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J. Appl. Phys. 131, 123102 (2022); https://doi.org/10.1063/5.0081237 131, 123102
© 2022 Author(s).
Rayleigh wave and super-shear evanescent
wave excited by laser-induced shock at
a soft solid–liquid interface observed by
photoelasticity imaging technique
Cite as: J. Appl. Phys. 131, 123102 (2022); https://doi.org/10.1063/5.0081237
Submitted: 08 December 2021 • Accepted: 05 March 2022 • Published Online: 28 March 2022
Thao Thi Phuong Nguyen, Rie Tanabe-Yamagishi and Yoshiro Ito
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Rayleigh wave and super-shear evanescent
wave excited by laser-induced shock at a soft
solid–liquid interface observed by photoelasticity
imaging technique
Cite as: J. Appl. Phys. 131, 123102 (2022); doi: 10.1063/5.0081237
View Online Export Citation CrossMar
k
Submitted: 8 December 2021 · Accepted: 5 March 2022 ·
Published Online: 28 March 2022
Thao Thi Phuong Nguyen,
1,2,a)
Rie Tanabe-Yamagishi,
3
and Yoshiro Ito
4
AFFILIATIONS
1
Institute of Research and Development, Duy Tan University, 550000 Danang, Vietnam
2
Faculty of Natural Science, Duy Tan University, 550000 Danang, Vietnam
3
Department of Intelligent Mechanical Engineering, Fukuoka Institute of Technology, 3-30-1 Wajiro-higashi, Higashi-ku,
Fukuoka 811-0295, Japan
4
Department of Mechanical Engineering, Nagaoka University of Technology, 1603-1 Kamitomioka, Nagaoka, Niigata 940-2188,
Japan
Note: This paper is part of the Special Topic on Shock Behavior of Materials.
a)
Author to whom correspondence should be addressed: thaonguyen@duytan.edu.vn
ABSTRACT
We investigated laser-induced shock excitation of elastic surface waves at a free surface and a soft solid–liquid interface using a custom-
designed photoelasticity imaging technique. Epoxy-resin and pure water were selected as the solid and liquid media. The elastic surface
waves were excited via a shock process induced by focusing a single nanosecond laser pulse on the solid surface. To confirm the experimen-
tal observations, the roots of the Rayleigh and Stoneley equations were calculated. For a free surface, we present an entire-field observation
of elastic surface waves, which includes a super-shear evanescent wave (SEW) that propagates faster than the shear wave but slower than the
longitudinal wave. For a soft solid–liquid interface, we demonstrate the presence of a non-leaky Rayleigh wave that corresponds to a real
root of the Stoneley equation. We also evidence the existence of a SEW that propagates 1.7 times faster than the shear speed in the solid and
corresponds to a complex conjugate root of the Stoneley equation. These results correct the previously accepted notion that the Scholte wave
is the only surface wave that can be generated at a soft solid–liquid interface.
Published under an exclusive license by AIP Publishing. https://doi.org/10.1063/5.0081237
INTRODUCTION
The excitation of elastic surface waves by pulsed laser has
received widespread attention in the last few decades.
1
Compared
to conventional methods, the laser beam has proved most useful
for generating and investigating elastic surface waves, thanks to its
non-contact feature and the ability to generate wide-frequency-
band acoustic disturbances.
2
The smallest wavelength of a laser-
excited surface elastic wave can reach a few micrometers, which
corresponds to a frequency approaching 1 GHz and limits the pulse
length to the nanosecond range.
3
This advantage provides opportu-
nities to separate different wave modes propagating on the interface
and, thus, is important to study the interface waves traveling at
only slightly different speeds.
2
The excitation of elastic surface
waves demonstrates great potential for a variety of applications,
including nondestructive testing,
4–6
investigation of the elastic
properties of surfaces and thin films,
7,8
evaluation of nanostructure
and nanomaterials,
9
and more recently, elastography.
10
Depending on the fluence of the laser pulse, we can classify
the mechanism for the laser excitation of elastic surface waves into
the thermoelastic regime and the ablative regime.
11
Thermoelastic
expansion is the most well-known mechanism for transforming a
laser pulse into acoustic waves. In this regime, low laser irradiation
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J. Appl. Phys. 131, 123102 (2022); doi: 10.1063/5.0081237 131, 123102-1
Published under an exclusive license by AIP Publishing
induces the localized thermal expansion of the target and generates
surface disturbances.
10,12
In the ablative regime, the laser pulse
ablates the target material to form a high-pressure plasma. Upon
expansion, the plasma induces a shock into the surrounding
media.
13
This laser-induced shock drives an impulse into the solid
and excites four kinds of waves: longitudinal wave (P-wave), shear
wave (S-wave), lateral waves, and elastic surface waves.
14
Compared
to the thermoelastic regime, the shock regime can induce much
stronger acoustic impulses and is a newly established non-contact
elastic surface wave excitation technology.
13
Although the laser-
induced shock process has been thoughtfully investigated in the
last decades, the laser-induced shock excitation of elastic surface-
waves has not been investigated much in the literature.
For a free surface, the Rayleigh wave is the most popular
elastic surface wave that can be excited. The Rayleigh surface wave
is first discovered theoretically as a root of the Rayleigh equation in
the late 19th century and was confirmed later, thanks to many
experimental investigations. In the absence of losses, the Rayleigh
wave is continuous, nondispersive, and exists in the entire fre-
quency range, where the elastic medium can be considered uniform
and isotropic.
