2. Equations and boundary conditions of the problem#
2.1. Boundary equations and conditions#
The equation to be solved in the frequency domain is the Helmholtz equation [bib2]:
\(p\) is a complex quantity designating sound pressure and \(s\), also complex, represents the source terms of the problem. The parameter \(k\) refers to the wave number of the problem being treated; it can be complex or real, depending on whether the propagation takes place or not in a domain with a certain attenuation modelled through a complex speed, for example if the material is « porous », for example if the material is « porous » [bib6]:
: label: eq-4
k=frac {omega} {c}
with \(c\) designating the speed of sound, which can be complex in the case of propagation in a « porous » or viscoacoustic environment and \(\omega\) is a real in all cases, which refers to the pulsation such as:
: label: eq-5
omega =2pi f
\(f\) is the working frequency of the harmonic problem. In the figure, we represent any confined domain where the Helmholtz equation () and border conditions apply. \(\Omega\) is an open bounded by \({ℝ}^{3}\) with a regular \(\partial \Omega\) border, partitioned into \(\partial {\Omega }_{v}\) and \(\partial {\Omega }_{z}\):
Figure 2.1-1: Configuration of the acoustic problem in a bounded domain
Equation () is to be solved in a limited domain \(\Omega\). The boundary conditions to be taken into account on border \(\partial \Omega\) of domain \(\Omega\) are expressed in their most general form as:
Conventionally, \(\partial /\partial n\) refers to the normal derivative operator. \(\alpha\), \(\beta\) and \(\gamma\) are complex operators, which can be scalars, or integral operators depending on whether the boundary of application of the boundary condition is a local reaction or a non-local reaction (case of fluid-structure interaction).
The current developments in Code_Aster concern only local reaction boundary conditions, for which \(\alpha\), \(\beta\), and \(\gamma\) are scalars; the specifiable cases are as follows:
The case \(\alpha =0\), \(\beta \ne 0\) and \(\gamma \ne 0\) designates a domain border with imposed vibratory speed. Indeed, there is a relationship connecting the complex sound pressure gradient to the complex particle normal vibratory speed. That is to say:
where \({\rho }_{0}\) designates the density of the fluid in question, and \({V}_{n}\) is imposed, the vibratory speed normal to the wall (\({V}_{n}=v\cdot n\) where \(n\) designates the unit vector of the external normal at the border \(\partial \Omega\)). A watertight wall thus verifies the natural condition: \(\frac{\partial p}{\partial n}=0\).
The case \(\alpha \ne 0\), \(\beta \ne 0\) and \(\gamma =0\) concerns a border with imposed acoustic impedance \(Z\). Acoustic impedance \(Z\) is defined as the ratio of pressure to the normal particle vibratory speed in the vicinity of the wall with imposed impedance:
The case \(\alpha \ne 0\), \(\beta =0\) and \(\gamma \ne 0\) represents the case where the sound pressure \(p\) is imposed at a border (most often \(\gamma =0\), corresponding to \(p=0\)).
The case \(\alpha \ne 0\), \(\beta =0\) and \(\gamma \ne 0\) represents the case where the sound pressure \(p\) is imposed at a border (most often \(\gamma =0\), corresponding to \(p=0\)).