A
WRITE UP
ON
PROPAGATION OF SOUND IN FLUIDS
BY
AJAYI DAVID OLUSHEYE
ARC/04/3171
AND
OYEGOKE ADENIYI SUNDAY ARC/04/3225
SUBMITTED TO
THE DEPARTMENT OF ARCHITCTURE,
SCHOOL OF ENVIRONMENTAL TECHNOLOGY,
FEDERAL UNIVERSITY OF TECHNOLOGY.
AKURE, ONDO-STATE.
COURSE TITLE-ENVIRONMENTAL CONTROL III (ACOUSTIC)
COURSE CODE-ARC 507
COURSE LECTURER-
PROF. O. O. OGUNSOTE.
MARCH 2009
TABLE OF CONTENTS
1.0
Introduction.
1.1
Sound propagation in fluid
2.0
The Propagation of Sound
3.0
Sound Propagation in a Cylindrical Duct with Compliant Wall
3.1
Analysis
4.0
Fast Sound Propagation in Binary Fluid Mixtures
5.0
Speed of Sound in Fluid
6.0
Stagnation and Sonic Properties of Sound Waves
6.1
Case Study 1: Adiabatic Flow
6.2
Case Study 2: Sonic Flow (Ma=1)
7.0
Conclusions
References
1.0
INTRODUCTION
1.1
Sound propagation in a fluid
Sound waves are traveling pressure waves in a fluid (gas or a liquid).
Px, t   P0  Px  vt Consequence of this is that there can be no sound propagation in
vacuum. We verify this by showing that an otherwise noisy personal alarm device
becomes completely silenced when placed in a vacuum. Corresponding to the pressure
wave is a displacement wave s(x-vt) of atoms from where they would be if there had been
no sound propagating. The relationship between the displacement wave and the deviation
from
average
PB
pressure
is
a
characteristic
property
of
the
fluid:
ds
dx
Note that ds/dx rather than s appears on the right hand side of the equation because ds/dx
measures the deformation of the fluid from equilibrium: A constant s simply corresponds
to an overall displacement of the fluid. The negative sign is there because a contraction
ds/dx<0 gives an increase in pressure and thus a positive p. B is called the bulk modulus
of the fluid and has dimensions of a pressure. It can be shown that
B  p
where
 
c
c
p
o
is the ratio of the constant pressure to the constant volume specific heat. We shall return
to Eq. 12 later in the course.
To derive the wave velocity in a fluid we write Newton's second law for a cylindrical
slice
of
thickness
and
F  APx   Px  x    Ax
Ma  PAx
area
A.
The
dp
d 2s
 AxB 2 We
dx
dx
force
on
also
this
slice
is
calculate
d 2s
dt 2
Newton tells us to equate F and ma and this gives us the wave-equation
F  Ma 
d 2s
d 2s

