The MSM in the governing equations of the problem
- Derivation of the NLS equation
Alternatively, the kinematic and dynamic boundary conditions of the free surface can be combined in [(Mei, Stiassnie, and Yue 2005)]. The smallness of those fields allows for the Taylor expansion of the velocity potential and the free-surface boundary conditions around z 0 [(Mei, Stiassnie, and Yue 2005)]. This section is devoted to the implementation of the MSM in the problem of weakly nonlinear, narrowband wavelines.
As for the dynamic and combined free surface boundary conditions, they are respectively shaped, at z 0, as. These boundary value problems, independently of the order of , are governed by linear operators. This, and the form of the nonlinear terms forced on the free surface, leads to higher order solutions that.
Similar results are obtained for other orders of the hierarchy, which are expressed through Eqs. We proceed with solving the boundary value problems of Eq. 15a), which happen to be homogeneous. Using, therefore, the above expression on the free surface boundary condition, we get
From the theory of the linearized WWP, and using Eq. 23) can alternatively be written as. Requiring the satisfaction of the above equation, the boundary value problem of Eq. 15b,ii) has a solution, which is the sum of 11 and a particular solution. So, equating the coefficients of the similar functions of z (hyperbolic sine and cosine), we get.
At this point, the consideration of third-order boundary value problems of the perturbation hierarchy remains. At this point, for ease of use, a summary of the results obtained in the previous section should be made. In this spirit, the asymptotic representation of the velocity potential of slowly modulated waves.
The Averaged Variational Principle in weakly nonlinear, narrow-banded wavetrains
- Introduction
- Derivation of the NLS equation
This allowed (Sedletsky, 2016) to derive the DS system of equations in terms of the AVP. Thus arises the apparent need for the a priori knowledge of the velocity potential's vertical dependence. Still, to meet the nature of the problem, we generalize them allowing the slow spatiotemporal modulation of the wave parameters.
Thus the x t, -derivatives of function as slow modulations of the basic values or k and (3). However, the phase difference between the respective terms in and is always the linear case ( / 2). So the use of cos ( ) in the first-order approximation of one field induces the presence of sin ( ) in the other field.
The absence of other similar terms is the result of exploiting the results of Chap. K is explicitly known, it is not possible to proceed with the study of the desired waves. Using the method of undetermined coefficients [(Boyce and DiPrima 2012)], the solution to Eq. where B{ }i are arbitrary, slowly varying functions.
Concretely, one system consists of the EL equations that correspond to the variations. Since the nature of the WWP requires to take positive values, it is obvious from Eq. However, a really important advantage of the method is that the variations with respect to the components that determine the first-order free surface height (fundamental amplitude and phase modulation) correspond to the solvency conditions we impose within MSM [(Whitham 1970 Kurylev 1981)].
Considering our findings, the question of future directions towards the utilization of AVP arises. Note on a Modification of the Nonlinear Schrödinger Equation for Application to Deep Water Waves." Proceedings of the Royal Society of London. On the Existence Theory of Irrotational Water Waves." In Mathematical Proceedings of the Cambridge Philosophical Society, 83:137–57.
On the nonlinear analogy of the WCB method and the method of the averaged Lagrangian." Notes of the POMI Scientific Seminars. Integral Properties of Periodic Finite Amplitude Gravitational Waves.” Proceedings of the Royal Society of London. A variational formulation of the problem of fluid movement in a container of finite dimensions." Journal of Applied Mathematics and Mechanics.
Approximation of the Korteweg-de Vries Equation by the Nonlinear Schrödinger Equation.” Journal of Differential Equations.
Appendices