[<< wikibooks] 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 Kw, 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:

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

== Fluid ==
Two simplified fluid equations will be considered here:

−

∂
u

∂
x

=
κ

∂
p

∂
t

…
…
.
E
q
(
1
)

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

and

−

∂
p

∂
x

=

ρ

f

∂
u

∂
t

…
…
.
E
q
(
2
)

{\displaystyle -{\frac {\partial p}{\partial x}}={\rho }_{f}{\frac {\partial u}{\partial t}}\dots \dots .Eq(2)}

where

κ
=

1

ρ

f

c

2

{\displaystyle \kappa ={\frac {1}{{\rho }_{f}c^{2}}}}
,

u

{\displaystyle u}
is the fluid particle velocity and

p

{\displaystyle 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
=
ρ
R
T

{\displaystyle p=\rho RT}
and the isentropic gas law

c

2

=
γ
R
T

{\displaystyle 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

−

∂
u

∂
x

=

(

κ
+

1
K

)

∂
p

∂
t

…
…
E
q

(
3
)

{\displaystyle -{\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, Mw, must be included.
The total stiffness impedance of the wall is then:

Z

w

=
j
ω

M

w

+

K

w

j
ω

=
j

M

w

ω

(

ω

2

−

ω

n

2

)

{\displaystyle Z_{w}=j\omega M_{w}+{\frac {K_{w}}{j\omega }}=j{\frac {M_{w}}{\omega }}\left({\omega }^{2}-{\omega }_{n}^{2}\right)}

Treating this wall impedance as a compliance term (i.e.

K
=
j
ω

Z

w

{\displaystyle K=j\omega Z_{w}}
), and substitute back into Eq 3 yields

−

∂
u

∂
x

=

(

κ
+

1

M

w

(

ω

n

2

−
ω

2

)

)

∂
p

∂
t

{\displaystyle -{\frac {\partial u}{\partial x}}=\left(\kappa +{\frac {1}{M_{w}({{\omega }_{n}^{2}-\omega }^{2})}}\right){\frac {\partial p}{\partial t}}}

Here,

ω

n

=

K

w

M

w

{\displaystyle {\omega }_{n}={\sqrt {\frac {K_{w}}{M_{w}}}}}
.
Furthermore, by taking out

κ

{\displaystyle {\kappa }}
, the expression becomes

−

∂
u

∂
x

=
κ

(

M

w

(

ω

n

2

−
ω

2

)

+

ρ

f

c

2

M

w

(

ω

n

2

−
ω

2

)

)

∂
p

∂
t

{\displaystyle -{\frac {\partial u}{\partial x}}=\kappa \left({\frac {M_{w}\left({{\omega }_{n}^{2}-\omega }^{2}\right)+{\rho }_{f}c^{2}}{M_{w}({{\omega }_{n}^{2}-\omega }^{2})}}\right){\frac {\partial p}{\partial t}}}

After some manipulation, it yields

−

∂
u

∂
x

=
κ

(

ω

1

2

−
ω

2

ω

n

2

−
ω

2

)

∂
p

∂
t

{\displaystyle -{\frac {\partial u}{\partial x}}=\kappa \left({\frac {{{\omega }_{1}^{2}-\omega }^{2}}{{{\omega }_{n}^{2}-\omega }^{2}}}\right){\frac {\partial p}{\partial t}}}

where

ω

1

2

=

ω

n

2

+

ρ

f

c

2

M

w

{\displaystyle {\omega }_{1}^{2}={\omega }_{n}^{2}+{\frac {{\rho }_{f}c^{2}}{M_{w}}}}

For rigid wall, as

K

w

→
∞

{\displaystyle K_{w}\to \infty }
,

ω

n

2

→
∞

{\displaystyle {\omega }_{n}^{2}\to \infty }
,

ω

1

2

→
∞

{\displaystyle {\omega }_{1}^{2}\to \infty }
, hence,

−

∂
u

∂
x

=
κ

∂
p

∂
t

{\displaystyle -{\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
=

κ

(

ω

1

2

−
ω

2

ω

n

2

−
ω

2

)

{\displaystyle C=\ \kappa \left({\frac {{{\omega }_{1}^{2}-\omega }^{2}}{{{\omega }_{n}^{2}-\omega }^{2}}}\right)}