15
For studying elastic surface wave excited at the interface
between two media, one must consider the Stoneley wave equation.
In the 1950s, Scholte described a specific case of the Stoneley wave
equation where one medium is liquid and discovered the Scholte
wave.
4
The Scholte wave travels along a solid–liquid interface, has
its energy mostly concentrated in the liquid, and propagates
without attenuation at a speed smaller than the Rayleigh speed if
the viscosity of the liquid is negligible.
4,16
For a general solid–liquid
interface, a Scholte wave can always be generated. To discuss the
excitation of the Rayleigh wave at a solid–liquid interface, we need
to distinguish two situations: a hard solid–liquid interface and a
soft solid–liquid interface. A hard solid–liquid interface refers to
the liquid sound speed smaller than the solid shear speed. Here,
the Rayleigh wave has a speed larger than the sound speed in the
liquid, thus radiating energy in the liquid and becoming leaky
(so-called leaky Rayleigh wave).
17–19
This wave is the least
damped, thus most observable of all the leaking wave types,
20
and
has been thoroughly discussed in the literature. A soft solid–
liquid interface refers to the interface where the liquid sound
speed is larger than the solid shear speed. In contrast to the hard
solid–liquid interface, the soft solid–liquid interface has been
rarely investigated. In 1999, Padilla et al. showed experimentally
and theoretically that for a soft solid–liquid interface, the complex
root that gives rise to the leaky Rayleigh wave becomes real and
gives rise to a non-Leaky Rayleigh wave.
18
However, a conflict
result was presented by Glorieux et al.
19
In recent years, the evidence of an inhomogeneous wave that
propagates along a free surface faster than the S-wave but slower
than the P-wave renews the research interest in this topic. This
elastic surface wave has a speed almost twice the S-wave speed and
couples into the S-wave during its propagation. Because of this cou-
pling, it loses energy and decays in its propagating direction.
14,21,22
Throughout the literature, this wave has been called the leaky
wave,
23
the super-shear Rayleigh wave,
21
the pseudo P-wave,
20
non-
geometric P-S wave,
24
or most recently, the super-shear evanescent
wave.
22
We will use “super-shear evanescent wave”(SEW) when
referring to this wave in this paper since it best describes the leaky
behavior and super-shear propagation speed of the wave.
The existence of SEWs attracts significant interest in seismol-
ogy, engineering, and nondestructive testing. Super-shear rupture, a
rupture that propagates at a speed faster than the shear wave, has
recently been observed in large seismic events.
25
These events relate
to the existence of the SEW and break the theoretical limit that
cracks should not propagate faster than the speed of the Rayleigh
wave.
21
The existence of the SEW can also lead to misidentification
of elastic properties of structures being investigated if one picking
phase velocities of the SEW as the Rayleigh wave phase velocities
for inversion.
23
In recent years, the SEW has also been proposed as
a non-contact tool to measure the elastic properties of soft mate-
rial.
22
Altogether, understanding the process driving SEW genera
tion and propagation characteristics is crucial for both minimizing
surface wave-induced fracture as well as for improving the efficacy of
surface elastic waves in nondestructive testing applications. Although
SEWs have been studied theoretically and experimentally for a free
surface, the existence of a similar wave at a solid–liquid interface has
not been discussed that much. In general, a SEW is hard to detect
because it may not separate from the Rayleigh wave until propagating
some distance from its source, at which point the evanescent nature
makes measurement difficult.
22
Moreover, the feasibility of acquiring
this type of wave may require the shear speed in the solid smaller
than the sound speed of the liquid
26
(i.e., soft solid–liquid interface),
a condition that has not been discussed much in the literature.
Under these circumstances, it is the purpose of this paper to
examine the laser-induced shock excitation of Rayleigh wave and
SEW at a soft solid–liquid interface experimentally. One difficulty
in studying elastic surface waves at a solid–liquid interface is the
requirement of a method that can capture the whole-field elastic
disturbances at a resolution sufficient to separate different wave
modes. The piezoelectric transducer is the conventional method for
studying elastic surface waves. In this technique, transducers are
connected to the sample surface and record the arrival time of the
wave trains.
18
The major disadvantage of this method is the mea-
suring accuracy depends on the contact conditions between the
transducer and the specimen.
27,28
Optical detection has been
known as a powerful non-contact method for investigating surface
waves. We can count here, for example, the beam reflection tech-
nique,
29
Brillouin scattering,
30
optical interferometry,
30
and inter-
ferometric method based on the photoelastic effect.
16,31
The
disadvantages of these methods are that they require a high quality
of the surface and rely on the spatial derivative of the surface dis-
placement. Various optical interferometric techniques have been
developed to improve the detection sensitivity and robustness for
industrial applications. For example, the signal quality and reliabil-
ity of laser Doppler vibrometers can be significantly improved by
applying diversity reception technique.
32
By integrating detector
arrays, a classic interferometric design such as a Michelson or
Mach–Zehnder interferometer can be transformed into a robust
system with high sensitivity and can endure a tough industrial envi-
ronment.
33
Another non-contact approach to detect elastic surface
waves with high stability and sensitivity in the harsh environment
is the interferometer schemes employing photorefractive crys-
tals.
34,35
Sagnac-based interferometers have also been developed to
provide an alternative, cost-effective, and easy-to-use scheme for
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J. Appl. Phys. 131, 123102 (2022); doi: 10.1063/5.0081237 131, 123102-2
Published under an exclusive license by AIP Publishing
industrial inspection of acoustic emission.
