Prediction of diffuse sound transmission through finite-sized periodic structures

The diffuse sound transmission through a wall or floor can be efficiently computed with a hybrid approach. The wall is then modelled deterministically as finite size effects and modal behaviour can be important in the considered frequency range, while the rooms carry a diffuse field and are modelled as stochastic subsystems. The finite element method is usually employed to compute the modal behaviour of the wall. At higher frequencies, the computational cost then increases significantly as a fine mesh is required due to the short wavelength of the structural deformation. For this reason, an alternative approach was recently developed for finite-sized thick and layered walls, which allows to replace the finite element model by an analytical model. However, the application of this approach is limited to layered structures such as sandwich panels or double glazing. In the present work, the extension towards more complicated, finite-sized building elements which exhibit spatial periodicity, is considered by invoking periodic structure theory. The propagating waves resulting from the free wave propagation analysis of the periodic unit cell, are combined into standing waves, which satisfy the simply supported boundary conditions. A Fourier transform of the system of equations allows for a fast conversion of the mode shapes of the unit cell to the entire finite-sized structure. The methodology is illustrated by comparing the predicted sound insulation for two cross laminated timber panels and two periodic rib-stiffened panels with alternative predictions involving a full finite element model of the entire structure, and experimental data.