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Theory of propagation of elastic waves in a
uid-saturated porous solid. II. Higher frequency range
Maurice A. Biot
To cite this version:
Maurice A. Biot. Theory of propagation of elastic waves in a uid-saturated porous solid. II. Higher
frequency range. Journal of the Acoustical Society of America, Acoustical Society of America, 1956,
28 (2), pp.179-191. �10.1121/1.1908241�. �hal-01368668�
THE JOURNAL OF THE ACOUSTICAL SOCIETY OF AMERICA VOLUME 28, NUMBER 2 MARCH, 1956
Theory of Propagation of Elastic Waves in a Fluid-Saturated Porous Solid.
II. Higher Frequency Range
M. A. BIOT*
Shell Development Company, RCA Building, New York, New York
(Received September 1, 1955)
The theory of propagation of stress waves in a porous elastic solid developed in Part I for the low-frequency
range is extended to higher frequencies. The breakdown of Poiseuille flow beyond the critical frequency
is discussed for pores of flat and circular shapes. As in Part I the emphasis of the treatment is on cases where
fluid and solids are of comparable densities. Dispersion curves for phase and group velocities along with
attenuation factors are plotted versus frequency for the rotational and the two dilational waves and for six
numerical combinations of the characteristic parameters of the porous systems. Asymptotic behavior at
high frequency is also discussed.
1. INTRODUCTION
A PREVIOUS paper • dealing with the subject of
propagation of elastic waves in a fluid saturated
porous solid was restricted to the low-frequency range.
By this it was meant a frequency range between zero
and a certain value beyond which the assumption of
Poiseuille flow broke down. The purpose of this paper
is to extend the theory to the full frequency range
without the limitation of the foregoing assumption.
There remains however an upper bound for the fre-
quency, namely, that at which the wavelength becomes
of the order of the pore size. Such a case must, of
course, be treated by a different method.
A theoretical study of the breakdown of Poiseuille
flow is presented in Secs. 2 and 3, by considering the
flow of a viscous fluid under an oscillatory pressure
gradient either between parallel walls or in a circular
tube. The case of the circular tube was originally treated
by Kirkhoff. This study yields a complex viscosity
correction factor function of the frequency through
the dimensionless ratio f/fc where fc is a characteristic
frequency of the material. The case of flow between
parallel walls and that of the circular tube indicate that
the effect of pore cross-sectional shape is well repre-
sented by taking the same function of the frequency for
the viscosity correction and simply changing the fre-
quency scale. As in Part I we are primarily concerned
with applications to liquids and we have neglected the
thermoelastic effects.
Application of these results to fluid friction in a
porous material is discussed in Sec. 4 and a "structural
factor" is introduced which represents the effect of
sinuosity and shape of the pores.
The propagation of rotational waves is discussed in
Sec. 5. Four numerical combinations of parameters are
considered. Group velocity, phase velocity, and attenua-
tion are plotted for these four cases as a function of the
frequency ratio f/f,. There is only one type of rotational
wave. The influence of the structural factor is also
* Consultant.
• M. A. Biot, J. Acoust. Soc. Am. 28, 168 (1956), preceding
paper.
evaluated by calculating phase velocity and attenua-
tion for a typical case.
The propagation of dilatational waves is discussed in
Sec. 6. Group velocity, phase velocity, and attenuation
curves are plotted for six numerical combinations of the
parameters. There are two types of such waves, desig-
nated as waves of the first and second kind. The latter are
characterized by high attenuation. An interesting plot
is that of the attenuation per cycle. Both the rotational
waves and the waves of the first kind exhibit a maxi-
mum value of this attenuation in a range of f/f• near
unity. In this range the inertia and viscous forces are of
the same order.
As discussed in Part I when the dynamic compati-
bility condition is satisfied or nearly satisfied (Zl•___•l)
the wave of the first kind has a very small attenuation.
This is shown by cases 2 and 5. The other two waves,
however, retain much higher attenuation. In such a case
only one type of wave may be observed unless special
attention is given to the others. Another aspect of this
phenomenon will be exhibited when a dilatational wave
is reflected at a surface of discontinuity. The reflected
energy is split up into three types of waves, two of which
may be unobserved because of their high attenuation.
The phenomenon then appears as the propagation of a
single-type body wave with small attenuation in the
body and a high absorption at the reflection surface.
Certain assumptions upon which the present theory is
based, such as perfect elasticity of the solid, limitations
on the nonuniformity of pore size, and the neglection of
thermal effects will determine the categories of materials
and frequency ranges for which it is applicable. It
should, however, be of value beyond its strict applica-
bility by indicating orders of magnitudes or qualitative
trends.
In applications to wave propagation in such ma-
terials as clay, silts, or muds, one should note that the
rotational wave is determined entirely by the shearing
rigidity of the solid. Since the latter may be small, the
rotational waves may, in this case, propagate with a
velocity which is considerably lower than that of the
dilatational waves of first and second kind.
