# Engineering Acoustics/Sound Propagation in a Cylindrical Duct With Compliant Wall

There have been many applications that deals with sound propagation in a
compliant wall duct. These range from hydraulic applications(i.e. water
hammer) to biomechanical
applications (i.e. pressure pulse in an
artery). In working with sound
propagation in a circular duct, the duct wall is often assumed to be
rigid so that any pressure disturbance in the fluid has no effect on the
wall. However, if the wall is assumed to be compliant, i.e. wall
deformation is possible when a pressure disturbance is encountered,
then, this will change the speed of the sound propagation. In reality,
the rigid wall assumption will be valid if the pressure disturbance in
the fluid, which is a function of the fluid density, is very small so
that the deformation of the wall is insignificant. However, if the duct
wall is assumed to be thin, i.e. \~ 1/20 of the radius or smaller, or if
the wall is made of plastic type of material with low Young\'s modulus
and density, or if the fluid contained is "heavy", the rigid wall
approximation is no longer true. In this case, the wall is assumed to be
compliant.

In the book by Morse & Ingard \[1\], the wall stiffness is defined as
K~w~, and this is the ratio between the pressure disturbances, p, to the
fractional change in cross-sectional area of the duct, produced by p. Of
course this pressure disturbance p is not static and the inertial of the
wall has to be considered. Because the deformation of the wall is due to
the pressure disturbances in the fluid, this is a typical
fluid-structure interaction problem, where the pressure disturbances in
the fluid cause the structural deformation, which in turn, modifies the
pressure disturbances.

Unlike sound propagate in a duct with a rigid wall where sound pressure
travels down the tube axially; part of the pressure is used to stretch
the tube radially. Clearly, because of the inclusion of tube wall
displacement, this becomes a fluid-structure interaction problem.

# Analysis

In this analysis, it is expected that the speed of the propagation will
depend on the material properties of the tube wall, i.e. Young\'s
modulus and the density. Also, as the analysis unfolds, it will become
apparently clear that the speed of propagation will vary with excitation
frequency, unlike wave propagation in a rigid-wall tube. Keep in mind
that the analysis presented here is a very simple one. Of course
depending on the fluid & structure model, one can reach a more accurate
results. In the analysis presented here, the goal is to provide a basic
physical understanding of sound propagation in a cylindrical duct with
compliant wall.

Assumptions:

1.  One-dimensional analysis
2.  All viscous & thermal dissipation are neglected
3.  Analysis are per unit length
4.  No mean flow velocity is considered

## Fluid

Two simplified fluid equations will be considered here:

$-\frac{\partial u}{\partial x}=\kappa \frac{\partial p}{\partial t}\dots \dots .Eq(1)$

and

$-\frac{\partial p}{\partial x}={\rho }_f\frac{\partial u}{\partial t}\dots \dots .Eq(2)$

where $\kappa =\frac{1}{{\rho }_fc^2}$, $u$ is the fluid particle
velocity and $p$ is the fluid pressure.

The first equation is the continuity equation, where the density term is
replaced by the pressure term by applying the ideal gas law $p=\rho RT$
and the isentropic gas law $c^2=\gamma RT$.

Because of the compliant wall, the fluid experiences an additional
compressibility effects, and according to Morse & Ingard \[1\] , this
additional compressibility is derived based on the wall stiffness (K).
The wall stiffness is defined as the ratio between the pressure and the
fractional change in cross-sectional area. By introducing K into Eq. 1,
it yields

$-\frac{\partial u}{\partial x}=\left(\kappa +\frac{1}{K}\right)\frac{\partial p}{\partial t}\dots \dots Eq\left(3\right)$

If the mass of the tube is considered, then additional mass parameter,
M~w~, must be included.

The total stiffness impedance of the wall is then:

$Z_w=j\omega M_w+\frac{K_w}{j\omega }=j\frac{M_w}{\omega }\left({\omega }^2-{\omega }^2_n\right)$

Treating this wall impedance as a compliance term (i.e.
$K=j\omega Z_w$), and substitute back into Eq 3 yields

$-\frac{\partial u}{\partial x}=\left(\kappa +\frac{1}{M_w({{\omega }^2_n-\omega }^2)}\right)\frac{\partial p}{\partial t}$

Here, ${\omega }_n=\sqrt{\frac{K_w}{M_w}}$.

Furthermore, by taking out ${\kappa }$, the expression becomes

$-\frac{\partial u}{\partial x}=\kappa \left(\frac{M_w\left({{\omega }^2_n-\omega }^2\right)+{\rho }_fc^2}{M_w({{\omega }^2_n-\omega }^2)}\right)\frac{\partial p}{\partial t}$

After some manipulation, it yields

$-\frac{\partial u}{\partial x}=\kappa \left(\frac{{{\omega }^2_1-\omega }^2}{{{\omega }^2_n-\omega }^2}\right)\frac{\partial p}{\partial t}$

where ${\omega }^2_1={\omega }^2_n+\frac{{\rho }_fc^2}{M_w}$

For rigid wall, as $K_w\to \infty$, ${\omega }^2_n\to \infty$,
${\omega }^2_1\to \infty$, hence,
$-\frac{\partial u}{\partial x}=\kappa \frac{\partial p}{\partial t}$,
which leads back to Equation 1 .

If the impedance analogy is used, i.e. pressure is the voltage and the
velocity the current, then

$C=\ \kappa \left(\frac{{{\omega }^2_1-\omega }^2}{{{\omega }^2_n-\omega }^2}\right)$
,where C is the compliance of the wall per unit length.

