Added description of boundary condition: incident plane wave on port in infinite wall.

archive/lrftubes_doc_2021-03-31.pdf

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+lrftubes_doc_820663a.pdf

archive/lrftubes_doc_latest.pdf

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-lrftubes_doc_820663a.pdf

lrftubes.lyx

@@ -1570,6 +1570,12 @@ To model absorption of sound, a one-dimensional porous material model should
  This work has been postponed to a later stage.
 \end_layout
 
+\begin_layout Standard
+Prismatic and spherical ducts filled with porous material are defined in
+ dbmduct.py.
+ These use the Delaney-Bazley-Miki model.
+\end_layout
+
 \begin_layout Section
 Overview of this documentation
 \end_layout
@@ -7075,7 +7081,7 @@ A membrane is a mechanical
 \end_layout
 
 \begin_layout Section
-Hole
+Holes in plate
 \end_layout
 
 \begin_layout Standard
@@ -7083,16 +7089,73 @@ series_impedance.py/class CircHoleNeck(SeriesImpedance)
 \end_layout
 
 \begin_layout Standard
-Behaves like an acoustic mass with losses.
+A plate with several holes can be modelled using CircHoleNeck.
- It represents holes in sheet material, which can form the neck of a Helmholtz
+ It behaves like an acoustic mass with losses and can represent the neck
- resonator.
+ of a Helmholtz resonator.
- Hole-hole interaction is neglected.
+ Typical uses are to connect volumes to eachother or volumes to ducts, to
- The resistance term is an approximation.
+ form Helmholtz resonators.
 \end_layout
 
 \begin_layout Standard
-Usable for connecting volumes to eachother or volumes to ducts, to form
+Limitations are that hole-hole interaction is neglected and that the resistance
- Helmholtz resonators.
+ term is an approximation for holes with diameter >> length.
+\end_layout
+
+\begin_layout Standard
+Impedance is given by the equation:
+\end_layout
+
+\begin_layout Standard
+\noindent
+\align center
+\begin_inset Formula 
+\begin{equation}
+Z_{holes}=\frac{1}{N_{h}}\left(R_{v}+i\omega M_{A}\right)
+\end{equation}
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+in which 
+\begin_inset Formula $N_{h}$
+\end_inset
+
+ is the number of holes, 
+\begin_inset Formula $R_{v}$
+\end_inset
+
+ the acoustic resistance as described in equation 
+\begin_inset CommandInset ref
+LatexCommand ref
+reference "eq:Rv_hole"
+plural "false"
+caps "false"
+noprefix "false"
+
+\end_inset
+
+, 
+\begin_inset Formula $\omega$
+\end_inset
+
+ the angular frequency and 
+\begin_inset Formula $m_{a}$
+\end_inset
+
+ the acoustic mass as described in equation 
+\begin_inset CommandInset ref
+LatexCommand ref
+reference "eq:acoustic_mass"
+plural "false"
+caps "false"
+noprefix "false"
+
+\end_inset
+
+, except without Karal's discontinuity factor.
 \end_layout
 
 \begin_layout Section
@@ -7278,7 +7341,7 @@ literal "true"
 
 \begin_inset Formula 
 \begin{equation}
-M_{A}=\chi(\alpha,k)\frac{8\rho_{0}}{3\pi^{2}a_{L}},
+M_{A}=\chi(\alpha,k)\frac{8\rho_{0}}{3\pi^{2}a_{L}},\label{eq:acoustic_mass}
 \end{equation}
 
 \end_inset
@@ -7904,12 +7967,347 @@ Filling this in, we obtain the following low-frequency approximation to
 \begin_layout Standard
 \begin_inset Formula 
 \begin{equation}
-Z_{\mathrm{rad}}=\frac{z_{0}}{S}\left[i\frac{8ka}{3\pi}+\frac{1}{2}\left(ka\right)^{2}+\mathcal{O}\left(\left(ka\right)^{3}\right)\right]
+Z_{\mathrm{rad}}=\frac{z_{0}}{S}\left[i\frac{8ka}{3\pi}+\frac{1}{2}\left(ka\right)^{2}+\mathcal{O}\left(\left(ka\right)^{3}\right)\right]\label{eq:Zrad-baffled-piston}
 \end{equation}
 