PA

x

dx 2
dt 2
d 2s P d 2s

dx 2 B dt 2
AxB
From
which
we
conclude
that
the
speed
of
sound
is
1
P
 
2
B
V
B
V 
P
P

P
 1
Neglecting the
we could immediately have written down this formula based on
dimensional analysis or an analogy with the previously derived expression for the wavevelocity on a taut string. The formula was first derived by Newton himself. Putting in
numbers we get for air at ambient pressure (
,
,
)
V 
7 5
10 N / M 2
5
 328M / S
1.3kg / M 2
which is indistinguishable from the measured value
. Note that we have
derived the interesting result that at constant pressure the velocity of sound is greater in a
light than a heavy gas.
2.0
The Propagation of sound
Sound is a sequence of waves of pressure which propagates through compressible media
such as air or water. (Sound can propagate through solids as well, but there are additional
modes of propagation). During their propagation, waves can be reflected, refracted, or
attenuated by the medium. The purpose of this experiment is to examine what effect the
characteristics of the medium have on sound.
All media have three properties which affect the behavior of sound propagation:
1. A relationship between density and pressure. This relationship, affected by
temperature, determines the speed of sound within the medium.
2. The motion of the medium itself, e.g., winds. Independent of the motion of
sound through the medium, if the medium is moving, the sound is further
transported.
3. The viscosity of the medium. This determines the rate at which sound is
attenuated. For many media, such as air or water, attenuation due to viscosity is
negligible.
What happens when sound is propagating through a medium which does not have
constant properties? For example, when sounds speed increases with height? Sound
waves are refracted. They can be focused or dispersed, thus increasing or decreasing
sound levels, precisely as an optical lens increases or decreases light intensity.
One way that the propagation of sound can be represented is by the motion of wave
fronts-- lines of constant pressure that move with time. Another way is to hypothetically
mark a point on a wave front and follow the trajectory of that point over time. This latter
approach is called ray-tracing and shows most clearly how sound is refracted.
In the simulation which follows, the effects of the medium on sound propagation can be
visualized. The user can generate a variety of sound-speed profiles and wind-speed
profiles by clicking on the profile choices and dragging the red dots to establish
amplitudes. Two sound sources are available: a spherical source, in which initial sound
waves emanate uniformly in all directions; and a planar source, in which initial sound
waves emanate in a single direction. The location of the source and it orientation can be
changed by dragging the red dots. Sound propagation in this simulation is in two
dimensions; and media profiles depend on height only. Pressing 'Start' will begin the
simulation. Propagation is represented both by rays (black) and wave fronts (red). Note
that the sound speed C0 is artificially low to accentuate the effects of the medium. (Sound
speed in air is nominally 340m/s; in water, 1500m/s.) Data, including sound speed, wind
speed, and derivatives, may be obtained by clicking anywhere within the orange
propagation field.
3.0
Sound propagation in a cylindrical duct with compliant wall
There have been many applications that deal 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.
3.1
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.
Fluid
Two simplified fluid equations will be considered here:
and
where
, 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 = ρRT and the isentropic gas law c2 = γRT.
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 (Kw). 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
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:
Treating this wall impedance as a compliance term (i.e. K = jω Zw), and substitute back
into Eqn 3 yields
Here,
.
Furthermore, by taking out kappa, the expression becomes
After some manipulation, it yields
Where
For rigid wall, as
,
,
, hence,
, which
leads back to Equation 1 .
If the impedance analogy is used, i.e. pressure is the voltage and the velocity the current,
then
, where C is the compliance of the wall per unit length.
The speed of the sound is then determined by
, hence
Here, cp is the phase velocity and it depends on the excitation frequency, ω, the acoustic
wave is dispersive. When the excitation frequency is below the natural frequency, ωn, 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 under formed tube has a diameter of D and after deformation, it
becomes
. The area change in this case is
ratio between the two is
. The
. Hence, the wall stiffness is then
.
Where the inverse of this is known as the compliance or distensibility (a bio-medical
term). To determine the term
, 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,
Applying Hook's Law, it is possible to determine the strain, , by