36
A fiber-optic Sagnac
interferometer developed by Pelivanov et al. has been demonstrated
to have the detection sensitivity approaching a piezoelectric trans-
ducer with a reduced sensitivity to roughnes,
37,38
thus enabling the
detection of surface acoustic waves propagating over rough surfaces
of composite materials.
39,40
To obtain a direct whole-field observation of the induced
elastic surface wave, we use the approach of dynamic photoelastic
imaging. In our previous works,
41–43
we have combined a high-
speed pump-and-probe imaging system and the photoelasticity
technique to develop a custom-designed photoelasticity imaging
system. This system can give a direct and whole-field visualization
of the transient stress distribution inside a solid with high temporal
and spatial resolutions. In addition, it provides a normal shadow-
graph image in the liquid phase at the same time. With these abili-
ties, the photoelasticity imaging technique can serve as a powerful
tool for non-contact, direct, and whole-field visualization of the
elastic surface waves. Recently, Zhang et al. have employed a
similar dynamic photoelastic/shadowgraph imaging system to study
the transient stress waves excited by the incident shock at a fluid–
solid boundary.
44
Although the resolution was low because of the
small intensity of the phenomenon under investigation, this tech-
nique provided a whole-field image of surface waves. Their work,
however, studied a hard solid–liquid interface and focused on
dynamic fracture mechanisms at the solid surface.
In this research, we selected epoxy-resin and water as solid
and liquid media, as they compose a soft solid–liquid interface and
Poison’s ratio (ν) of epoxy-resin which is within the domain to
generate the SEW (ν.0:263).
23,24
Furthermore, epoxy-resin has a
high photoelasticity constant that allows clear images of the
induced elastic waves. The elastic surface waves were excited by
focusing a single nanosecond laser pulse on the solid surface. We
first observed the elastic surface waves at a free epoxy-resin surface
to show the ability of the photoelasticity imaging technique as a
potential method for investigating the elastic surface waves. Then,
we used this method to observe the surface disturbances induced at
the interface between epoxy-resin and water. The roots of the
Rayleigh and Stoneley equations were calculated to verify the exper-
imental observations. Our results provide direct and visual evidence
that a non-leaky Rayleigh wave and a SEW can be excited by a
laser beam at a soft solid–liquid interface.
CHARACTERISTIC EQUATIONS FOR SURFACE WAVES
The presentation of the Rayleigh equation and Stoneley equa-
tion in this section follows the work of Ansell.
45
Rayleigh equation
The Rayleigh equation for a wave number kalong the interface
can be expressed as
45
R;4uffiffiffiffiffiffiffiffiffiffi
ul
pffiffiffiffiffiffiffiffiffiffiffi
un
pþ(2un)2¼0, (1)
where u¼k2,n¼ω2
c2
s,l¼ω2
c2
p, with c
s
and c
p
are the S-wave speed
and P-wave speed in the solid, respectively, and ωis the angular
frequency of the waves.
The square roots in R give rise to two branch points at u¼l
and u¼n. The top sheets (referred to as “+ sheets”) are defined by
π
2arg ffiffiffiffiffiffiffiffiffiffi
ul
p,arg ffiffiffiffiffiffiffiffiffiffiffi
un
p
π
2. The Riemann surface for R
has four sheets designated by a group of two signs (+,+). Each of
the signs stands for the Riemann sheet of ffiffiffiffiffiffiffiffiffiffi
ul
p,ffiffiffiffiffiffiffiffiffiffiffi
un
p
. The
Rayleigh equation, thus, can take two forms on the four sheets,
4uffiffiffiffiffiffiffiffiffiffi
ul
pffiffiffiffiffiffiffiffiffiffiffi
un
pþ(2un)2¼0
and
þ4uffiffiffiffiffiffiffiffiffiffi
ul
pffiffiffiffiffiffiffiffiffiffiffi
un
pþ(2un)2¼0:
On squaring (1), filtering out the frequency dependence by writing
the equation intern of wave speed c¼ω
k, and extracting the irrele-
vant root c¼0, the Rayleigh equation becomes the well-known
polynomial,
46
Ψ¼c2
c2
s
3
8c2
c2
s
2
þ24 16 c2
s
c2
p
!
c2
c2
s16 1 c2
s
c2
p
!
¼0:(2)
Equation (2) always has three roots for cthat have the real
part larger than zero.
47
One of these three roots is always real and
gives rise to the well-known Rayleigh wave.
45,47
Depending on
Poisson’s ratio of the solid medium, the other two roots can be real
or complex.
45,47
For Poisson’s ratio larger than 0.263, the two roots
are complex conjugates of each other and give rise to the SEW.
22,23
Stoneley equation
The form of the Stoneley equation given for the wave number
kalong the interface is
45
S;ffiffiffiffiffiffiffiffiffiffiffiffiffi
um
p4uffiffiffiffiffiffiffiffiffiffi
ul
pffiffiffiffiffiffiffiffiffiffiffi
un
pþ(2un)2
hi
þρn2ffiffiffiffiffiffiffiffiffiffi
ul
p¼0,
(3)
where
u¼k2,ρ¼ρliquid
ρsolid
,n¼ω2
c2
s
,m¼ω2
c2
l
,l¼ω2
c2
p
,
where csand cpare the S-wave and P-wave speeds in the solid and
clis the sound speed in the liquid. ρliquid and ρsolid are the densities
of the liquid and solid, respectively. kis the wave number in the
horizontal direction, and ωis the angular frequency of the waves.
