This page is a complementary material to the paper "Some comments on the expressions of the end corrections of ducts and perforations for linear acoustics and large wavelengths" authored by Luc Jaouen & Fabien Chevillotte and published in Acta Acustica United with Acustica in March 2018 (Vol. 104(2) pp. 243 – 250).
section: Home | Basics of acoustics

Some comments on the expressions of the end corrections of ducts and perforations for linear acoustics and large wavelengths

In this page we derive the expressions of the length corrections, as presented by Ingard [Ing53b], for a rectangular perforation in a rectangular pattern and a circular perforation in a rectangular pattern.

The perforation schemes and the nomenclature used are the ones reported in figures below.

[Ing53b] U. Ingard, On the Theory and Design of Acoustic Resonators, J. Acoust. Soc. Am. 25, 1953, 1037-1061 DOI

The length corrections derived here, $\varepsilon$, are determined by calculating the impedances at $z=0$ and at low pulsation $\omega$, seen by a wave propagating towards $z>0$. This impedances writes (see e.g. [Ray45],[Ing53b]):

\[ Z_{B} = \bar{p} / U_0 = \phi Z_0 - j\omega \rho_0 \varepsilon \]

$\bar{p}$ is the mean value of the pressure over the perforation surface area while $U_0$ is, following J. W. Strutt's first assumption, the uniform velocity amplitude of the particle velocity over the perforation surface area, i.e. the particle motion is assumed to be a piston-like motion with $\varepsilon_{0}$ being given by: $\varepsilon_{0}^{\textrm{constant},\ h>r} = 8/(3 \pi^{3/2}) \sqrt{A_0}$}. $Z_0$ is the characteristic impedance of the air: $Z_0=\rho_0 c_0$ where $\rho_0$ is the mass density of air at rest ($\sim$ 1.2 kg.m$^{-3}$) and $c_0$ is the speed of sound in the air ($\sim$ 340 m.s$^{-1}$).

[Ray45] J. W. Strutt (3rd lord Rayleigh), The theory of sound, volume 2 (second edition), Macmillan & Co. Ltd. (New York), 1896 (Appendix A, pp. 487-491).

Rectangular perforation in a rectangular pattern

Due to perforation periodicity, the velocity components in the same planes as the interfaces between two patterns should be equal to zero. Thus, a wave propagating towards $z>0$ appears to propagates in a rectangular waveguide with the dimensions of the pattern: $2a \times 2b$.

The expression of the pressure field in the Representative Elementary Volume (REV) for $z>0$ is given by:

\[ p(x,y,z)=\sum_{m}\sum_{n}{P_{mn} \cos\left(\frac{\pi m x}{a}\right) \cos\left(\frac{\pi n y}{b}\right) e^{-jk_{mn}z}} \]

where the expression of the wavenumber magnitude for mode ($m,n$) is:

\[ k_{mn}=\sqrt{k^2-k_x^2-k_y^2} \]

with

\[ k=\omega/c_0 , \ k_x=\frac{m\pi}{a}\ \textrm{and}\ k_y=\frac{n\pi}{b} \]

The expression for the z-component of the particle velocity, $u_z$, is deduced from Euler's equation:

\[ u_z(x,y,z)=\sum_{m}\sum_{n}{U_{mn} \cos\left(\frac{\pi m x}{a}\right) \cos\left(\frac{\pi n y}{b}\right) e^{-jk_{mn}z}} \]

where

\[ U_{mn}=\frac{k_{mn} P_{mn}}{\omega\rho_0} \]

Considering a uniform velocity amplitude of the particle velocity over the perforation (i.e. piston-like motion), $U_0$, the normal velocity at $z=0$ is written:

\[ u_z\left(x,y,0\right)=d\left(x,y\right) U_0=\sum_{m,n}U_{mn} \cos\left(\frac{\pi m x}{a}\right) \cos\left(\frac{\pi n y}{b}\right) \]

where $d(x,y)$ is the distribution function equal to 1 for $-a_1\leq x\leq a_1$ and $-b_1\leq y\leq b_1$, while $d(x,y)=0$ everywhere else.

