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Guided circumferential shear horizontal waves in an isotropic
hollow cylinder
Xiaoliang Zhaoa) and Joseph L. Roseb)
Department of Engineering Science & Mechanics, The Pennsylvania State University, University Park,
Pennsylvania 16802
共Received 30 March 2003; revised 31 October 2003; accepted 22 December 2003兲
Guided time-harmonic shear horizontal 共SH兲waves propagating in the circumferential direction of
an isotropic hollow cylinder are studied. The dispersion equation as well as the displacement and
stress field across the wall thickness is derived analytically. Compared with the SH waves in a plate,
a quantitative guideline of how well a plate model can approximate a pipe in the circumferential
direction is given for defect characterization purpose. The work is also crucial for initiating work
efforts on three-dimensional wave scattering for pipeline inspection. © 2004 Acoustical Society of
America. 关DOI: 10.1121/1.1691037兴
PACS numbers: 43.20.Bi, 43.20.Mv 关SFW兴Pages: 1912–1916
I. INTRODUCTION
Axial cracks and corrosion defects are often found in
both industrial and military hollow cylindrical structures like
pipelines and cylindrical containers. Reliable and easy-to-use
inspection systems are in great need to locate the defects and
to be able to characterize and size them efficiently. Hirao and
Ogi1proposed a circumferential SH-wave Electromagnetic
Acoustic Transducer 共EMAT兲technique for detecting corro-
sion defects on the outer surface of steel pipelines with and
without protective resin coating. Gauthier2used multimode
SH waves generated by EMATs to form B-scan images of a
defect on a pipe. The reflection and transmission coefficients
of SH waves passing through a two-dimensional surface-
breaking defect or a stringer-like internal inclusion in a pipe
was reported by the present authors.3However, all of these
publications were based on an empirical plate-model ap-
proximation for a pipe of large diameter-to-wall-thickness
ratio. A rigorous theory of guided SH waves propagating in
the circumferential direction of a hollow cylinder needs to be
established.4
Gazis5,6 theoretically investigated guided waves that
propagate in the axial direction and are resonant in the cir-
cumferential direction of a hollow cylinder. The case when
the axial wave number is zero decouples into axially motion-
independent plane–strain vibration and longitudinal shear vi-
bration, both of which are standing waves in the circumfer-
ential direction. Liu and Qu7developed the model of guided
plain–strain waves propagating in the circumferential direc-
tion of a hollow cylinder, with emphasis on the dispersion
relation and displacement profile derivation and discussion.
In this paper, the guided time-harmonic SH wave propagat-
ing in the circumferential direction of a hollow cylinder is
studied. The dispersion equation as well as the displacement
and stress distribution across the wall of the hollow cylinder
is derived analytically. They are compared with that of the
SH waves in a plate of the same thickness numerically. Rela-
tive errors of the phase and group velocities when using a
plate model to approximate a pipe are given as a quantitative
measure of whether that approximation is valid. With the
approximated plate model, the guided wave interaction and
scattering from a three-dimensional defect in a hollow cylin-
drical structure of large diameter-to-wall-thickness ratio
could be tackled without extensive eigenmode and wave
structure calculations in each scattering direction.8
II. THEORETICAL MODEL
Consider steady-state time-harmonic waves propagating
in the circumferential direction of a hollow cylinder of inner
radius aand outer radius b, as shown in Fig. 1. Assume the
material is linearly elastic and isotropic, and the wave mo-
tion is independent of z. Two types of guided waves are
possible in this hollow cylinder: one is the plane–strain vi-
bration wave similar to the Lamb wave in a plate;7the other
is the longitudinal shear wave, which will be studied in detail
in this paper.
