Natural vibration of two-dimensional slender structure-water interaction systems subject to Sommerfeld radiation condition

The dynamic behaviour of 2-dimensional flexible slender structure-water interaction systems subject to a Sommerfeld radiation condition at the infinity boundary of the water domain is investigated. A new parameter, the speed of radiation wave, is introduced into the Sommerfeld radiation condition to consider the influences of both of the pressure wave and the free surface wave of the water, which is an extension of the original Sommerfeld condition. The governing equations describing the dynamic behaviour of the system are analysed and solved using a separation of variables method. It is demonstrated that the natural vibration of the 2-dimensional slender structure-water interaction system subject to a Sommerfeld radiation condition is governed by a complex eigenvalue equation which has only pairs of complex conjugate eigenvalues. The number of the pairs of complex conjugate natural frequencies equals the number of the natural modes of the corresponding dry structure and is independent of the continuous fluid domain which has infinite degrees of freedom. The examples, including four cases of shallow water, deep water, no free surface wave and incompressible water, demonstrate and illustrate the developed theoretical and numerical method