To express $U_{mn}$, the last equation is multiplied by $\cos(\pi m x/a) \cos(\pi n y/b)$ and integrated over the cross-section of the REV:

\[ \begin{multline*} \int^{a}_{-a}\int^{b}_{-b}d(x,y) U_0 \cos\left(\frac{\pi m x}{a}\right) \cos\left(\frac{\pi n y}{b}\right)dxdy=\\ \int^{a}_{-a}\int^{b}_{-b} \sum_{m',n'}U_{m'n'} \cos\left(\frac{\pi m' x}{a}\right) \cos\left(\frac{\pi n' y}{b}\right)\cos\left(\frac{\pi m x}{a}\right) \cos\left(\frac{\pi n y}{b}\right)dxdy \label{eq.u2} \end{multline*} \]

Using the orthogonality property of modes $\cos(\pi m x/a) \cos(\pi n y/b)$, one writes:

\[ \begin{multline} \int^{a_1}_{-a_1}\int^{b_1}_{-b_1} U_0 \cos\left(\frac{\pi m x}{a}\right) \cos\left(\frac{\pi n y}{b}\right)dxdy= \int^{a}_{-a}\int^{b}_{-b} U_{mn} \cos^2\left(\frac{\pi m x}{a}\right) \cos^2\left(\frac{\pi n y}{b}\right)dxdy \end{multline} \] \[ 4 U_0 \frac{ab}{\pi^2mn} \sin\left(\frac{\pi m a_1}{a}\right) \sin\left(\frac{\pi n b_1}{b}\right) = U_{mn} \frac{ab}{\nu_{mn}} \]

where $\nu_{00}=1/4$, $\nu_{0n}=\nu_{m0}=1/2$ and $\nu_{mn}=1$ when $m\neq0$ and $n\neq0$.

$U_{mn}$ now writes:

\[ U_{mn}=4 \nu_{mn} U_0 \frac{\sin\left(\pi m \xi\right) \sin\left(\pi n \eta\right)}{\pi^2mn} \]

where two dimensionless shape ratios have been introduced: $\xi=a_1/a$ and $\eta=b_1/b$ (the product $\xi\eta$ is thus equal to the perforation rate $\phi$).

From what's above, $P_{mn}$ writes:

\[ P_{mn}=\frac{4 \nu_{mn} \omega\rho U_0}{k_{mn}} \frac{\sin(\pi m \xi) \sin(\pi n \eta)}{\pi^2 m n} \]

The mean value for the pressure over the perforation surface area, $\bar{p}$, is obtained integrating the equation of the pressure over this perforation:

\[ \bar{p}=\frac{1}{4a_1b_1}\int^{a_1}_{-a_1}\int^{b_1}_{-b_1}\sum_{mn}P_{mn}\cos\left(\frac{\pi m x}{a}\right) \cos\left(\frac{\pi n y}{b}\right)dxdy \] \[ \bar{p}=\frac{1}{4a_1b_1}\sum_{mn}\frac{16 \nu_{mn} \omega\rho_0 U_0}{k_{mn}} \xi \eta a_1 b_1 \left[\displaystyle\frac{\sin(\pi m \xi)}{\pi m \xi} \frac{\sin(\pi n \eta)}{\pi n \eta}\right]^2 \] \[ \bar{p}=\sum_{mn}\frac{4\nu_{mn} \omega\rho_0 U_0}{k_{mn}} G_{mn} \]

with

\[ G_{mn}= \Big[\frac{\sin(\pi m \xi)}{\pi m \xi} \frac{\sin(\pi n \eta)}{\pi n \eta}\Big]^2 (\xi\eta) \]

At low frequencies, $\omega \ll \omega_{mn}$, $k_{mn}$ values can be approximated with:

\[ k_{mn}\approx\sqrt{-\left(k_x^2+k_y^2\right)}\approx j\sqrt{\left(\frac{m\pi}{a}\right)^2+\left(\frac{n\pi}{b}\right)^2} \]

