APMR
Acoustical Porous Material Recipes
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.
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$^{-1}$).
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:
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 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
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.