Three square roots in S give rise to three branch points in the
complex uplane at u¼l,u¼m, and u¼n. The branch cuts
were taken as (1,l) for ffiffiffiffiffiffiffiffiffiffi
ul
p,(1,m) for ffiffiffiffiffiffiffiffiffiffiffiffiffi
um
p, and
(1,n) for ffiffiffiffiffiffiffiffiffiffiffi
un
p. The top Riemann sheet (referred to as the
“+ sheet”) for each of the square roots is given by
π
2(arg ffiffiffiffiffiffiffiffiffiffi
ul
p,arg ffiffiffiffiffiffiffiffiffiffiffiffiffi
um
p,arg ffiffiffiffiffiffiffiffiffiffiffi
un
pπ
2:
The Riemann surface for the function S has eight sheets, each
of which are defined by a combination of sheets
ffiffiffiffiffiffiffiffiffiffi
ul
p,ffiffiffiffiffiffiffiffiffiffiffiffiffi
um
p,ffiffiffiffiffiffiffiffiffiffiffi
un
p
. And so, the general equation (3) can
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Applied Physics ARTICLE scitation.org/journal/jap
J. Appl. Phys. 131, 123102 (2022); doi: 10.1063/5.0081237 131, 123102-3
Published under an exclusive license by AIP Publishing
be written as eight separate equations,
45
S1;ffiffiffiffiffiffiffiffiffiffiffiffiffi
um
p4uffiffiffiffiffiffiffiffiffiffi
ul
pffiffiffiffiffiffiffiffiffiffiffi
un
pþ(2un)2
hi
þρn2ffiffiffiffiffiffiffiffiffiffi
ul
p¼0,
S2;ffiffiffiffiffiffiffiffiffiffiffiffiffi
um
p4uffiffiffiffiffiffiffiffiffiffi
ul
pffiffiffiffiffiffiffiffiffiffiffi
un
pþ(2un)2
hi
ρn2ffiffiffiffiffiffiffiffiffiffi
ul
p¼0,
S3;ffiffiffiffiffiffiffiffiffiffiffiffiffi
um
pþ4uffiffiffiffiffiffiffiffiffiffi
ul
pffiffiffiffiffiffiffiffiffiffiffi
un
pþ(2un)2
hi
ρn2ffiffiffiffiffiffiffiffiffiffi
ul
p¼0,
S4;ffiffiffiffiffiffiffiffiffiffiffiffiffi
um
pþ4uffiffiffiffiffiffiffiffiffiffi
ul
pffiffiffiffiffiffiffiffiffiffiffi
un
pþ(2un)2
hi
þρn2ffiffiffiffiffiffiffiffiffiffi
ul
p¼0,
S1;S1¼0, S2;S2¼0, S3;S3¼0, S4;S4¼0:
(4)
To solve the set of equations (4), we use the function T intro-
duced by Ansell,
45
where T is defined by T ¼S1S2S3S4,
T¼n4(ul)ρ2(um)þ4uffiffiffiffiffiffiffiffiffiffi
ul
pffiffiffiffiffiffiffiffiffiffiffi
un
pþ(2un)2
no
2
n4(ul)ρ2(um)4uffiffiffiffiffiffiffiffiffiffi
ul
pffiffiffiffiffiffiffiffiffiffiffi
un
pþ(2un)2
no
2
:
(5)
By writing equation T ¼0 in term of wave speed c¼ω
k,wehave
c
cs
8
1c
cp
2
!
ρ21c
cl
2
!
þ4ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1c
cs
2
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1c
cp
2
sþ2c
cs
2
!
2
8
<
:9
=
;
2
2
43
5
c
cs
8
1c
cp
2
!
ρ21c
cl
2
!
4ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1c
cs
2
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1c
cp
2
sþ2c
cs
2
!
2
8
<
:9
=
;
2
2
43
5¼0:(6)
The Stoneley wave equation can give rise to either real roots
or complex roots.
45
The slowest real root of the Stoneley equa-
tion for a solid–liquid interface corresponds to the Scholte
wave.
48
For a hard solid–liquid interface, one of the complex
roots gives rise to the leaky Rayleigh wave.
17
While the roots that
correspond to the Scholte wave and leaky Rayleigh wave are rec-
ognized and treated extensively in the literature, the other roots
have not been discussed this much and have usually been con-
sidered trivial or extraneous.
EXPERIMENTAL METHODS
Laser-induced shock excitation of elastic surface
waves
We excited the elastic surface waves by focusing a single
1064-nm laser pulse with a full width at half maximum (FWHM)
of 13 ns through a 5× objective lens onto an epoxy-resin surface
to a spot approximate 50 μm in radius. When interacting with the
target, the laser beam gasifies and ionizes the target material to
form a plasma. This plasma continues to absorb the laser pulse
energy to get a high-temperature, high-pressure status. Upon
expansion, this plasma drives a shock wave into the upper
medium and an impulse on the target surface that excites the
elasticwavesintothesolid.Figures 1(a) and 1(b) show the two
configurations of the experiment in this research. In the free
surface configuration [Fig. 1(a)],thetargetisputintheair.
We covered the epoxy-resin surface with a thin layer of black
paint to enhance the absorptivity because laser ablation of
epoxy-resin in air generates weak elastic waves that cannot be
observed.
