A MODULAR MULTI-GAUSSIAN BEAM MODEL FOR ISOTROPIC
AND ANISOTROPIC MEDIA
Lester W. Schmerr Jr.1'2 and Alexander Sedov3
!
Center for NDE, Iowa State University, Ames, IA, 50011, USA
2
Dept. of Aerospace Eng. and Eng, Mech., Iowa State University, Ames, IA, 50011, USA
3
Dept. of Mechanical Eng., Lakehead University, Thunder Bay, Ontario,Canada, P7B 5E1
ABSTRACT. A highly compact model is described for propagating a Gaussian beam in a multilayered
medium where the layers are either isotropic or anisotropic in their material properties. This formulation
for a single Gaussian can be used as the basis for a multi-Gaussian beam model that can model the fields
of a circular planar piston transducer (planar or focused) present after multiple reflections/ transmissions
from curved interfaces.
INTRODUCTION
Multi-Gaussian beam models can be used to describe the propagation of sound beams
from planar and focused piston transducers in a variety of testing situations [1], One of the
attractive features of multi-Gaussian models is that they are numerically very efficient. This
is because these models rely on the superposition of a small number (10-15) of Gaussian
beams whose properties can be described in analytical terms even after propagation through
general anisotropic media and after interactions with multiple curved interfaces [2]. As the
number of interfaces involved increases, however, the analytical expressions for the
amplitude and phase of a Gaussian beam become increasingly complex. Such cases can arise
in practice, for example, when using angle beam shear waves and one or more "skips" in
testing plate and pipe geometries. Here, we will show that by representing these interactions
in a modular matrix form and invoking some general transformation relations it is possible
to express a Gaussian beam, even after multiple curved interface interactions, in a form that
is analogous to the propagation of the Gaussian in a single medium. The elements of this
modular model will be applicable to propagation in both isotopic and general anisotropic
media. In particular, we will describe the modular Gaussian approach for the immersion
setup shown in Fig. 1 where a Gaussian beam is radiated at oblique incidence through a
curved fluid-solid interface and indicate how that case can be easily extended to multiple
interfaces. The modular model, however, is also directly applicable to a variety of contact
and angle-beam testing configurations as well.
CP657, Review of Quantitative Nondestructive Evaluation Vol. 22, ed. by D. O. Thompson and D. E. Chimenti
© 2003 American Institute of Physics 0-7354-0117-9/03/S20.00
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FIGURE 1. Propagation of a Gaussian beam through a curved interface between a fluid and an anisotropic
solid. The distances sv and s2 are taken as the propagation distances along the central axis of the Gaussian
beam. In the solid, s2 is measured along the group velocity direction for a particular refracted wave (only one
of three possible refracted waves is shown). The y3 coordinate is taken along the direction of the slowness
vector in the anisotropic solid and ( y l 9 y 2 ) are coordinates orthogonal to that y3 axis, with yl in the plane of
incidence and y2 normal to that plane.
PROPAGATION OF A GAUSSIAN BEAM - FLUID
For the geometry of Fig.l, we will assume that a Gaussian velocity profile is present at
the "transducer" and propagates as a Gaussian beam into the fluid as shown in Fig. 2. The
velocity in the Gaussian beam can be written as
(see Fig. 2), where x = (^, x2 ) , e is a unit vector in the direction of propagation and Mj is a
complex- valued symmetrical 2x2 matrix. The distance sl is along the central axis of the
Gaussian beam and cpl is the wave speed of the fluid (medium l).This Gaussian beam can
be shown to satisfy the wave equation in the fluid (in the paraxial approximation) if [3]
(2)
and
M, (Sl) = [Df'M, (0) + cr" ] [Bf^M, (0) + Af"
(3)
where V} (0),M, (0) are the known starting amplitude and phase values in the Gaussian at
the "transducer" location (sj = 0).
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FIGURE 2. Propagation of a Gaussian beam in a fluid from a "transducer" at sl = 0 where the Mt (0) phase
matrix and amplitude, Vl (0), are assumed to be known.
The four 2x2 "propagation" matrices
1 0
yvp .
0 1
0 0
0 0
C
plSl
0
T\prop _
\
—
U
c
plsl
1 0
0
(4)
TRANSMISSION OF A GAUSSIAN BEAM ACROSS AN INTERFACE
Equation (1) describes the changes in the Gaussian beam as it propagates in the fluid.