The analytical derivation of the frequency equation of
the guided circumferential SH waves begins with Navier’s
equation of motion as Eq. 共1兲, where uis the displacement
vector,
is the density, and
are Lame
´’s constants, re-
spectively
ⵜ2u⫹共⫹
兲ⵜⵜ•u⫽
共
2u/
t2兲.共1兲
Consider only the displacement in the zdirection in the cy-
lindrical coordinates, i.e., ur⫽u
⫽0 and uz⫽0; Eq. 共1兲can
be written as
2uz
t2⫽
冉
2uz
r2⫹1
r
uz
r⫹1
r2
2uz
2
冊
.共2兲
Note that uzis a function of r,
, and tonly. Since the wave
is time harmonic and propagates along the
direction, let
uz⫽⌿共r兲ei共kb
⫺
t兲,a⭐r⭐b,共3兲
in which the wave number kis defined by k⫽
/c(b). Here,
c(r) is the linear phase velocity for material particles located
at distance rfrom the axis of the hollow cylinder. Equation
共2兲can be transformed into
a兲Presently with Intelligent Automation, Inc. Rockville, MD 20855. Elec-
tronic mail: xzhao@i-a-i.com
b兲Electronic mail: jlresm@engr.psu.edu
1912 J. Acoust. Soc. Am. 115 (5), Pt. 1, May 2004 0001-4966/2004/115(5)/1912/5/$20.00 © 2004 Acoustical Society of America
⌿⬙⫹1
r⌿⬘⫹
冋
2
cT
2⫺
冉
kb
r
冊
2
册
⌿⫽0, 共4兲
Here, ⬘and ⬙denotes the first- and second-order derivative
with respect to r, respectively; cT⫽
冑
/
is the shear wave
velocity. This equation is easy to recognize as a Bessel equa-
tion. Follow the convention used in Refs. 7 by introducing
the nondimensional variables
r
¯
⫽r
b,k
¯
⫽kh,
¯
⫽
h
cT,h⫽b⫺a共5兲
k
&
⫽k
¯
1⫺
,
&⫽
¯
1⫺
⫽
b
cT,
⫽a
b.共6兲
The solutions can be written as
⌿共r兲⫽AJk
&共
&r
¯
兲⫹BYk
&共
&r
¯
兲,a⭐r⭐b,共7兲
where Jk
&(x) and Yk
&(x) are, respectively, the first and second
kind of Bessel functions of order k
&.Aand Bare arbitrary
constants.
It is assumed that the surface of the hollow cylinder is
traction free, i.e.,
rz
兩
r⫽a,b⫽
uz
r
冏
r⫽a,b
⫽0. 共8兲
Substituting Eqs. 共3兲and 共6兲into Eq. 共8兲, we can get
AJk
&
⬘
冉
cTr
冊
⫹BYk
&
⬘
冉
cTr
冊
兩
r⫽a,b⫽0, 共9兲
or in the expanded form
冦
A
冋
Jk
&
⫺1
冉
cTa
冊
⫺Jk
&
⫹1
冉
cTa
冊
册
⫹B
冋
Yk
&
⫺1
冉
cTa
冊
⫺Yk
&
⫹1
冉
cTa
冊
册
⫽0
A
冋
Jk
&
⫺1
冉
cTb
冊
⫺Jk
&
⫹1
冉
cTb
冊
册
⫹B
冋
Yk
&
⫺1
冉
cTb
冊
⫺Yk
&
⫹1
冉
cTb
冊
册
⫽0
.共10兲
This is a set of linear homogeneous algebraic equations with
Aand Bthe unknown variables. For nontrivial solutions, the
determinant of the coefficient matrix of this system of equa-
tions must vanish. Thus, the dispersion relation between the
wave number and the frequency of the circumferential
guided SH wave in a hollow cylinder can be written as
关Jk
&
⫺1共kTa兲⫺Jk
&
⫹1共kTa兲兴关Yk
&
⫺1共kTb兲⫺Yk
&
⫹1共kTb兲兴
⫺关Jk
&
⫺1共kTb兲⫺Jk
&
⫹1共kTb兲兴关Yk
&
⫺1共kTa兲
⫺Yk
&
⫹1共kTa兲兴⫽0, 共11兲
where kT⫽
/cTis the wave number of the shear wave.
Since both k
&and kTare frequency dependent, for a given
frequency f共or kT), k
&can be obtained from Eq. 共11兲, and
consequently the phase velocity c(b)⫽2
f/k
&(1⫺
) and
group velocity cg⫽d
/dk. For each frequency point f, Eq.
共11兲may give many real roots, each of which corresponds to
a possible wave mode that can propagate in the circumferen-
tial direction. Therefore, a family of phase velocity and
group velocity versus frequency dispersion curves can be
obtained. They provide a map of which kind of mode can be
present in the structure and at what phase and group velocity,
which is of great help in designing guided wave sensors and
interpreting acquired inspection data. Note that in Gazis’
article,5resonant modes in the circumferential direction of a
hollow cylinder were discussed. They are special cases when
k
&are integers.