179
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180 M.A. BIOT
2. OSCILLATORY FRICTION FORCE IN A
TWO-DIMENSIONAL DUCT
We are interested in the motion of a fluid in a two-
dimensional duct, i.e., a space limited by two parallel-
plane boundaries when these boundaries are subject
to an oscillatory motion and when an oscillatory pres-
sure gradient acts at the same time on the fluid.
We consider only the two-dimensional motion and
neglect all pressure gradients and velocity components
normal to the boundaries. The x-direction is parallel to
the boundaries and the y-axis is normal to it with the
boundaries represented by y= 4-a. The x-component
of the velocity of the boundary is/• and that of the fluid
• (Fig. 1). The latter component has a distribution
along y which is to be determined.
The equation of motion of the fluid in the x-direction
is
op
oi•) .... {-•' , (2.1)
Ox Oy 2
where u is the ,Ascosity, os the mass density of the fluid,
and p the pressure. Introducing the relative velocity
of the fluid,
we write,
PrO1 --
We may consider
Xps-
(2.2)
Op 02U1
.... os//+u ß (2.3)
Ox Oy 2
op
.... pt// (2.4)
Ox
to be equivalent to an external volume force and Eq.
(2.3) becomes
02U1
Of (2.5)
•' = t•/ps.
Assuming that all quantities are sinusoidal functions
of time with a factor e •t, and rewriting Eq. (2.5) with-
out this factor we have
d•U1
•----/a UI= --X. (2.6)
df
The general solution to this equation is
UI=--+C cosh (2.7)
y
y--o
O-fi = U•
Fro. 1. Two-di-
mensional flow be-
tween parallel walls.
with the condition that the function be symmetric in y.
The constant C is determined by the condition
at the wall, i.e., for y= 4-a•
X 1
/a cosh[(•) «alJ
The velocity distribution is
(2.8)
X
__
cosh[(ia/r)«y3
coshi (/a/v) lal]
(2.9)
For use in the general theory, we shall need both the
average velocity of the fluid through the cross section
and the friction force at the wall. The average velocity
U1 (A,) is given by
coshi (/a/•)«y] }
2alUl(•,)=----•X f •i I 1-cosh[•(ia/r),a13 dy. (2.10)
Hence,
X{ 1(•)« [(•)« ]}
UI(A,)=• 1---- tanh a• . (2.11)
a• •
The friction stress at the wall (y-- a0 is
T /'[L-•-y jy=_al •--• (:)
(2.12)
In applying these results we need the expression for
the total friction force 2r excited by the fluid on the
wall, per unit average velocity of the fluid relative to
the wall, i.e., we must calculate the ratio
]
2r 2t• al -• tanh al
UI(A,) al 1 • • ia « . (2.13)
1--•11(•) tanh[(-•)a, 1]
In this expression we have a nondimensional variable.
and we write
(2.14)
2r 2t• i«K1 tanh(ilK1)
__
UI(Av) a;1 1
1--- tanh (i«gl)
(2.15)
Let us examine the limiting case when the frequency
tends to zero, i.e., for gl--*0. We have
2r/Ul(a,) = 6t•/a1. (2.16)
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ELASTIC WAVES IN POROUS SOLIDS. II 181
This corresponds to Poiseuille flow. In this case the
velocity profile for U• if parabolic. If we put
1 i•Kx tanh(i•K•)
F•(g)=- (2.17)
3 1
1--• tanh (i•gx)
we have F•(0) = 1, and we write
2r 6•
•=--FI(K1), (2.18)
UI(Av) al
where F•(K•) is a complex quantity which is equal to
unity for g•=0 and represents the deviation from
Poiseuille friction as the frequency increases. There is a
difference of phase between the velocity and the friction
force. For large values of the frequency, i.e.,
Fro. 2. Frequency
correction functions
for the viscosity
in two-dimensional
flow.
we have
2.0
1.5
1.0
0 2 4 6 8 I0
F•(g0 =--/ =--
3 3 \-•- / '
(2.19)
therefore, the friction force at large frequencies and for
constant velocity increases like the square root of the
frequency and is 45 degrees out of phase with the ve-
locity. Everything happens as if the static viscosity
coefficient u were replaced by a dynamic value.
uF•(g•). (2.20)
We separate the real and imaginary part of F•(gx)
as follows'
Fx(K•)=Frx(•x)+Fix(•x). (2.21)
The values of Frx(gx) and Fix(g•) are plotted in Fig. 2.
It is seen that in accordance with Eq. (2.19) these curves
become asymptotically parallel to straight lines of
slope K•/3V2=0.234 gx for large values of •.
