2. Benchmark solution#

2.1. Calculation method used for the reference solution#

The general analytical solution for a waveguide is written as:

  • for pressure:

(2.1)#\[ p (x, y, z) =A\ mathrm {exp} (\ mathit {ikx}) +B\ mathrm {exp} (-\ mathit {ikx})\]

  • for the vibratory speed:

(2.2)#\[ V=-\ frac {1} {{\ rho} _ {0} {0} {c}} _ {0}}\ left (A\ mathrm {exp}) (\ mathit {ikx}) -\ mathit {Bexp} (-\ mathit {bexp}} (-\ mathit {ikx})\ left (A\ mathrm {exp}) (\ mathit {ikx}) -\ mathit {bexp} (-\ mathit {bexp}} (-\ mathit {ikx}) (-\ mathit {ikx}) (-\ mathit {ikx})\]

  • A and B are determined by the boundary conditions:

(2.3)#\[ \ text {En} x=0\ to {V} _ {n} = {V} _ {n0}\]
\[\]

: label: eq-4

text {En} x=Lto p (L, y, z) =ZV (L, y, z)cdot {n} _ {L}

This results in:

\[\]

: label: eq-5

A=Bleft (frac {Z- {rho} _ {0} {rho} _ {0}} {Z+ {rho} _ {0}} _ {0}}right)mathrm {exp} (-2mathit {iL})

And:

(2.4)#\[ B=\ frac {{\ rho} _ {0} {c} _ {0} {c} _ {V}}} {\ left (\ frac {Z- {\ rho} _ {0} {c} _ {0} _ {0}} _ {0}} {0}}\ right)\ mathrm {exp} (-2\ mathit {iL}}} {0}}\ right)\ mathrm {exp} (-2\ mathit {iL}}}) -1}\]

In the case studied, the output of the guide is anechoic, \(Z={\rho }_{0}{c}_{0}\) and therefore:

(2.5)#\[ p (x, y, z) = {\ rho} _ {0} {c} _ {0} {V} _ {n}\ mathrm {exp} (-\ mathit {ikx}) V (x, y, z) =- {V} _ {n}\ mathrm {exp} (-\ mathit {ikx}) {e} _ {x}\]

For acoustic intensity:

(2.6)#\[ I=\ frac {1} {2} p {V} ^ {\ text {*}}} =\ frac {1} {2} {\ rho} _ {0} {c} _ {0} {c} _ {0} {V}} _ {V} _ {V} _ {V} _ {V} _ {V} _ {V} _ {V} _ {V} _ {V} _ {V} _ {V} _ {V} _ {V} _ {V} _ {V} _ {V} _ {V} _ {V} _ {V}\]

That is, the active acoustic intensity is uniform throughout the guide and parallel to the axis. The natural frequencies are given for the guide closed at both ends by:

(2.7)#\[ {f} _ {m, n, p} =\ frac {{c} _ {0}} {0}} {0}} {2} {\ left (\ frac {{m} ^ {2}}} {{I} _ {x} ^ {2}}} +\ frac {{p} ^ {2}}} +\ frac {{p} ^ {2}}} +\ frac {{p} ^ {2}}} +\ frac {{p} ^ {2}}} {{I} _ {z} ^ {2}}}\ right)} ^ {\ frac {1} {2}}\]

2.2. Benchmark results#

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

Acoustic intensity at points \(A,B,C,D\) (for models A and C).

Natural frequencies No. 2 to No. 9.

2.3. Uncertainty about the solution#

Analytical solution

2.4. Bibliographical references#

  1. BOUIZI A. Solving linear acoustic equations by a mixed finite element method - Thesis (1989).