Wave radiation and diffraction by a circular cylinder submerged below an ice sheet with a crack are considered based on the linearized velocity potential theory together with multipole expansion. The solution starts from the potential due to a single source, or the Green function satisfying both the ice sheet condition and the crack condition, as well as all other conditions apart from that on the body surface. This is obtained in an integral form through Fourier transform, in contrast to what has been obtained previously in which the Green function is in the series form based on the method of matched eigenfunction expansion in each domain on both sides of the crack. The multipole expansion is then constructed through direct differentiation of the Green function with respect to the source position, rather than treating each multipole as a separate problem. The use of the Green function enables the problem of wave diffraction by the crack in the absence of the body to be solved directly. For the circular cylinder, wave radiation and diffraction problems are solved by applying the body surface boundary condition to the multipole expansion, through which the unknown coefficients are obtained. Extensive results are provided for the added mass and damping coefficient as well as the exciting force. When the cylinder is away from the crack, a wide spacing approximation method is used, which is found to provide accurate results apart from when the cylinder is quite close to the crack.

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Abstract

The monotonic convergence (MC) property of discrete two-dimensional (2-D) systems described by the Roesser model is studied. The MC problem of the 2-D system is firstly converted to two H∞ disturbance attenuation problems of the traditional one-dimensional system. Then, the sufficient condition is derived for the MC, which is given by two linear matrix inequalities (LMIs). Furthermore, it can be shown that either of the LMIs can also guarantee the Bounded-Input Bounded-Output (BIBO) stability of the 2-D system. Finally, a simulation example is given to show the effectiveness of the LMIs condition.