.
With some substitutions,
For small strain,
Hence,
This is the wall stiffness, a function of only the tube elastic properties.
Mass of the tube per unit length is considered, then
Mw = ρsπh(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,
is 0.1, the material is steel with
and E = 2x1011Pa. The fluid contained inside is assumed to be air with
and the free wave speed of
.
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
. In this plot, propagation of wave is possible only if the
phase speed is real. There are two important frequencies that deserve a close attention.
The first is the natural frequency of the empty structure, i.e. ωn and the natural frequency
of the fluid loaded structure, ω1. In this plot, ωn = 450Hz while ω1 = 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 ωn. Between ωn and ω1, the phase
speed is imaginary, which means no wave can propagate in between these two
frequencies. As soon as the frequency increases pass ω1, 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 are used to
excite the tube, until the excitation frequency matches ωn. Beyond ωn, no true wave
propagation is possible because in between these two frequencies.
For a very rigid tube, i.e. E = 2x1020Pa , 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.
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 ωn and ω1.
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.
4.0
FAST SOUND PROPAGATION IN BINARY FLUID MIXTURES
Binary fluid mixtures are fluids consisting of two different components. As for
simple (one component) fluids, the main experimental probes for the study of these
properties of binary mixtures are light and neutron scattering. To connect the dynamics
with scattering experiments one can use the kinetic theory of fluids. I have applied the
kinetic theory to binary mixtures where the atomic masses of the molecules of the two
components are very different (disparate-mass binary mixtures). I have studied both
dilute (gaseous) mixtures and dense mixtures. In the description of the dynamics of the
fluid through the density-density correlation functions, one can introduce modes, which
can be thought of as the different channels by which the correlations decay in time. Some
modes are propagating, in the sense that they describe propagating, and damped
processes. Others are not propagating, and describe diffusive, purely damped processes.
The results concern the appearance of a fast propagating mode, in disparate-mass binary
mixtures, in a vast range of densities, from dilute gas mixtures to rather high (liquid)
densities. This fast mode appears beyond the hydrodynamic regime. One can call this
mode fast sound, because, like ordinary sound, it propagates, but it is faster. The most
important point is that the fast sound is associated with dynamics of the light component
only. In the thesis I explain how this phenomenon could be observed in light and neutron
scattering experiments. If it is detected in actual scattering experiments on disparate-mass
binary mixtures it would be the first time that a non hydrodynamic mode in a fluid is
clearly "seen".
5.0
SPEED OF SOUND IN FLUID
The so-called sound speed is the rate of propagation of a pressure pulse of
infinitesimal strength through a still fluid. It is a thermodynamic property of a fluid.
A pressure pulse in an incompressible flow behaves like that in a rigid body. A displaced
particle displaces all the particles in the medium. In a compressible fluid, on the other
hand, displaced mass compresses and increases the density of neighboring mass which in
turn increases density of the adjoining mass and so on. Thus, a disturbance in the form of
an elastic wave or a pressure wave travels through the medium. If the amplitude and
therefore the strength of the elastic wave is infinitesimal, it is termed as acoustic wave or
sound wave.
Figure 39.1(a) shows an infinitesimal pressure pulse propagating at a speed " a " towards
still fluid (V = 0) at the left. The fluid properties ahead of the wave are p,T and
the properties behind the wave are p+dp, T+dT and
, while
. The fluid velocity dV is
directed toward the left following wave but much slower.
In order to make the analysis steady, we superimpose a velocity " a " directed towards
right, on the entire system (Fig. 39.1(b)). The wave is now stationary and the fluid
appears to have velocity " a " on the left and (a - dV) on the right. The flow in Fig. 39.1
(b) is now steady and one dimensional across the wave. Consider an area A on the wave
front. A mass balance gives
Fig
1.1:
Propagation
(a) Wave Propagating into still Fluid
of
a
sound
wave
(b) Stationary Wave
This shows that
(a)
if dρ is positive.
(b) A compression wave leaves behind a fluid moving in the direction of the wave
(Fig. 1.1(a)).
(c) Equation (1.1) also signifies that the fluid velocity on the right is much smaller
than the wave speed " a ". Within the framework of infinitesimal strength of the wave
(sound wave), this " a " itself is very small.

