2. Benchmark solution#

2.1. Calculation method used for the reference solution#

The frequencies of the excitation are quite low and at the same time the waveguide is long enough in relation to its lateral dimensions to be limited to plane waves: the phenomenon is then identical at all points on a wave plane, i.e. does not depend on the coordinates describing the points of this plane, i.e. it does not depend on the coordinates describing the points of this plane, for example \(y\) and \(z\).

In this hypothesis, we give the well-known general solution of the acoustic equations for the two quantities pressure \(p\) and acoustic velocity \(v\):

(2.1)#\[ v=f (t-\ frac {x} {c}) +g (t+\ frac {x} {c})\]

And:

(2.2)#\[ p=\ rho c\ left [f (t-\ frac {x} {c}) -g (t+\ frac {x} {c})\ right]\]

The guide is assumed to be closed at the end of abscissa \(L\) on an impedance \({Z}_{L}\); a reflection occurs at this impedance, which gives a return wave \(g\).

At each point of the guide, there is then superposition of the two functions \(f\) and \(g\); by definition even the terminal impedance \({Z}_{L}\) imposes on the abscissa point \(L\), between \(p\) and \(v\), the following relationship:

(2.3)#\[ \ frac {{p} _ {L}}} {{v}} _ {L}} = {Z} _ {L}\]

In the harmonic case \(f\) and \(g\) are written:

\[\]

: label: eq-4

begin {array} {} f (t-frac {x} {c}) =I {e} {x} {c}) =I {e} ^ {iomega (t-frac {x} {c}) =R {e} {c}) =R {e} {c}) =R {e} {c}) =R {e} ^ {iomega (t+frac {x} {c})}end {array})}end {array}

where \(I\) and \(R\) are determined by boundary conditions.

In the calculation of the impedance \(Z=\frac{p}{v}\) at any point \(x\) the variable time this time is eliminated, in accordance with the calculation of the impedances itself and is written:

\[\]

: label: eq-5

Z (x)mathrm {=} {Z} _ {0} _ {0}frac {I {e} ^ {mathrm {-} iomegafrac {x} {c}}mathrm {-} R {-} R {e}} _ {0}}frac {0}} {0}frac {x} {c}} +R {e} ^ {iomegafrac {x} {c}}}}

The terminal impedance becomes:

(2.4)#\[ {Z} _ {L} = {Z} _ {0}\ frac {I {e}} ^ {-i\ omega\ frac {L} {c}} -R {e} ^ {i\ omega\ frac {L} {L} {L} {L} {0}}\ frac {L} {c}} {i\ omega\ frac {c}}} +R {e} ^ {i\ omega\ frac {L} {c}} {i\ omega\ frac {L} {c}}}\]

We call \({Z}_{0}=\rho c\) iterative impedance.

On the fluid border at the entrance to the guide, the incident wave limit condition imposed on \({P}_{i}={P}_{0}{e}^{i\omega t}\) is obtained by writing the following linear relationship at the border:

(2.5)#\[ p-\ rho c {v} _ {n} = {P} _ {i}\]

where \({v}_{n}=\mathrm{v.n}\) is the speed following the unit normal \(n\) outgoing of the fluid.

In addition, a terminal impedance value \({Z}_{L}={Z}_{0}\) is imposed at the output of the guide, which makes it an anechoic end.

The terminal impedance is equal to the iterative impedance \({Z}_{0}\) when \(R=0\), i.e. when there is no return wave; we then have a progressive wave that is pure in the direction of the incident wave, i.e.:

(2.6)#\[\begin{split} \ begin {array} {} v=i {e} ^ {i\ omega (t-\ frac {x} {c})}\\ p=\ rho cI {e} ^ {i\ omega (t-\ frac {x} {c})}\ end {array} ^ {i\ omega (t-\ frac {x} {c})}\ end {array}\end{split}\]

Thus the imposed incident wave relationship () is written:

(2.7)#\[ p-\ rho c {v} _ {n} =p (x=0) +\ rho cv (x=0) =2\ rho cI {e} ^ {i\ omega t} ^ {i\ omega t}\]

from which we identify \(2\rho cI{e}^{i\omega t}={P}_{i}\); we deduce the expression of the progressive wave of pressure in the guide when \({P}_{i}\) is imposed at the entrance of the guide:

(2.8)#\[ p=\ frac {{P} _ {i}}} {2} {e}} {e} ^ {-i\ omega\ frac {x} {c}}} =\ frac {{P} _ {0}} {0}}} {0}}} {2} {e}} {e}} ^ {e}} ^ {i\ omega (t-\ frac {x} {c})}\]

2.2. Benchmark results#

Pressure at points \(A\), \(B\), \(C\), \(D\) (for A, B, C, D, E, and F models).

2.3. Uncertainty about the solution#

Analytical solution.

2.4. Bibliographical references#

    1. STIFKENS « Introduction to the Code_Aster of the incident wave type boundary condition in vibro-acoustics - Report HP-61/95/026/