Solutions of the characteristic equations for an immersed plate in a liquid in the cut-off region

The characteristic equations for an immersed plate in a liquid have been solved by many authors using different methods. In the region near cut-off for the higher order modes no information has been published about the roots of these equations and it is generally assumed that they have purely imaginary values, based on an assumption from the free plate case. Our numerical results for the roots of these equations show that in fact they always have complex values. The phase velocity reaches a maximum, but finite, value at cut-off and is a continuous function of frequency for a given plate thickness. The imaginary parts of those roots are so large that their related modes are very highly damped and practically non-propagating. The effect of finite plate loss is included to give more realistic solutions, particularly near the cut-off region. These results are presented and their possible effects on other plate modes are discussed