41,49
In the solid–liquid configuration [Fig. 1(b)], the
target is put in a glass vessel filled with pure water. The water–air
interface is 5 mm above the target. The laser beam is directed
through the water and focused on the target surface. In this
experiment, we did not apply an absorptive coating to the target
because the impulse was significantly increased, thanks to the
confinement effect of the liquid.
41,50
The target was put on an
X–Y translational stage and was moved after each shot to refresh
the surface. The pulse energy was 60 mJ.
Photoelasticity imaging system
We used a custom-designed time-resolved photoelasticity
imaging technique to observe the laser-induced elastic surface
waves. In this technique, a circular polariscope and fast imaging
system are used to achieve a whole-field visualization of the tran-
sient stress inside the solid [Fig. 1(c)]. The imaging system was
based on a pump-and-probe system with an intensified charge-
coupled device (ICCD) camera operated in a gated mode to
capture a single-shot image for each ablation pulse.
49
This
system captured photoelasticity images for epoxy-resin block
while it gave normal shadowgraph images for non-birefringent
media, water phase, in our case. We used a Nd:YAG laser
(Powerlite 8000) to induce the elastic waves, and another
Nd:YAG laser (NY 82) to provide the illuminating light
(532 nm, FWHM = 6 ns). We defined the delay time as the inter-
val between the pump pulse and the probe pulse. Although this
technique has a high temporal resolution of nanoseconds, it
allows only single-shot observations where one image is cap-
tured per ablation pulse,
49
thus measurements with varying
delay durations must be conducted to capture the whole time-
dependent evolution of the elastic waves.
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J. Appl. Phys. 131, 123102 (2022); doi: 10.1063/5.0081237 131, 123102-4
Published under an exclusive license by AIP Publishing
RESULTS AND DISCUSSION
Elastic surface waves observed at a free surface
In Fig. 2, we present a visualization of laser-induced elastic
surface waves at a free surface. Figure 2(a) shows the evolution of a
shock wave in the air and elastic waves induced in the solid,
observed from 100 to 1900 ns after irradiation. Figure 2(b) shows
detailed images of the waves observed at 700 ns. Although each
frame was taken for a different event, the propagation of excited
waves can be well followed, thanks to the good reproducibility of
the events.
As early as 100 ns after irradiation, we can observe a shock
wave that expanded fast into the air. At the laser fluence used in
our experiment, the induced plasma expands following a
laser-supported-radiation-wave model, and thus the shock front
does not have a semi-circular shape but is slightly elongated in the
vertical direction.
50
Inside the shock wave in the air, there was a
shadow of the ablated material. Inside the solid, we could observe
elastic wavefronts. From 300 ns, the longitudinal wave (P-wave)
and the shear wave (S-wave) can be distinguished. From 700 ns,
two surface waves are visible [Fig. 2(b)]. The 1st surface wave is
characterized as a dark pattern propagating along the interface
slightly slower than the S wave. The 2nd surface wave is character-
ized as a tangent line to the S-wave front with one end is on the
interface. This wave vanished fast and its location on the interface
is difficult to be determined after 900 ns. From 1000 ns, the P-wave
was almost undetected, but the S-wave, 1st, and 2nd surface waves
can be observed until 1900 ns.
We tracked the distance traveled by the P-wave, S-wave, and
the 1st surface wave with time. To measure the distances traveled
by the P-wave and S-wave, we centered a circle at the laser-focal
point and fit it to the wavefronts. The distances traveled by these
wavefronts were elucidated from the radius of the circle. The results
are presented in Fig. 3 with the plots presenting the mean value
measured at each delay time, and the average deviation at each
delay time is within 1%. To estimate the wave speeds, we fitted a
linear regression model to each data set. High goodness of fit
(R
2
> 0.99) confirmed the reproducibility of the result. We found
that the P- and S-wave propagated at constant speeds of 2550 +40
and 1110 +20 ms1, respectively. The 1st surface wave propagated
at a constant speed of 1030 +20 ms1, which is about 0:93 times
the S-wave speed.
The 2nd surface wave had one end traveling along the
solid–air boundary and appeared as a tangent line to the S-wave
front. The propagation of the 2nd surface wave on the interface was
difficult to track because it vanished at a large delay time and its
position on the boundary was not clear to locate. On the other
hand, the angle θbetween it and the interface showed good
stability and was measured to be θ¼32+1. Thus, from the
S-wave speed of cs¼1110 +20 ms1, we estimated that the 2nd
surface wave propagates along the free surface at the speed
c2¼cs
sinθ¼2100 +100 ms1, which approximates 1.9 times the
S-wave speed.
To confirm the nature of the observed elastic surface waves,
we solved the Rayleigh equation using Eq. (2) for epoxy-resin with
cs¼1110 ms1and cp¼2550 ms1. The results return a real root
c¼0:94cs, which is close to the speed of the 1st wavefront, and a
complex conjugate root c0¼(1:92 +0:37i)cs, of which the real
part is close to the speed of the 2nd wavefront.
FIG. 1. Simplified schematic of the experimental setup. Laser-induced shock excitation of elastic waves at a free surface (a) and at a soft solid–liquid interface (b).
(c) Schematic diagram of the photoelasticity imaging system (BE: beam expander, P: polarizer, Q: quarter wave plate, A: analyzer, ND: neutral density filter, and BP: band-
pass filter).
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J. Appl. Phys. 131, 123102 (2022); doi: 10.1063/5.0081237 131, 123102-5
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It is well known that the real root of the Rayleigh equation
gives rise to the Rayleigh wave speed. We, thus, can conclude that
the 1st wavefront is the Rayleigh wave.