When this Gaussian beam strikes a curved fluid-solid interface (Fig.l), reflected and
transmitted Gaussian beams are generated. In the anisotropic solid, there is a transmitted
quasi P-wave (qp) and two transmitted quasi S-waves (qsl9qs2). The velocity field for a
transmitted Gaussian beam of type a (a = qp, qsl, qs2) can be written as
(5)
where ua2 is the magnitude of the group velocity, ua2, of the anisotropic solid (medium 2)
for a wave of type a, <T* is the corresponding polarization vector and y =(yl9y2) are
coordinates perpendicular to y3 which is taken in the direction of the slowness vector in
medium 2. The distance s2 is taken along the central ray of the refracted Gaussian beam in
the direction of the group velocity. For an isotropic solid, of course, s2 is measured along
y3 and ua2 = ca2 where ca2 is the phase velocity for a wave of type a (see Fig. 1).
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FIGURE 3. Interaction of a Gaussian beam with a curved fluid/anisotropic solid interface. The angle Oa2 is
the angle measured from the interface normal to the direction of the slowness vector, sa2, in the anisotropic
solid for a wave of type a (only one refracted wave shown). The axis of the refracted Gaussian beam,
however, is along the group velocity direction (not shown).
To obtain the amplitude, V", and polarization vector, da, of the wave transmitted in the
solid at the interface (s2 =0)in the paraxial approximation is relatively easy since in that
approximation both those quantities can be found by solving for the problem of the
transmission of a plane wave at a planar interface [3]. Thus, we find
(6)
where T^ is the plane wave transmission coefficient for an incident wave of type J3 and a
transmitted wave of type a. In Eq. (6) the distance sl is now the distance to the interface
(see Fig. 1). This result is valid for both isotropic and anisotropic elastic solid problems as
long as the appropriate plane wave transmission coefficient and polarization are used.
Obtaining M^(0)at the interface is more complicated. It involves matching the phases of
the incident and transmitted waves at the interface and approximating the interface surface
(if it is curved) to second order near the point where the central ray of the incident Gaussian
strikes the interface. Space does not allow us to give those details here, but they can be
found in [3]. The result is that we can find M" (0) in terms of M} (sl) in a form identical to
Eq. (3), i.e.
(7)
For an fluid/anisotropic solid interface the 2x2 transmission matrices appearing in Eq. (7)
are given by (see Fig. 3)
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A trans _
o o
cos0
C trans
r
2
-<sin<9a2 /ca2)cos0 pl
-<sin<9a2 1 ca2)
~ Ja
Trx/raw _
cos 9a2 - u° sin 0a2 1 ca2
0
cos 0a2 - u" sin 0a2 I ca2
1
(8)
where
(9)
and (/211,^12 = h2l,h22) are the curvatures of the interface (in and perpendicular to the plane
of incidence) where the central ray of the incident wave strikes that interface. The^wf,^)
in Eq. (8) are the components of the group velocity vector, u a2 , along the (yl,y2) axes,
respectively, for a wave of type a. For an isotropic solid u" = u" = 0. It can be seen from
Eq. (8) that interface curvature effects and beam skewing effects get intermixed for
anisotropic materials. This makes these expressions more complex than the isotropic case.
PROPAGATION OF A GAUSSIAN BEAM - ANISOTROPIC ELASTIC SOLID
Waves traveling in an isotropic or anisotropic solid do not satisfy wave equations.
However, using high frequency asymptotic ray theory, it can be shown that a Gaussian beam
travels along the group velocity direction and satisfies forms similar to the fluid case [3],
namely
det
(10)
and
(11)
where
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1 0
0 1
C prop
i
_
—
0
0
0 0
\ca2-2Ca)s2
-Das2
-D"s2
(ca2-2E"}s2
ua2
D prop
o
_
—
(12)
1 0
0 1
In Eq. (12) ua2 is again the magnitude of the group velocity. The terms
a
(C ,Da,Ea}represent deviations of the slowness surface curvatures (as measured in the
slowness coordinates ( y i 9 y 2 ) ) from the isotropic case along the refracted ray [4]. In the
isotropic case Ca = Da = Ea = 0. These curvature terms can be obtained by expanding the
y3 component of the slowness vector, s a2 , (s") to second order in the (y19y29y3)
coordinates in the form [4]
(13)
where the matrix K" is given by
\\ca2-2Ca
2\ -D°
-Da
(14)
For some simple types of anisotropic media the curvature terms can be expressed in
analytical form. In general, however, they must be obtained numerically from the values of
the slowness surface in the neighborhood of the refracted ray.