The expression of the end correction $\varepsilon$ is obtained from expression of the impedance $Z_B$.

\[ \varepsilon_{\Box\subset\blacksquare} =\frac{4}{\pi}\sum_{mn^*}\frac{\nu_{mn}G_{mn}}{\sqrt{\left(\displaystyle\frac{m}{a}\right)^2+\left(\displaystyle\frac{n}{b}\right)^2}} \]

$\sum_{mn^*}$ indicates a sum over $m$ and $n$ where the mode ($m=0$, $n=0$) is not accounted for (this mode contributes to the real part of the impedance $Z_{B}$).

The periodicity of the perforations is further discussed in the paper.

Circular perforation in a rectangular pattern

The derivation presented in the section section is now applied for a circular perforation in a rectangular pattern. For this second configuration, the normal velocity at $z = 0$ writes:

\[ u_z(x,y,0)=d\left(x,y\right) U_0 = \sum_{m,n}U_{mn} \cos\left(\frac{\pi m x}{a}\right) \cos\left(\frac{\pi n y}{b}\right) \label{eq.ur1} \]

where the distribution function $d(x,y)$ equals $1$ for $r=\sqrt{x^2+y^2}\leq r_1$ and $0$ everywhere else.

The expression of $u_z(x,y,0)$ is multiplied by $\cos(\pi m x/a) \cos(\pi n y/b)$ and integrated over the perforation surface area. Finally, using the orthogonality property of modes $\cos(\pi m x/a) \cos(\pi n y/b)$ and the polar coordinates system, one can write:

\[ \begin{multline} \int^{2\pi}_{0}\int^{r_1}_{0} U_0 \cos\left(\frac{\pi m r \cos(\theta)}{a}\right) \cos\left(\frac{\pi n r \sin(\theta)}{b}\right)rdrd\theta=\\ \int^{a}_{-a}\int^{b}_{-b} U_{mn} \cos^2\left(\frac{\pi m x}{a}\right) \cos^2\left(\frac{\pi n y}{b}\right)dxdy \end{multline} \]

The oscillatory part of the left hand side in the previous equation can be re-written as:

\[ \begin{multline} \cos\left(\frac{\pi m r \cos(\theta)}{a}\right) \cos\left(\frac{\pi n r \sin(\theta)}{b}\right)=\\ \frac{1}{2}\cos\left(r\sqrt{\displaystyle\left(\frac{\pi m}{a}\right)^2+\left(\frac{\pi n}{b}\right)^2}\sin\left(\theta+\gamma\right)\right)+\\ \frac{1}{2}\cos\left(r\sqrt{\displaystyle\left(\frac{\pi m}{a}\right)^2+\left(\frac{\pi n}{b}\right)^2}\sin\left(\theta-\gamma\right)\right) \end{multline} \]

with $\gamma=\arctan{\displaystyle\left(\frac{na}{mb}\right)}$.

Using

\[ \int_{0}^{2\pi}\cos\left(r\sqrt{\displaystyle\left(\frac{\pi m}{a}\right)^2+\left(\frac{\pi n}{b}\right)^2}\sin\left(\theta\pm\gamma\right)\right)=2\pi J_0\left(r\sqrt{\displaystyle\left(\frac{\pi m}{a}\right)^2+\left(\frac{\pi n}{b}\right)^2}\right) \]

and the property of Bessel functions of the first kind: $\int_{0}^{X} J_0(x)xdx=XJ_1(X)$, one writes:

\[ \begin{multline} \int_{0}^{r_1}J_0\left(r\sqrt{\displaystyle\left(\frac{\pi m}{a}\right)^2+\left(\frac{\pi n}{b}\right)^2}\right)rdr=\\ \int_{0}^{r_1'}J_0\left(r'\right)\frac{r'dr'}{\left(\frac{\pi m}{a}\right)^2+\left(\frac{\pi n}{b}\right)^2}=\\ r_1 \frac{J_1\left(r_1\sqrt{\displaystyle\left(\frac{\pi m}{a}\right)^2+\left(\frac{\pi n}{b}\right)^2}\right)}{\sqrt{\displaystyle\left(\frac{\pi m}{a}\right)^2+\left(\frac{\pi n}{b}\right)^2}} \end{multline} \]