The complex conjugate root of the Rayleigh equation has been
demonstrated theoretically to give rise to a SEW which propagates
faster than the shear wave and is weakly transverse in polariza-
tion.
14,20
Due to matching tangential wave vectors at the surface, it
couples into the plane shear wave and decays as it propagates along
the surface.
14
Although the physical admissibility of this wave has
been previously debated, current experimental observations in the
near field of excitation source confirm its existence.
21,22
It is also
demonstrated that the non-geometric wave that was reported in
many shallow high-frequency seismic surveys is a SEW.
23
In our
observation, the 2nd surface wave corresponds to a complex conju-
gate root of the Rayleigh equation. It travels faster than the shear
wave, thus radiating the energy into the solid at the S-wave speed
and appearing in the image as a tangent line to the S-wave front.
We, thus, conclude that the 2nd wavefront observed in our image is
the SEW.
Although the existence of SEW has been reported,
21–23
this
result presents a direct whole-field visualization of the SEW at a
free surface. Such a whole-field image of SEW was only provided
by simulation before. Thus, we propose the photoelasticity imaging
technique as an effective and unique method for whole-field and
direct monitoring of elastic surface waves.
Elastic surface waves observed at a soft solid–liquid
interface
Figure 4 presents the propagation of elastic waves observed at
the soft solid–liquid interface. Because of the confinement effect of
the liquid, the impulse has been significantly increased. As a result,
we observed a complicated fringe system inside the solid.
41
In the
liquid medium, we can observe the primary shock wave, reflected
wave, and the cavitation bubble, which had been described in our
previous work.
43
In the solid phase, we can identify the P-wave,
which is the fastest wavefront. Inside this wavefront are photoelastic
fringes that semi-quantitatively show the stress distribution in the
FIG. 2. Laser-induced shock excitation of elastic surface waves at a free surface observed by the photoelasticity imaging technique. (a) Evolution of elastic waves induced
in the solid. (b) Detailed image of elastic waves observed at 700 ns after irradiation.
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solid phase.
41
Because of the presence of many fringes, the
S-wavefront cannot be distinguished. However, the P-S lateral wave
can be identified. The angle at which the P-S lateral wavefront
makes to the interface can be used to calculate the S-wave speed.
Our observation at 1000ns shows that the P-S lateral wave made
an angle 28to the solid–liquid boundary, revealing that the
S-wave speed was 1180 ms
−1
(with P-wave speed was 2550 ms
−1
).
This speed is higher than the S-wave speed observed for the free
surface. This difference can be explained by the fact that that the
shear wave speed increased as the induced stress was increased.
52
In the interval of 1200–2000 ns, we can separate three waves at
the boundary. We named them #1, #2, and #3 wavefronts as stated
in Fig. 4(a). The #1 wavefront has the shape of near-vertical fringes.
The #2 wavefront was recognized as concentric semicircle fringes.
The #3 wavefront was recognized as fringes that were inclined to
the solid–liquid interface. In the photoelastic images, the number
of fringes semi-quantitatively shows the strength of the wave, and
the width of the fringe group in the propagation direction can
show the wavelength. We, thus, used the center point of each fringe
group as the track point to measure the distance traveled by each
wavefront. A closer look into the near-boundary area at earlier
delay time revealed the #4 wavefront that appeared between P-wave
and S-wave. The #4 wavefront was distinguished from the #3 wave-
front in that it travels at a higher speed. We show a magnified
observation of this wavefront from 600 to 900 ns in Fig. 4(b).
Within this period, the #4 wavefront is well distant from the group
of #1, #2, and #3 wavefronts. Before 500 ns, the #4 wave could not
be separated from other surface waves. After 900 ns, this wave was
getting dim and its position on the interface was difficult to locate.
Figure 4(c) shows the travel distance over time of the observed
wavefronts. The plots present the mean value measured at each
delay time. The average deviation at each point is within 3%. By
applying a linear fit to each data set, we can estimate the speed of
these waves. The #1 wavefront travels at 880 +40 ms1, which is
about 0.75 times the S-wave speed. The #2 wavefront travels at
1150 +60 ms1, which is slightly slower than the S-wave speed.
The #3 wavefront travels at 1590 +40 ms1, which is about 1.35
times the S-wave speed. The #4 wavefront travels at 2000 +100 ms1,
which is about 1.7 times the S-wave speed.
To discuss the nature of these observed waves, we solved the
Stoneley equation [Eq. (6)] for the interface between water and
epoxy-resin, with cl¼1480 ms1and ρliquid ¼998 kgm3for water
and cs¼1180 ms1,cp¼2550 ms1,andρsolid ¼1100 ms1kg m3
for epoxy-resin. The result gave two real roots (ignore the negative
ones) at c1¼0:779csand c2¼0:996cs. When comparing to our
observation, we found that the #1 wavefront fit root c1.The#2wave-
front fit root c2,asshowninFig. 4(c).
Besides the two real roots, Eq. (6) also gave three complex
conjugates (ignore the negative ones): c3¼(1:36 +0:313i)cs,
c4¼(1:66 +0:296i)cs, and c5¼(2:50 +0:197i)cs. In a complex
conjugate root, the real part reveals propagating speed, and the
imaginary part reveals the attenuation along the propagation direc-
tion of the wave. Among the three above conjugates, we temporar-
ily consider c5as an extraneous root since it results in a wave speed
that is larger than the P-wave speed. We then compared the roots
c3and c4with the observed wavefronts and found that wavefront
#3 fit root c3and wavefront #4 fit root c4[Fig. 4(c)].
