Vibroacoustic Simulation

4 Fluid Systems

In Chapters 1–3, wave propagation is considered in unbound media, and boundaries occurred only in the context of scattering, reflection, or radiation. Beams or shells were characterized by their boundaries, but their effect was the creation of different forms of wave propagation. For example, the dimension (thickness) of a plate causes the different wave types, bending, in-plane shear, and dilation, but in principle there was always at least one infinite dimension. In Chapter 4 we will approach the system from Chapter 1, meaning that we consider bounded systems, for example a closed cavity with specific input and output positions.

4.1 One-dimensional Systems

A one-dimensional subsystem can be a small tube with radius $R$ small enough that the wavelength is larger than the diameter $2R$ for the highest considered frequency (Pierce, 1991). The tube has rigid surfaces and is filled with a fluid, e.g. air with given density ${\rho }_0$ and speed of sound $c_0$.

The solution of the wave Equation (2.42) in the tube is given by the one-dimensional solution of the wave equation for pressure (2.34) and velocity (2.36)

  (4.1)

  (4.2)

Interaction with the tube fluid takes place at the end and the beginning of the tube. So, we look for a representation similar to a 2DOF system (1.73) ...