Kinematics in Regular and Irregular Waves based on a Lagrangian Formulation

Kinematics in two-dimensional regular and irregular swface waves is described based on the Lagrangian form of the equations of motion, with particular emphasis 011 the conditions in the so-called splash zone in irregular waves. A practical method for accurate calculation of kinematics in broad-banded irregular waves is developed based on Gerstner's wave theory, and theoretical calculations are compared with laborat01:y wave data. A review of basic hydrodynamics has also been called for, and is presented from a Lagrangian as well as Eulerian point of view. The results of the analytical study and the study of the wave data question the applicability of certain universally accepted fluid dynamical principles. The basic equations of fluid motion are presented on Eulerian and Lagrangian form, including the general Lagrangian form of the Laplacian. The relations governing vortex motion are also presented, including the theorems of Helmholtz, Kelvin and others on the rate of change of vorticity and circulation. Rotation of fluid elements is also studied from a Lagrangian point of view, showing that vorticity is not suited to express how a fluid element actually rotates about itself. It is found reason to question the common Lagrangian form of the continuity equation, namely that the Jacobian must be constant and that it in general can be set equal to 1, since this requirement results in some ambiguities and fundamental inconsistencies. Further, when considered in a Lagrangian frame of reference, we have that the theorems of Helmholtz, Kelvin and others require that a given Lagrangian point always represents the same identifiable material "particle", i.e. that the Jacobian is constant and equals 1. Hence, there is also reason to question the common assumption of irrotational (potential) flow in motions generated by conservative (potential) forces only, since this requirement is based on a material (Lagrangian) interpretation of the theorems on vortex motion. The weaker requirement of zero curl of the acceleration in such flows still applies, irrespective of the behaviour of the Jacobian. The Lagrangian wave theories of Gerstner and Miehe are presented, pertammg to regular waves in deep and intermediate water, respectively. These wave theories represent closed orbital particle motion, i.e. without any net transport of mass. They also contain vorticity (rotationality) at second order, and are therefore traditionally considered invalid beyond first order. The classical solution for surface waves is Stokes 2d order wave theory. The difference between this theory and the two above is Stokes drift; a second order forward transport of mass. Stokes waves and Stokes drift are here also studied from a Lagrangian point of view. It is found that Stokes waves violate continuity and cause a vorticity at second order within less than one wave period, even for waves of small amplitude. Stokes waves are therefore theoretically inconsistent in the Lagrangian frame of reference, which in turn questions the arguments rendering Gerstner's (and Miche's) theory invalid, i.e. the above-mentioned assumption of irrotational motion. The wave theories of Gerstner and Miehe are concluded to be applicable basic solutions for regular waves in the limit of negligible viscosity. Irregular waves are here modelled as a sum of linear regular Gerstner or Miehe waves, superposed in the Lagrangian frame of reference. The Lagrangian approach is better suited to show the physics of the wave motions than the Eulerian approach, and the linear Lagrangian model of irregular waves automatically includes what are known as nonlinear interactions