Applying the momentum balance on the same control volume in Fig. 1.1 (b). It
says that the net force in the x direction on the control volume equals the rate of
outflow of x momentum minus the rate of inflow of x momentum. In symbolic
form, this yields
In the above expression, Aρa is the mass flow rate. The first term on the right hand side
represents the rate of outflow of x-momentum and the second term represents the rate of
inflow of x momentum.

Simplifying the momentum equation, we get
If the wave strength is very small, the pressure change is small.
Combining Eqs (1.1) and (1.2), we get
The larger the strength
of the wave, the faster the wave speed; i.e., powerful
explosion waves move much faster than sound waves. In the limit of infinitesimally small
strength,
we can write
Note that
(a) In the limit of infinitesimally strength of sound wave, there are no velocity gradients
on either side of the wave. Therefore, the frictional effects (irreversible) are confined to
the interior of the wave.
(b) Moreover, the entire process of sound wave propagation is adiabatic because there is
no temperature gradient except inside the wave itself.
(c) So, for sound waves, we can see that the process is reversible adiabatic or isentropic.
So the correct expression for the sound speed is
For a perfect gas, by using of
, and
, we deduce the speed of sound
as
For air at sea-level and at a temperature of 150C, a=340 m/s
6.0
STAGNATION AND SONIC PROPERTIES OF SOUND WAVES

The stagnation properties at a point are defined as those which are to be obtained
if the local flow were imagined to cease to zero velocity isentropically. As we will
see in the later part of the text, stagnation values are useful reference conditions in
a compressible flow.
Let us denote stagnation properties by subscript zero. Suppose the properties of a flow
(such as T, p , ρ etc.) are known at a point, the stagnation enthalpy is, thus, defined as
Where h is flow enthalpy and V is flow velocity

For a perfect gas , this yields,
Which defines the Stagnation Temperature
Now,
can be expressed as
Since,
If we know the local temperature (T) and Mach number (Ma) , we can find out the
stagnation temperature T0 .

Consequently, isentropic(adiabatic) relations can be used to obtain stagnation
pressure and stagnation density as
Values of
and
as a function of Mach number can be generated using
the above relationships and the tabulated results are known as Isentropic Table .
Note that in general the stagnation properties can vary throughout the flow field.
Let us consider some special cases:6.1
Case study 1:
Adiabatic Flow:
(From eqn 1.1) is constant throughout the flow. It follows that the
are
constant throughout an adiabatic flow, even in the presence of friction.
Hence, all stagnation properties are constant along an isentropic flow. If such a flow
starts from a large reservoir where the fluid is practically at rest, then the properties in the
reservoir are equal to the stagnation properties everywhere in the flow Fig (2.1)
Fig 2.1: An isentropic process starting from a reservoir
6.2
Case study 2: Sonic Flow (Ma=1)
The sonic or critical properties are denoted by asterisks: p*, ρ*, a*, and T* . These
properties are attained if the local fluid is imagined to expand or compress isentropically
until it reaches Ma = 1.
ImportantThe total enthalpy, hence T0 , is conserved as long as the process is adiabatic, irrespective
of frictional effects. From Eq. (2.1), we note that
This gives the relationship between the fluid velocity V, and local temperature (T), in an
adiabatic
Considering the condition, when Mach number, Ma=1, for a compressible flow we can
write from Eq. (40.2), (40.3) and (40.4),

For

The fluid velocity and acoustic speed are equal at sonic condition and is
diatomic
gases,
like
air
,
the
numerical
values
are
0r
7.0
CONCLUSION
An understanding of the nature of sound waves is essential to discussion on
acoustics. Sound waves are longitudinal waves originating from a source and conveyed
by a medium. Sound is a disturbance, or wave, which moves through a physical medium
(such as air, water or metal) from a source to cause the sensation of hearing in animals.
Sound is the sensation of the medium acting on the ear. The source can be a vibrating
Solid body such as the string of a guitar or the membrane of a drum, but it can also be a
vibrating gaseous medium, such as air in a whistle. The medium may be either a fluid or
a solid.
18
REFERENCES
Morse & Ingard (1968):
"Theoretical
Acoustics",
Princeton
University Press,
Princeton, New Jersey
Aments, W. S., (1953):
Sound Propagation in Gross Mixtures. Acoustic. Sot.
Amer., v.25.P.638.
Biot, M. A., (1956a):
Theory of Propagation of Elastic waves in a FluidSaturated Porous Solid, Part I: J.
Freudenthal, A. M., (1958): The Mathematical Theories of the Inelastic Continuum,
First Part: Encyclopedia of physics, v.6, p.267.
"http://en.wikibooks.org/wiki/engineering acoustics/sound propagation in a cylindrical
duct with compliant wall