Standing Gravity Waves in a Horizontal Circular Eccentric Annular Tank

Standing gravity waves in half-full horizontal cylindrical containers with eccentric tube are analyzed using the linear theory of water waves. The problem solution is obtained by the method of conformal coordinate transformation, leading to standard truncated matrix Eigen-value problem from which fluid motion characteristics (Eigen-frequencies and wave modes) are calculated. The effects of tube eccentricity and radius ratio upon the three lowest antisymmetric and symmetric sloshing frequencies and the associated hydrodynamic pressure mode shapes are examined. Also, convergence of the adopted approach with respect to the eccentricity condition, and radius ratio is discussed. Accuracy of the present analysis is checked by comparison with the known results of the limiting cases.

The Transformation Between the Eulerian and Lagrangian Solutions for Irrotational Standing Gravity Waves

This study reports the transformations between the third-order Eulerian and Lagrangian solutions for the standing gravity waves on the uniform depth. Regarding the motion of a marked fluid particle, the instantaneous velocity, the mass conservation and the free surface must be the same for either Eulerian or Lagrangian methods. We impose the assumption that the Lagrangian wave frequency is a function of wave steepness. 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 particle trajectories and the period of particle motion can be derived directly in Lagrangian form. It shows that the given Eulerian solutions are capable of being transformed into the completely unknown Lagrangian solutions and the reversible process is also identified.

Asymptotic forms for jets from standing waves

Standing gravity waves forced beyond the maximum height for perfect periodicity can produce vertical jets with sharp-pointed tips. In this paper, canonical forms for the wave crests are derived which display sharp cusps in the limit as the time t tends to infinity. The theoretical profile is in general quartic in the space coordinates, and can describe the smooth transition of a fairly low wave crest to a cusped form. There is no singularity as the surface slope passes through 45°.