3. OSCILLATORY FRICTION FORCE IN A
THREE-DIMENSIoNAL DUCT
We shall now solve the same problem as in Sec. 2
except that instead of a two-dimensional motion be-
r=a =Ui '
r ,////////////////////////"
Fro. 3. Three-dimensional flow in a circular duct.
tween two plane boundaries we now consider a straight
duct of circular cross section, (Fig. 3) of radius a.
As in the foregoing, we consider the components of the
motion and the pressure gradient along the direction x
of the axis. In this case Eq. (2.1) is replaced by
op
p• .... FtzV2g r, (3.1)
Ox
where V 2 is the Laplacian operator. We assume that
• is independent of x, and that the flow is axially
symmetric so that the operator is
02 1 0
V 2= ! (3.2)
Or 2 r Or'
Putting
op
----p•=Xo• (3.3)
Ox
as before, and introducing the relative velocity Ux
= U--• of the fluid with respect to the wall, we may
write for Eq. (3.1)
v(O•U•+l OUI• OU1
...... X. (3.4)
\ Or • r • ! O t
All quantities being sinusoidal functions of time contain
a factor e •t. By rewriting Eq. (3.4) without this factor,
we find
dU• 1 dU• ia X
[ U•= ---- (3.5)
dr • r dr v v
This is a Bessel's differential equation for U•. The
general solution of this equation, which is finite at
r=0, is
[ ]
U•=---FCJo i r , (3.6)
ia
where C is a constant and
Jo(iV'iz) = berz+i beiz (3.7)
with Kelvin functions of the first kind and zero order.
Introducing the boundary condition Ux=0 for r-a
i12/U1
x
,01i( '--•• )«a]
(3.8)
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182 M.A. BIOT
Again, here we need the average velocity for the cross
section. The average velocity Ui(A,) is given by
or
with
iaUl(Av)
x
U•(^v) = U•rdr (3.9)
2 •ok
•= 1--• Jo(i•)•d• (3.10)
k2Jo(ilk)
g=a . (3.11)
The value of the integral in expression (3.10) is known.
We have 2
k
fo Jo(i•)•d•=tc(bei'K-iber'K). (3.12)
Hence,
/a U•(^v) 2 ber'tcq-ibei'tc
--=1- . (3.13)
X iK ber•+ibei•
In these expressions
d
ber'z=--berz
dz
d
bei'z =--beiz.
dz
We also evaluate the friction between the fluid and the
wall. The stress r at the wall is
[ dr a•o ia x,v)'
(3.14)
The total friction force is 2•rar, and the ratio of this
force to the average velocity is
with
2•rar 2•r•tcT (tc)
Ux(^v) 2
! !
ber tc+iber •
(3.15)
T(tc) = . (3.16)
bertc q-ibeitc
This formula is anologous to Eq. (2.15) where T(K)
plays the role of v'i tanh (x/'itc•). We now consider the
limiting value of expression (3.13) for g--•0, i.e., for
very low frequency. We have
berg+ibeitc= Jo(iV'ig) = 1 q----
itc K 3
ber' tc q-ibei' tc .....
2 16
itc 2 1
--K4-]L ß . .
2 2 2e4 e
(3.17)
• N. W. McLachlan, Bessel functions for Engineers (Clarendon
Press, Oxford, England, 1934), p. 125.
Hence, in the limiting case
and
2 itc •
1----T(tc) -) ,
itc 8
2;rat
UI(Av)
This expression checks with that obtained from
Poiseuille flow. Again here we introduce a function
1
F(tc) =- (3.19)
4 2
1--T½)
itc
such that F (0)= 1, and write
2•rar
•= 8riff (tc). (3.20)
The function F(tc) measures the deviation from Poiseuille
flow friction as a function of the frequency parameter
tc. For large values of tc, i.e., at high frequency the asym-
ptotic values are
1 exp[tc(lq-i' •
bertc + ibeitc--)•- --
(3.21)
1 /1-Fi\ [K(1-+-i) ifil
ber'tc+ibei%->•[--! exp -..
(2•g) • \ • ]
Hence,
1+i
T(g) > , (3.22)
and
4\•-/' (3.23)
As in the two-dimensional case it is found that the
friction at high frequency and for constant velocity
is proportional to the square root of the frequency and
is 45 degrees out of phase with the velocity. Every-
thing happens as if the static viscosity coefficient tz
were replaced by a dynamic complex value.
We put
(3.24)
F (tc) = Fr (tc) +iFi (tc). (3.25)
The real and imaginary parts of this function are
plotted in Fig. 4. For large value of g the curve becomes
parallel to the straight lines. tc/4V2=0.177g.
4. CALCULATION OF THE OSCILLATORY FRICTION
FORCE IN A POROUS MATERIAL
In applying the results of the previous section to a
porous material, we introduce the assumption that the
variation of friction with frequency follows the same
laws as found in the foregoing for the tube of uniform
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