so that the double integration equation now writes:

\[ 2\pi r_1 U_0 \frac{J_1\left(r_1\sqrt{\displaystyle\left(\frac{\pi m}{a}\right)^2+\left(\frac{\pi n}{b}\right)^2}\right)}{\sqrt{\displaystyle\left(\frac{\pi m}{a}\right)^2+\left(\frac{\pi n}{b}\right)^2}}=U_{mn} \frac{ab}{\nu_{mn}} \]

$U_{mn}$ and $P_{mn}$ can be expressed as:

\[ U_{mn}=\frac{2\pi r_1 \nu_{mn} U_0}{ab} \frac{J_1\left(r_1\sqrt{\displaystyle\left(\frac{\pi m}{a}\right)^2+\left(\frac{\pi n}{b}\right)^2}\right)}{\sqrt{\displaystyle\left(\frac{\pi m}{a}\right)^2+\left(\frac{\pi n}{b}\right)^2}} \]

and

\[ P_{mn}=\frac{2\pi r_1 \nu_{mn} U_0}{ab}\frac{\rho \omega}{k_{mn}} \frac{J_1\left(r_1\sqrt{\displaystyle\left(\frac{\pi m}{a}\right)^2+\left(\frac{\pi n}{b}\right)^2}\right)}{\sqrt{\displaystyle\left(\frac{\pi m}{a}\right)^2+\left(\frac{\pi n}{b}\right)^2}} \]

The mean pressure over the perforation area $\bar{p}$ writes:

\[ \bar{p}=\frac{1}{\pi r_1^2}\int_{0}^{2\pi}\int_{0}^{r_1}\sum_{mn}P_{mn}\cos\left(\frac{\pi m r \cos(\theta)}{a}\right) \cos\left(\frac{\pi n r \sin(\theta)}{b}\right)rdrd\theta \] \[ \bar{p}=\frac{1}{\pi r_1^2}\sum_{mn}\frac{4\pi^2 r_1^2 \nu_{mn} U_0}{ab}\frac{\rho \omega}{k_{mn}} \left[\frac{J_1\left(r_1\sqrt{\displaystyle\left(\frac{\pi m}{a}\right)^2+\left(\frac{\pi n}{b}\right)^2}\right)}{\sqrt{\displaystyle\left(\frac{\pi m}{a}\right)^2+\left(\frac{\pi n}{b}\right)^2}}\right]^2 \]

From the approximate expression for $k_{mn}$ recalled in the previous section:

\[ \bar{p}=\frac{4\pi}{ab} U_0 \rho \omega\sum_{mn}\nu_{mn} \frac{J_1^2\left(r_1\sqrt{\displaystyle\left(\frac{\pi m}{a}\right)^2+\left(\frac{\pi n}{b}\right)^2}\right)}{j\left[\displaystyle\left(\frac{\pi m}{a}\right)^2+\left(\frac{\pi n}{b}\right)^2\right]^{3/2}} \]

Finally, from the expression of the impedance $Z_B$, the expression of the length correction $\varepsilon_{\circ\subset\blacksquare}$ is derived:

\[ \varepsilon_{\circ\subset\blacksquare}=\frac{4\pi}{ab} \sum_{mn*} \nu_{mn} \frac{J_1^2\left(r_1\sqrt{\displaystyle\left(\frac{\pi m}{a}\right)^2+\left(\frac{\pi n}{b}\right)^2}\right)}{\left[\displaystyle\left(\frac{\pi m}{a}\right)^2+\left(\frac{\pi n}{b}\right)^2\right]^{3/2}} \]

For this configuration, the shape ratios $\xi$ and $\eta$ are re-defined as $r_1/a$ and $r_1/b$ respectively.