Once k
&is obtained from Eq. 共11兲, it can be substituted
back to Eq. 共10兲so that the constant Bcan be expressed as a
function of constant A. Thus, displacement amplitude ⌿(r)
can be expressed as
⌿共r兲⫽A
冋
Jk
&共kTr兲⫺Jk
&
⫺1共kTa兲⫺Jk
&
⫹1共kTa兲
Yk
&
⫺1共kTa兲⫺Yk
&
⫹1共kTa兲Yk
&共kTr兲
册
,
a⭐r⭐b.共12兲
Note that Eq. 共12兲actually describes the displacement am-
plitude variation with respect to rin the hollow cylinder
wall. Once this displacement field uzis obtained, the stress
components of the circumferential SH waves can be calcu-
lated by Hooke’s law. In the cylindrical coordinate system,
the only nonzero stress components are
rz⫽
uz
r⫽AkT
2
再
Jk
&
⫺1共kTr兲⫺Jk
&
⫹1共kTr兲
⫺Jk
&
⫺1共kTa兲⫺Jk
&
⫹1共kTa兲
Yk
&
⫺1共kTa兲⫺Yk
&
⫹1共kTa兲关Yk
&
⫺1共kTr兲
⫺Yk
&
⫹1共kTr兲兴
冎
ei共kb
⫺
t兲共13兲
z⫽
1
r
uz
⫽ikbA
r
冋
Jk
&共kTr兲
⫺Jk
&
⫺1共kTa兲⫺Jk
&
⫹1共kTa兲
Yk
&
⫺1共kTa兲⫺Yk
&
⫹1共kTa兲Yk
&共kTr兲
册
ei共kb
⫺
t兲.
共14兲
Note that both stress components vary along the rdirection
and propagate in the
direction. They are an indispensable
part of the elastic wave field in the hollow cylinder. One may
notice that if we keep the fh value unchanged and increase
to 1, i.e., either both aand bgo to ⬁while b⫺a⫽hremains,
FIG. 1. Cylindrical coordinates of the hollow cylinder and dimensions.
1913J. Acoust. Soc. Am., Vol. 115, No. 5, Pt. 1, May 2004X. Zhao and J. L. Rose: Circumferencial shear horizontal waves in hollow cylinders
or b⫺a→0 while frequency f→⬁, Eq. 共4兲will yield
⌿⬙⫹
冋
2
cT
2⫺k2
册
⌿⫽0, 共15兲
which is the governing equation of SH waves in a plate.9
This observation indicates that when
⬇1, guided SH waves
in a hollow cylinder can be approximated as SH waves in a
plate, which is practically much easier to handle.
III. NUMERICAL COMPUTATION AND DISCUSSION
In this section, numerical examples are presented for
guided circumferential SH waves in carbon steel hollow cyl-
inders of
⫽0.2, 0.5, and 0.8, respectively. The results of SH
waves in a plate 共corresponds to
⫽1兲are also shown for
comparison. In those calculations, the longitudinal and shear
velocities of the carbon steel were chosen as cL⫽5900m/s,
cT⫽3200m/s. For a given frequency f, numerical solutions
of the nondimensional wave numbers k
&were obtained by
applying the commonly used bisectional root-search method
to the dispersion equation 共11兲. The phase and group veloci-
FIG. 2. 共a兲Phase velocity and 共b兲group velocity dispersion curve for SH
wave in the circumferential direction of a hollow cylinder. In the legend, a
is the inner radius, bis the outer radius of the cylinder.
FIG. 3. 共a兲Phase velocity and 共b兲group velocity plate model approximation
error for SH waves in the circumferential direction of a 10-in. schedule-40
pipe.
FIG. 4. Circumferential SH wave particle displacement distribution 共a兲n0mode at fh⫽1 and 共b兲n1mode at fh⫽3 in the wall of a hollow cylinder of
a/b⫽0.2, 0.5, 0.8, and 0.932, respectively. SH waves in a plate are also shown for comparison.
1914 J. Acoust. Soc. Am., Vol. 115, No. 5, Pt. 1, May 2004X. Zhao and J. L. Rose: Circumferencial shear horizontal waves in hollow cylinders
ties of the circumferential SH waves were obtained subse-
quently. Their dispersion curves are shown in Figs. 2共a兲and
共b兲, respectively. It is seen that the phase velocity dispersion
curves of the circumferential SH waves resembles those in a
plate, e.g., the lowest-order SH modes all start from a finite
value, while the higher-order modes start from infinity with
the cutoff frequency being the same as the corresponding SH
mode in a plate. They all asymptotically approach the shear
wave velocity of steel as fh tends to infinity. However, each
SH mode in the hollow cylinder is shifted up and towards the
right compared with that of a plate, and the lowest-order
mode is no longer nondispersive as in the case of a plate. The
group velocity dispersion curves of the hollow cylinder
changed more dramatically with the change of
values. The
lowest-order mode is no longer the fastest mode in energy
propagation at high frequencies, and each high-order mode
has a maximum value at a finite frequency.The smaller the
value, the higher the peak value and lower the fh product
where that maximum occurs.