 \end_inset
 
 
+\end_layout
+
+\begin_layout Subsection
+Incident plane wave on small port in infinite baffle
+\end_layout
+
+\begin_layout Standard
+Situation: an acoustic system, which is connected to the outside world though
+ a port, ending in an infinite wall
+\begin_inset CommandInset ref
+LatexCommand ref
+reference "fig:bc_planewave_port"
+plural "false"
+caps "false"
+noprefix "false"
+
+\end_inset
+
+.
+ There is an incident plane wave with specified amplitude and frequency.
+ It would be beneficial for computing time to replace the outside world
+ by a boundary condition on the port.
+ Here it is approached as a scattering problem.
+ More information is described in 'Sound absorbing materials' (1949) Zwikker
+ et al., pp.
+ 132-134.
+ The pressure field can be written as:
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula 
+\begin{equation}
+p_{t}=p_{i}+p_{s}\label{eq:scattering-problem}
+\end{equation}
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+in which 
+\begin_inset Formula $p_{t}$
+\end_inset
+
+ is the total pressure field, 
+\begin_inset Formula $p_{i}$
+\end_inset
+
+ the incident pressure field and 
+\begin_inset Formula $p_{s}$
+\end_inset
+
+ the scattered pressure field.
+ All depend on both position and time.
+ If only the infinite wall is taken into account and the port and system
+ behind it are ignored, the amplitude of the incident plane wave and its
+ reflection can be described as:
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula 
+\begin{equation}
+p_{i}(x,t)=\begin{cases}
+P_{i}\cdot\cos(kx) & x<0\\
+undefined & x=0\\
+0 & x>0
+\end{cases}
+\end{equation}
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+in which 
+\begin_inset Formula $P_{i}$
+\end_inset
+
+ is half the amplitude of the incident plane wave (resulting in sound pressure
+ 
+\family roman
+\series medium
+\shape up
+\size normal
+\emph off
+\bar no
+\strikeout off
+\xout off
+\uuline off
+\uwave off
+\noun off
+\color none
+
+\begin_inset Formula $P_{i}$
+\end_inset
+
+
+\family default
+\series default
+\shape default
+\size default
+\emph default
+\bar default
+\strikeout default
+\xout default
+\uuline default
+\uwave default
+\noun default
+\color inherit
+ on the surface of a reflecting wall), 
+\begin_inset Formula $k$
+\end_inset
+
+ is the wavenumber and 
+\begin_inset Formula $x$
+\end_inset
+
+ the position into the wall.
+ There is no scattered pressure field, so this is the total pressure field
+ right away.
+ When the port and system behind it are added, the total pressure field
+ no longer is equal to the incident pressure field: a correction must be
+ added, which is captured in 
+\begin_inset Formula $p_{s}$
+\end_inset
+
+.
+ The correction is due to the air slug within the port moving.
+ At 
+\begin_inset Formula $x<0$
+\end_inset
+
+, this has the same effect als a baffled piston.
+ On the condition that the wavelength is much larger than the port size,
+ the scattered field near the boundary (but still outside of the port) is
+ given by:
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula 
+\begin{equation}
+p_{s}(x=0^{-})=-Z_{rad}U
+\end{equation}
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+in which 
+\begin_inset Formula $Z_{rad}$
+\end_inset
+
+is the radiation impedance of a baffled piston and 
+\begin_inset Formula $U$
+\end_inset
+
+ is the acoustic volume flow rate.
+ Note the minus sign, which stems from the direction in which 
+\begin_inset Formula $U$
+\end_inset
+
+ is defined.
+ The same convention is taken as in COMSOL: velocity 
+\begin_inset Formula $v$
+\end_inset
+
+ is positive when inwards, so inwards 
+\begin_inset Formula $U$
+\end_inset
+
+ is positive.
+ Filling in equation 
+\begin_inset CommandInset ref
+LatexCommand ref
+reference "eq:scattering-problem"
+plural "false"
+caps "false"
+noprefix "false"
+
+\end_inset
+
+, just outside of the port at 
+\begin_inset Formula $x=0^{-}$
+\end_inset
+
+, yields:
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula 
+\begin{equation}
+p_{t}(x=0^{-})=P_{i}-Z_{rad}U
+\end{equation}
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+It is questionable whether the port acoustically ends at the boundary, so
+ this might be an approximation.
+ In COMSOL, the pressure is continuous, to it is fine to apply it at 
+\begin_inset Formula $x=0$
+\end_inset
+
+ instead of 
+\begin_inset Formula $x=0^{-}$
+\end_inset
+
+.
+ 
+\begin_inset Formula $U$
+\end_inset
+
+ can be found by integrating the inner product of velocity and the normal
+ vector over the boundary, while adding a minus sign because the normal
+ vector points outwards.
+ In COMSOL it is more convenient to use 
+\emph on
+specific
+\emph default
+ impedances and 
+\emph on
+velocities
+\emph default
+.
+ Then the equation is slightly modified to:
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula 
+\begin{equation}
+p_{t}(x=0)=P_{i}-z_{rad}v\label{eq:bc-planewave-port-pressure}
+\end{equation}
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+in which 
+\begin_inset Formula $z_{rad}$
+\end_inset
+
+is the specific radiation impedance of a baffled piston and 
+\begin_inset Formula $v$
+\end_inset
+
+ the acoustic velocity (inwards).
+ This equation can be applied as a 
+\emph on
+pressure 
+\emph default
+boundary condition in COMSOL.
+ The required 
+\begin_inset Formula $v$
+\end_inset
+
+ can be 'measured' by averaging the normal component of the velocity and
+ adding a minus sign to make it inwards.
+ Alternatively, the equation can be solved for 
+\begin_inset Formula $v$
+\end_inset
+
+ to obtain a 
+\emph on
+velocity 
+\emph default
+boundary condition:
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula 
+\begin{equation}
+v=\frac{P_{i}-p_{t}(x=0)}{z_{rad}}\label{eq:bc-planewave-port-velocity}
+\end{equation}
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+in which 
+\begin_inset Formula $p_{t}(x=0)$
+\end_inset
+
+ can be 'measured' by averaging it over the port's boundary.
+\end_layout
+
+\begin_layout Standard
+\begin_inset Float figure
+wide false
+sideways false
+status open
+
+\begin_layout Plain Layout
+\align center
+\begin_inset Graphics
+	filename img/bc_planewave_port.jpg
+	lyxscale 10
+	width 50text%
+
+\end_inset
+
+
+\begin_inset Caption Standard
+
+\begin_layout Plain Layout
+Schematic view of incident wave (green) on an infinite wall (blue) containing
+ a port with a system connected to it.
+ The location of the boundary condition is shown in red.
+\end_layout
+
+\end_inset
+
+
+\begin_inset CommandInset label
+LatexCommand label
+name "fig:bc_planewave_port"
+
+\end_inset
+
+
+\end_layout
+
+\end_inset
+
+
 \end_layout
 
