On the Irregular Frequencies Appearing In Wave Diffraction-Radiation Solutions

The detrimental effects of irregular frequencies are well known in naval hydrodynamics when using the boundary integral equation (BIE) method to solve the diffraction-radiation problem. These frequencies do not represent any kind of physical resonance but are due to a peculiarity of the BIE method involving the fictitious interior problem which has no unique solution at some eigen-frequencies. The finite element method, for example, and, as we will see, even some type of BIE methods are free of this effect. There are today two methods most often in use to remove the effects of irregular frequencies for a three dimensional body of arbitrary shape. The first one, which can be called the modified BIE method [5] consists in combining the original BIE for the potential with a BIE for normal derivative. Theoretically the new BIE is free of irregular frequencies. However there is some numerical disadvantages introduced by this innatural coupling. The first one is the necessity to evaluate the second order derivatives of the Green function which need a special treatment, and the second one is the choice of the coupling constant on which greatly depends the efficiency of the method. The second method commonly used is the so-called extended BIE method which consists in adding a kind of artificial lid on the fictitious free surface inside the body. This method also eliminates the effects of all irregular frequencies at the cost of modeling the interior free surface, which increases the number of unknowns, and the necessity for special treatement of the logarithmic singularity which appears in the expression for the Green function when two points are close to each other and close to the free surface. However this method seems to be more efficient than the previous one because of its generality and absence of any arbitrary constant in the formulation.