Root c1of the Stoneley equation has been well recognized as
the Scholte wave,
2,18
and, thus, we can confirm that the #1 wave-
front observed in our photoelastic images is the Scholte wave.
When the viscosity of the liquid is negligible (like water in our
case), the Scholte wave travels along a solid–liquid interface
without attenuation.
4,16
In our photoelasticity images, the Scholte
wave is represented by a group of parallel fringes. Because the wave
was not attenuated, the number of fringes was almost unchanged
with time within the observation period.
It has been well known that for a hard solid–liquid configura-
tion, a complex conjugate root of the Stoneley equation gives rise to
a leaky Rayleigh wave. In 1999, Padilla et al. proposed that for a
soft solid–liquid interface, this complex root becomes a real root
and indicates the existence of a non-leaky Rayleigh wave on the
solid–liquid interface.
18
Our observation agrees with the result of
Padilla et al. that real roots c2of the Stoneley equation correspond
to the speed of a physical wave. We, thus, conclude that the #2
wavefront observed in our images is the Rayleigh wave. Because
this wave is not leaky (corresponding to a real root), it attenuates
weakly along the propagation direction and can still be detected in
our images until several hundred nanoseconds after irradiation.
The #4 wavefront was well fit by root c4and shows the same
properties of a SEW that is recorded at a free boundary. (1) It
arises from a complex root of the wave equation. (2) It travels faster
than the S-wave but slower than the P-wave. (3) It is tangent to the
S-wave front. (4) It attenuates quickly that could not be detected
FIG. 3. Distances traveled by the P-wave, S-wave, and the 1st surface wave
plotted against time, observed for a free surface. The plots present the mean
value measured at each delay time. The average deviation at each delay time is
within 1%. The dashed lines show the linear fit applied to each data set.
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J. Appl. Phys. 131, 123102 (2022); doi: 10.1063/5.0081237 131, 123102-7
Published under an exclusive license by AIP Publishing
after a short interval. We, thus, propose that the #4 wavefront is
a SEW.
Although the #3 wavefront fit root c3of the wave equation, the
fact that it is much stronger than the #4 wavefront and still can be
detected until 2000 ns after irradiation makes us hesitate to con-
clude that the #3 wavefront corresponds to root c3, which should
give rise to an evanescent or leaky wave that attenuates quickly. We
noticed that the speed of the #3 wavefront is very close to the speed
of the primary shock wave that is induced in the liquid.
50
We, thus,
propose a possibility that the #3 wavefront is the combination of a
lateral wave that is caused by the energy irradiation of the primary
shock wave into the solid and a surface wave that arises from the c3
FIG. 4. (a) Laser-induced shock excitation of elastic surface waves at a soft solid–liquid interface. (b) Magnified observation of the #4 wavefront. (c) A comparison
between experimentally determined (symbols) and theoretically calculated (lines) distance traveled by each elastic surface wave. c1,c2,c3, and c4are the roots of the
Stoneley equation. The plots present the mean value measured at each delay time. The average deviation at each delay time is within 3%.
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J. Appl. Phys. 131, 123102 (2022); doi: 10.1063/5.0081237 131, 123102-8
Published under an exclusive license by AIP Publishing
root. This combination may explain why the #3 wavefront is strong
and attenuates very slow. From the experiments presented in this
paper, we cannot give a solid conclusion about the #3 wavefront
and suggest that more research is needed to confirm its origin.
In short, our observation evidence that a Rayleigh wave and a
SEW can be excited by a laser beam at a soft solid–liquid interface.
For a solid–liquid interface, it is a widespread belief that only
two typical types of elastic surface waves may exist: Scholte wave
and leaky Rayleigh wave.
48,52
While the Scholte wave always exists
for a solid–liquid interface, the leaky Rayleigh wave is considered to
exist only for a hard solid–liquid interface.
17
In 1999, Padilla et al.
showed that for a soft solid–liquid interface, a Rayleigh wave can
exist.
18
The admissibility of this wave was debated later by Glorieux
et al.,
19
who concluded that a Rayleigh-type wave could only exist
when the sound speed in the liquid is at least 1.3 times smaller
than the shear speed in the solid. In the works of Padilla et al.
18
and Glorieux et al.,
19
plexiglass was used as solid, and water
was used as the liquid medium. Plexiglass has a shear speed of
1430 m/s and sound speed in water can change from 1430 ms
−1
at
5 °C to 1480 ms
−1
at 20 °C.
54
This selection of interface is within
the unstable region where csapproximate cl, and, thus, the results
can be conflict. In our experiment, the shear speed in the solid is
well under the value of sound speed in the water and, therefore,
can confirm the existence of a non-leaky Rayleigh wave at a soft
solid–liquid interface where the liquid sound speed is larger than
the solid shear speed. Because this wave is not leaky, it is difficult
to be detected by methods based on reflection at the interface or
transmission in the liquid phase.
18
This can explain why a Rayleigh
wave was normally not detected in the previous reports.
For a liquid–solid interface, the generation of a wave that
arrives between the S-wave and the P-wave has been theoretically
predicted. In 1961, Phinney described this wave mode as an intrin-
sic leaky vibration of a solid half surface that does not “see”the
liquid half-space in a significant wave.