A MODULAR GAUSSIAN BEAM MODEL
If we combine all of our previous results, we now have an explicit expression for the
Gaussian beam in the solid where the velocity is given by
det
(15)
+s2 Iua2 +yTMa2(s2)y /2Equation (15) can be effectively used to model the Gaussian beam in the solid for the
geometry of Fig. 1 . Unfortunately, if the same formulation just outlined is continued for
Gaussian beams that are again transmitted or reflected multiple times at additional
interfaces, the calculation of the requisite M^ matrices becomes increasingly complex
algebraically. This complexity, however, can be avoided by writing terms such as M" (s2)
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directly in terms of the known Mj(0) by introducing "global" matrices (A G ,B G ,C G ,D G )
where
AG
CG
BG
DG
A prop
Ttprop ~] [~ A trans
Titrans ~11~ A prop
ir»/
jyprop
^trans
j^
Qtrans
Qprop
(16)
In terms of these global matrices, then one can show that
1
(17)
Similar forms identical to Eq. (17) can be written for M^ (0) and M, (sl )provided that the
appropriate global matrix is used. For example, Eq. (3) is already in the form of Eq. (17)
with the global matrices in that case just the propagation matrices for the fluid. This modular
way of expressing the solution is very convenient to generalize Eq. (15) for M
transmissions/reflections. In that case we have
r (M-H
\]
V m=1
/J
•exp ia>\
ia> ^sm/um-t
L
where Tmm+l is the appropriate transmission or reflection coefficient for the mth interface
and um is the magnitude of the group velocity for the appropriate wave in the mth medium.
In this case we have
"
(19)
in terms of the global matrices for the entire set of multiple propagation and
transmission/reflection matrices with similar relations holding for all the other Mm matrix
terms appearing in Eq. (18) in terms of the appropriate global matrices relating them to
M, (0) . It is also possible to use this modular approach to replace the product of terms
appearing in the "amplitude" part of Eq. (18) with a single square root term, thus reducing
Eq.(18) to a form that is quite similar to that for the propagation of a Gaussian beam in a
single medium. However, for numerical purposes it is more convenient to leave these terms
in the form of Eq. (18) since in any form that is used the square roots must be taken of
quantities that are complex and those square roots need to be properly interpreted. For all the
M w matrices that appear in Eq. (18) this is not a problem since one can show that in the
principal coordinates of those matrices, the determinant can be written as the product of two
complex numbers, both with positive imaginary parts [3]. Thus, the square roots of these
complex numbers also must be positive.
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SUMMARY AND DISCUSSION
We have described a highly modular model that describes the propagation of a Gaussian
beam in a general anisotropic solid with multiple interfaces. Using the approach of Wen and
Breazeale, by the superposition of as few as 10 such Gaussian beams, one can model the
corresponding wave field of a circular planar or focused piston source [5]. Thus, in this
manner one can also arrive at a highly modular multi-Gaussian beam model and obtain a
highly efficient formulation for modeling the wave fields of ultrasonic transducers in very
complex testing configurations.
ACKNOWLEDGEMENTS
For L.W. Schmerr this work was supported by the National Science Foundation
Industry/University Cooperative Research Program at the Center for NDE, Iowa State
University. A. Sedov was supported by the Natural Sciences and Engineering Research
Council of Canada.
REFERENCES
1.
2.
3.
4.
5.
Minachi, A., Margetan, F. J., and R. B. Thompson," Reconstruction of a piston
transducer beam using multi-Gaussian beams (MGB) and its applications," Review
of Progress in Quantitative Nondestructive Evaluation, D. O. Thompson and D. E.
Chimenti, Eds., Plenum Press, N.Y., 17A, 907-914,1998.
Schmerr, L.W.," A multi-Gaussian ultrasonic beam model for high performance
simulations on a personal computer," Matls. EvaL, 58, 882-888,2000.
Cerveny, V., Seismic Ray Theory, Cambridge University Press, Cambridge, UK,
2001.
Schmerr, L. W., and A. Sedov, "Gaussian beam propagation in anisotropic,
inhomogeneous elastic media," Review of Progress in Quantitative Nondestructive
Evaluation, D. O. Thompson and D. E. Chimenti, Eds., American Institute of
Physics, Melville, N.Y., 21 A, 123-129, 2002.
Wen, J. J., and M. A. Breazeale, "A diffraction beam field expressed as the
superposition of Gaussian beams," J. Acoust. Soc. Am., 83,1752-1756, 1988.
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