,where C is the compliance of the wall per unit length.
The speed of the sound is then determined by

1

L
C

{\displaystyle {\sqrt {\frac {1}{LC}}}}
, hence

c

p

=
c

(

ω

n

2

−
ω

2

ω

1

2

−
ω

2

)

{\displaystyle c_{p}=c{\sqrt {\left({\frac {{{\omega }_{n}^{2}-\omega }^{2}}{{{\omega }_{1}^{2}-\omega }^{2}}}\right)}}}

Here,

c

p

{\displaystyle c_{p}}
is the phase velocity and it depends on the excitation frequency,

ω

{\displaystyle {\omega }}
, the acoustic wave is dispersive. When the excitation frequency is below the natural frequency,

ω

n

{\displaystyle {\omega }_{n}}
, the phase speed is lower than that of the free wave speed in fluid.
The next step is to identify Kw and Mw, which will be determined through structural response.

== Structure ==
Assume the undeformed tube has a diameter of D and after deformation, it becomes

D
+
△
D

{\displaystyle D+\triangle D}
. The area change in this case is

π
4

(

2
D
△
D
+

(

△
D

)

2

)

{\displaystyle {\frac {\pi }{4}}\left(2D\triangle D+{\left(\triangle D\right)}^{2}\right)}
. The ratio between the two is

△
A

A

=

2
△
D

D

+

(

△
D

D

)

2

{\displaystyle {\frac {\triangle A}{A}}={\frac {2\triangle D}{D}}+{\left({\frac {\triangle D}{D}}\right)}^{2}}
. Hence, the wall stiffness is then

K

w

=

△
p
D

2
△
D

{\displaystyle 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

△
D

D

{\displaystyle {\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,

The hoop stress in a cylindrical tube is given by,

σ
=
p

D

2
h

{\displaystyle \sigma =p{\frac {D}{2h}}}

Applying Hook's Law, it is possible to determine the strain,

ε

{\displaystyle \varepsilon }
, by

ε
=

σ
E

{\displaystyle \varepsilon ={\frac {\sigma }{E}}}
.
With some substitutions,

ε
=

p
D

2
h
E

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

For small strain,

Δ
ε
=

Δ
p
D

2
h
E

=

Δ
D

D

{\displaystyle \Delta \varepsilon ={\frac {\Delta pD}{2hE}}={\frac {\Delta D}{D}}}

Hence,

K

w

=
E

h
D

{\displaystyle 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

=

ρ

s

π
h
(
D
+
h
)

{\displaystyle 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,

h
D

{\displaystyle {\frac {h}{D}}}
is 0.1, the material is steel with

ρ
=
7800

k
g

m

3

{\displaystyle \rho =7800{\frac {kg}{m^{3}}}}
and

E
=
2
x

10

11

P
a

{\displaystyle E=2x10^{11}Pa}
. The fluid contained inside is assumed to be air with

ρ
=
1.21

k
g

m

3

{\displaystyle \rho =1.21{\frac {kg}{m^{3}}}}
and the free wave speed of

c
=
343

m
s

{\displaystyle c=343{\frac {m}{s}}}
.
600px
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

m
s

{\displaystyle 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.

ω

n

{\displaystyle {\omega }_{n}}
and the natural frequency of the fluid loaded structure,

ω

1

{\displaystyle {\omega }_{1}}
. In this plot,

ω

n

=
450
H
z

{\displaystyle {\omega }_{n}=450Hz}
while

ω

1

=
550
H
z

{\displaystyle {\omega }_{1}=550Hz}
.
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

ω

n

{\displaystyle {\omega }_{n}}
. Between

ω

n

{\displaystyle {\omega }_{n}}
and

ω

1

{\displaystyle {\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

ω

1

{\displaystyle {\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

ω

n

{\displaystyle {\omega }_{n}}
. Beyond

ω

n

{\displaystyle {\omega }_{n}}
, no true wave propagation is possible because in between these two frequencies.
For a very rigid tube, i.e.

E
=
2
x

10

20

P
a

{\displaystyle 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.
600px
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

ω

n

{\displaystyle {\omega }_{n}}
and

ω

1

{\displaystyle {\omega }_{1}}
.
600px
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