 \begin_layout Chapter
@@ -8542,7 +8940,7 @@ Z_{h}=\left(\frac{\rho_{0}z_{0}}{i\omega V}+R_{v}+i\omega m_{\mathrm{neck}}\righ
 where
 \begin_inset Formula 
 \begin{equation}
-m_{\mathrm{neck}}=\frac{\rho_{0}\ell_{\mathrm{eff},\mathrm{neck}}}{S_{\mathrm{neck}}},
+m_{\mathrm{neck}}=\frac{\rho_{0}\ell_{\mathrm{eff},\mathrm{neck}}}{S_{\mathrm{neck}}},\label{eq:acoustic_mass_neck}
 \end{equation}
 
 \end_inset
@@ -9580,18 +9978,6 @@ For circular large holes with diameter
 \end_layout
 
 \begin_layout Standard
-For circular large holes with diameter 
-\begin_inset Formula $D$
-\end_inset
-
-, the end correction for both sides is
-\begin_inset Formula 
-\begin{equation}
-2\delta=\frac{8}{3\pi}D\approx0.85D.
-\end{equation}
-
-\end_inset
-
 Here we use a more advanced model, which includes the shear wave number.
  For unrounded edges and a perforate thickness of 
 \begin_inset Formula $t_{p}$
@@ -10388,7 +10774,7 @@ Z_{\mathrm{hole}}=i\omega\rho_{0}\frac{4}{\pi D^{2}}\left[\frac{t_{w}}{\left(1-f
 \begin_layout Standard
 \begin_inset Formula 
 \begin{equation}
-\Re[z_{\mathrm{hole}}]=\frac{2D\delta_{\nu}\omega\rho_{0}t_{w}}{\left(4\delta_{\nu}^{2}+\left(D-2\delta_{\nu}\right)^{2}\right)},
+\Re[z_{\mathrm{hole}}]=\frac{2D\delta_{\nu}\omega\rho_{0}t_{w}}{\left(4\delta_{\nu}^{2}+\left(D-2\delta_{\nu}\right)^{2}\right)},\label{eq:Rv_hole}
 \end{equation}
 
 \end_inset