Ljf Lambert Broer

Approximate equations for long water waves

In the first part of this paper the Hamiltonian theory of water waves is used to obtain some equations in local coordinates. These equations are approximations of the Boussinesq type. They are stable with respect to short wave perturbations, e.g. rounding off errors in digital computing. In the second part the relation of Boussinesq equations to Korteweg-de Vries and Benjamin-Bona-Mahony equations is investigated.

On the hamiltonian theory of surface waves

It is shown that the classical theory of gravity driven waves on the surface of a non-viscous liquid can be derived from a set of canonical equations. Various approximate equations then can be found by introducing suitable approximations to the kinetic and potential energy functionals. The stability of these approximate equations then can be insured beforehand by using positive definite approximate energy functionals. For fairly long, fairly low waves a stable equation of Boussinesq type is derived in this way. This equation is also valid for waves which are not approximately simple.

STABLE MODEL EQUATIONS FOR LONG WATER WAVES

In this paper, a sequel to two others [1, 2], some extensions and improvements of this earlier work are presented. Among these are: A more precise version of the proof of the basic canonical theorem, some considerations on conservation laws and their relation, a more complete treatment of the stability of the models, especially with respect to the wave amplitude, a short treatment of the Lagrangian version of the theory, a stable discrete model which might be useful for numerical experiments and an extension of the method to the case of slowly varying water depth.

Conversion from material to local coordinates as a canonical transformation

It is shown that the conversion from material to local coordinates in continuum mechanics can be considered as a restricted canonical transformation. As a simple example the longitudinal motion of an elastic bar is discussed.

Canonical transformations and generating functionals

It is shown that canonical transformations for field variables in hamiltonian partial differential equations can be obtained from generating functionals in the same way as classical canonical transformations from generating functions. A simple proof of the relation between infinitesimal invariant transformations and constants of the motion is obtained. The formalism is extended to cover finite and nonlocal transformations of the spatial variables.

Longitudinal motion of an elastic bar

SummaryThe exact equations of motion of an elastic bar are discussed, both in material and local coordinates. It is shown that for an ideal elastic material the former, but not the latter, are linear. An infinite number of conservation laws is shown to exist.

On some solutions of the wave equation

Methods of solution for the wave equation in terms of series of integrals are simplified by means of an operator formalism. In this way the equivalence of series solutions given by Bremmer and Broer is easily proved. A new solution, valid for a large gradient of the refractive index, is given. The splitting of the solution in two partial waves travelling in opposite directions is discussed in connection with the expressions for energy density and intensity.

Hidden hamiltonians of first-order equations

There are equations, like the KDV equation, of which the solutions behave like conservative systems although the equation is of first order in time. It is shown how equations of this kind can originate by a direct-product like process of fusion of two canonical conjugate variables. Conversely, for a class of dynamically well-behaved first-order equations a splitting of the independent variable into two conjugate parts and a corresponding hamiltonian functional can be found. It is shown how the action principle and the Noether theorem transform during this fusion or splitting process. A number of examples are discussed. It is shown how a KDV approximation can be derived directly from the hamiltonian of a second-order system without using the second-order wave equations.

Some comments on linear wave equations

Some aspects of linear wave equations are considered. For each wave mode of a system an asymptotic expansion of the angular frequency ω in terms of the wave number k has been derived. A set of balance equations has been obtained and finally the stability of linear waves has been investigated by means of the usual harmonic analysis and by Lyapunovs second method. The two methods are compared.

Potential waves in one dimension

Some properties of one-dimensional potential waves are discussed. In particular it is shown that wave equations of this kind admit an infinity of local conservation laws with densities and fluxes depending on the fields only. This set of conserved quantities has a simple structure. In some cases this structure can be made explicite by means of a principle of conserved flux. A close relation between this set of conserved quantities and solutions of the hodograph equations has been found.