The speed of the sound is then determined by $\sqrt{\frac{1}{LC}}$,
hence

$c_p=c\sqrt{\left(\frac{{{\omega }^2_n-\omega }^2}{{{\omega }^2_1-\omega }^2}\right)}$

Here, $c_p$ is the phase velocity and it depends on the excitation
frequency,${\omega }$, the acoustic wave is dispersive. When the
excitation frequency is below the natural frequency, ${\omega }_n$, the
phase speed is lower than that of the free wave speed in fluid.

The next step is to identify K~w~ and M~w~, which will be determined
through structural response.

## Structure

Assume the undeformed tube has a diameter of D and after deformation, it
becomes $D+\triangle D$. The area change in this case is
$\frac{\pi }{4}\left(2D\triangle D+{\left(\triangle D\right)}^2\right)$.
The ratio between the two is
$\frac{\triangle A}{A}=\frac{2\triangle D}{D}+{\left(\frac{\triangle D}{D}\right)}^2$.
Hence, the wall stiffness is then

$K_w=\frac{\triangle pD}{2\triangle D}$.

where the inverse of this is known as the compliance or
distensibility(a
bio-medical term).

To determine the term $\frac{\triangle D}{D}$, it is necessary to look
at the structural response by using Newtown\'s Law and Hook\'s Law.

Consider a cross section of a half tube with diameter D, thickness h and
tension T,

!A cut away cylindrical
tube{width="300"}

The hoop stress in a cylindrical tube is given by,

$\sigma =p\frac{D}{2h}$ Applying Hook\'s Law, it is possible to
determine the strain, $\varepsilon$, by

$\varepsilon =\frac{\sigma}{E}$.

With some substitutions,

$\varepsilon =\frac{pD}{2hE}$

For small strain,

$\Delta \varepsilon =\frac{\Delta pD}{2hE}=\frac{\Delta D}{D}$ Hence,

$K_w=E\frac{h}{D}$

This is the wall stiffness, a function of only the tube elastic
properties.

Mass of the tube per unit length is considered, then

$M_w={\rho}_s\pi h(D+h)$.

Finally, it is possible to plot the phase speed and the wall impedance
verse the excitation frequency.

# Discussions

In the simulation the thickness to diameter ratio,$\frac{h}{D}$ is 0.1,
the material is steel with $\rho = 7800 \frac{kg}{m^3}$ and
$E = 2x10^{11} Pa$. The fluid contained inside is assumed to be air with
$\rho = 1.21 \frac{kg}{m^3}$ and the free wave speed of
$c = 343 \frac{m}{s}$.

![](Tws1.jpg "Tws1.jpg"){width="600"}

In this diagram, the \'o\' denotes the real part of the phase speed and
the \'+\' denotes the imaginary part of the phase speed. The straight
line shows the sound speed in air with a numerical value of
$c = 343 \frac{m}{s}$. In this plot, propagation of wave is possible
only if the phase speed is real. There are two important frequencies
that deserves a close attention. The first is the natural frequency of
the empty structure, i.e. ${\omega }_n$ and the natural frequency of the
fluid loaded structure, ${\omega }_1$. In this plot,
${\omega }_n = 450 Hz$ while ${\omega }_1 = 550 Hz$.

Unlike 1-D wave propagation in a rigid duct where the propagation speed
is a constant, the phase speed depends on the excitation frequency. It
shows that the propagation speed decreases as the excitation frequency
approaches to ${\omega }_n$. Between ${\omega }_n$ and ${\omega }_1$,
the phase speed is imaginary, which means no wave can propagate in
between these two frequencies. As soon as the frequency increases pass
${\omega }_1$, the phase speed is greater than the free wave speed of
343 m/s. As the excitation frequency increases, the phase speed
approaches to the free wave speed.

When the excitation frequency is increased, the parts of the fluid
energy is used to excite the tube, until the excitation frequency
matches ${\omega }_n$. Beyond ${\omega }_n$, no true wave propagation is
possible because in between these two frequencies.

For a very rigid tube, i.e. $E = 2x10^{20} Pa$ , the phase speed is
exactly the free wave speed in air, which is a constant. This agrees
with what have been discussed before for a 1-D wave propagation in a
rigid duct.

![](Tws2.jpg "Tws2.jpg"){width="600"}

When the stiffness is reduced to 1/100 of the steel, there are numerous
differences than the steel tube. First, at the low frequency, the phase
speed is slower. This is because the lower wall stiffness, the more the
wall can be stretched, hence can absorb more energy. Also, the system
has a much lower ${\omega }_n$ and ${\omega }_1$.

![](Tws3.jpg "Tws3.jpg"){width="600"}

From the above analysis, it is possible to conclude the following:

1\. The stiffness, the wall thickness and the density of the tube
affects the phase speed dramatically.

2\. A reduction in stiffness, reduces the propagation speed at low
frequencies. The wave also becomes evanescent at much lower frequency.
This is because the natural frequency is reduced. As the stiffness
increases, the propagation speed approaches to that of the free wave
speed regardless of the frequency region. This is the case of rigid
wall.

3\. The propagation speed of a wave in a duct with compliant wall is
dispersive as it depends greatly on the frequency. The phase speed
differs significantly than that of in a rigid wall.

4\. The non-propagating zone is known as the stop band. Here, the wall
properties can be modified to create a larger stop band. Hence, a duct
with compliant wall can be considered as a type of band filter.

# References

\[1\]. Morse & Ingard (1968), \"Theoretical Acoustics\", Princeton
University Press, Princeton, New Jersey