It is also seen from Figs. 2共a兲and 共b兲that when the
value approaches 1, the circumferential guided SH wave dis-
persion curves approach those of a flat plate with the same
wall thickness. To be quantitative, relative approximation er-
rors (cpipe⫺cplate)/cpipe , where cdenotes phase or group ve-
locity, were calculated for each wave mode. For a 10-in.
schedule-40 pipe 共inner radius 127.25 mm, outer 136.53 mm,
the corresponding
⬇0.932兲, the relative errors of using the
plate model to calculate the phase and group velocities of the
circumferential guided SH waves in the frequency range
from 0.005 to 1 MHz are plotted in Figs. 3共a兲and 共b兲, re-
spectively. It is seen that the relative errors are within 5% in
the frequency region considered, although the error curves
for the group velocity seem a bit rough due to the numerical
derivation. Liu and Qu7did the same study for the circum-
ferential guided plane–strain waves in a circular annulus ver-
sus Lamb waves in a plate. They drew very similar conclu-
sions on using the plate model dispersion equation to
approximate a cylindrical shell of large
value.
To further confirm the above observation and understand
the propagation characteristics of the guided circumferential
SH waves, the displacement field and stress field distribution
in the hollow-cylinder wall of the first two modes are calcu-
lated from Eqs. 共12兲,共13兲, and 共14兲for various
values.
They are plotted in parts 共a兲and 共b兲of Figs. 4, 5, and 6,
respectively. In these calculations, the fh value for the n0
mode is arbitrarily chosen as 1 and the n1mode is 3; the
practical case
⫽0.932 is also included in the plot. For better
illustration, displacement field distributions are normalized
FIG. 5. Circumferential SH wave stress component
rz distribution 共a兲n0mode at fh⫽1 and 共b兲n1mode at fh⫽3 in the wall of a hollow cylinder of
a/b⫽0.2, 0.5, 0.8, and 0.932, respectively. SH waves in a plate are also shown for comparison.
FIG. 6. Circumferential SH wave stress component
zdistribution 共a兲n0mode at fh⫽1 and 共b兲n1mode at fh⫽3 in the wall of a hollow cylinder of
a/b⫽0.2, 0.5, 0.8, and 0.932, respectively. SH waves in a plate are also shown for comparison.
1915J. Acoust. Soc. Am., Vol. 115, No. 5, Pt. 1, May 2004X. Zhao and J. L. Rose: Circumferencial shear horizontal waves in hollow cylinders
within ⫾1 by each absolute maximum displacement value
for each
; the stress components are then calculated corre-
sponding to the normalized displacement. Figures 4共a兲and
共b兲show the amplitude distribution of the particle displace-
ments of the n0and n1mode guided circumferential SH
waves, respectively. With
→1, the distribution was seen to
approach monotonically that of a plate of the same fh value.
The same trend can also be noticed in Figs. 5 and 6共a兲and
共b兲, which plot the stress components
rz and
zof n0and
n1modes, respectively. Once again, the plate model is a very
accurate approximation to a hollow cylinder of
⬇1.
On the other hand, when
becomes smaller, all the
displacement and stress field distributions become more
asymmetric with respect to the midplane of the hollow-
cylinder wall. The lateral displacement of the n0mode cir-
cumferential SH wave has a larger value at the outer surface
of the hollow cylinder 关see Fig. 4共a兲兴, while its stress com-
ponent
z, which measures the interaction between particles
in the wave propagation direction, is smaller 关see Fig. 6共a兲兴.
The interaction between particles in the radius direction
rz
is no longer zero as is the case for a plate 关see Fig. 5共a兲兴;it
has maximum shifting toward the inner surface as
→0 and
remains zero at the inner and outer free surfaces. For the n1
mode, the zero crossings of both the displacement field uz
and the stress field
zshift toward the outer surface as
→0
and more energy is concentrated near the inner surface 关see
Figs. 4共b兲and 6共b兲兴; thus, the n1mode SH wave should have
a better chance to detect defects located near the inner sur-
face of a hollow cylinder.10 The radius dependence of the
stress component
rz also becomes complex 关see Fig. 5共b兲兴
as
→0, which is the result of a complex displacement pro-
file across the wall thickness.
IV. CONCLUSIONS
Guided SH wave propagating in the circumferential di-
rection of a hollow cylinder was studied. The dispersion
equation as well as the displacement and stress field distri-
bution across the wall of the hollow cylinder were derived
analytically. Both the phase and group velocity dispersion
curves and wave-field distribution are shown to be highly
dependent on the wall-thickness-to-radius ratio of the hollow
cylinder. When the ratio opts close to 1, they asymptotically
approach that of a plate of the same frequency–thickness
product. The relative error of phase and group velocities can
serve as a quantitative measure of how well the plate ap-
proximation to the hollow cylinder is in wave analysis. When
the ratio is small, precautions should be made on the mode
and frequency selection in using the circumferential SH
waves in defect detection due to the dispersion relation and
wave-field distribution difference from that of a plate.
ACKNOWLEDGMENT
The authors acknowledge support from the Gas Technol-
ogy Institute, Chicago IL.
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