Radiation boundary conditions for finite element solutions of generalized wave equations

On the basis of the dispersion relation of the generalized linear wave equation we derive a radiation boundary condition (RBC) that explicitly incorporates the physical parameters of the governing equation into the form of the boundary condition. Using finite element techniques we investigate the properties of the generalized RBC by examining forced and unforced solutions to the telegraph and Klein-Gordon equations in one dimension. The results show that within the limits of the physical parameters of the problem the generalized RBC is an improvement over the Sommerfeld RBC when the governing equation contains additional terms that influence the propagation. These gains are achieved without introducing any computational overhead. A two-dimensional example suggests that the 1D findings can generalize to higher dimensions.     

1H- und 13C-NMR-spektroskopische Untersuchungen zur Ladungsverteilung in substituierten 1,3-Dioxan-2-ylium-Salzen

The ¹H and ¹³C NMR spectra of a series of 1,3-dioxan-2-ylium salts substituted at C-2 and C-4, C-5, C-6 have been measured. The atomic charge distribution in 1,3-dioxan- and 1,3-dioxolan-2-ylium cations has been calculated with the MINDO/3 method, leading to a linear correlation between π-charge and ¹³C chemical shift at C-2. The strong C-21-proton downfield shift is caused by the anisotropic influence of the mesomeric cation system. The conformations of substituted 1,3-dioxan-2-ylium cations are discussed.