20
This wave may not always
be physically separable on experimental records because of the
close association with the body waves.
20
Moreover, this wave is
hard to observe because the acquisition of them is challenging.
22
In
fact, the records on surface waves that arrive between the P-wave
and the S-wave on a solid–liquid interface are very limited. In seis-
mology, there were field reports on an inhomogeneous wave with
the real part of the horizontal slowness lying in the range
1/cP,p,1/cS. These waves were observed in the studies of the
non-geometric P-S wave conversion at water bottom.
54,55
The theo-
retical analysis verified that this particular mode of conversion is
excited only when the acoustic source is close to the interface and
the solid S-wave speed is lower than the liquid sound speed,
26
which is equivalent to a soft solid–liquid configuration.
In many experiments of elastic surface waves generated at a
solid–liquid interface, the SEW has not been recorded (see, for
example, Refs. 16,18,19,29,44, and 56). In these works, the SEW
can be neglected because of their exponentially decaying amplitude.
It would not separate from the other elastic surface waves until it
had traveled some distance from its sources, at which point the
evanescent nature of it makes measurement difficult.
22
Moreover,
the feasibility of acquiring this type of wave is characterized by
cs,cl,cPand a high ratio of cp/cs. Thus, a proper selection of
liquid–solid pairs is essential to detect this wave. To the best of our
knowledge, our result verifies for the first time the existence of the
SEW excited by a laser beam on a soft solid–liquid interface, thus
correcting the previously accepted notion that the Scholte wave is
the only surface wave that can be excited at such a boundary. We
propose that the dynamical feature of SEWs on a solid–liquid inter-
face should be investigated more because it has the potential to
infer S-wave information and would be useful for applications in
seismology, engineering, and nondestructive testing.
In this research, we used Rayleigh and Stoneley equations for
a plane interface problem to compute the speed of surface distur-
bances induced by a laser-induced shock impulse. This approach
may not fully account for all the interface waves which propagate in
the near field of the excitation source. Future efforts should follow
a more efficient approach using Green’s function to define all wave
features propagating at the speed between S-wave and P-wave.
CONCLUSIONS
We investigated the laser-induced shock excitation of elastic
surface waves at a free surface and a soft solid–liquid interface
using a custom-designed photoelasticity imaging technique.
For the free surface, we provided a whole-field image of the
super-shear evanescent wave and the Rayleigh waves excited by a
laser-induced shock. Such direct and whole-field visualization has
not been presented in the literature, thus demonstrating photoelas-
ticity imaging technique as a unique and potential tool for analyz-
ing elastic surface waves.
We differentiate three elastic surface waves that are excited by
the laser-induced shock at the soft solid–liquid interface. (1) The
Scholte wave travels without attenuation along the interface at a
speed equal to 0.75 times the solid shear speed. (2) The Rayleigh
wave propagates slightly slower than the shear wave and attenuates
weakly along the propagation direction. It corresponds to a real
root of the Stoneley equation and, thus, is not in a leaky mode.
(3) The super-shear evanescent wave travels 1.7 times faster than the
solid shear speed and couples into the shear wave plane. This wave
correlates to a complex conjugate root of the Stoneley equation.
It attenuates quickly and can only be observed in a brief interval.
Our result confirmed that there is a non-leaky Rayleigh wave at a
soft solid–liquid interface, the existence of which has been debated in
the literature. We also provide the first direct evidence of a super-
shear evanescent wave excited by a pulsed laser at a solid–liquid
interface. This work gives new insight into the elastic surface waves
generated at a soft solid–liquid interface by a laser-induced shock,
suggesting that more theoretical and experimental studies are needed
to clarify the dynamical features of these phenomena.
ACKNOWLEDGMENTS
The authors wish to acknowledge Dr. Nguyen Van Yen and
Dr. Tina Mai from Duy Tan University for their inspiring discus-
sions and valuable comments.
AUTHOR DECLARATIONS
Conflict of Interest
The authors have no conflicts to disclose.
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J. Appl. Phys. 131, 123102 (2022); doi: 10.1063/5.0081237 131, 123102-9
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Ethics Approval
Ethics approval is not required.
DATA AVAILABILITY
The data that support the findings of this study are available
from the corresponding author upon reasonable request.
REFERENCES
1
P. A. Doyle, J. Phys. D: Appl. Phys. 19, 1613 (1986).
2
V. Gusev, C. Desmet, W. Lauriks, C. Glorieux, and J. Thoen, J. Acoust. Soc.
Am. 100, 1514 (1996).
3
P. Hess, Phys. Today 55,42–47 (2002).
4
F. Jenot, M. Ouaftouh, M. Duquennoy, and M. Ourak, J. Appl. Phys. 97,
094905 (2005).
5
M. Ochiai, J. Phys. Conf. Ser. 278, 012009 (2011).
6
F. Mirzade, J. Appl. Phys. 110, 064906 (2011).
7
D. Schneider, in IEEE International Ultrasonics Symposium (IUS) (IEEE, 2012),
p. 269.
8
H. Zhang, A. Antoncecchi, S. Edward, P. Planken, and S. Witte, Phys. Rev. B
104, 205416 (2021).
9
G. Chow, E. Uchaker, G. Cao, and J. Wang, Mech. Mater. 91, 333 (2015).
10
S. Das, A. Schill, C.-H. Liu, S. Aglyamov, and K. V. Larin, J. Biomed. Opt. 25,
035004 (2020).
11
P. Grasland-Mongrain, Y. Lu, F. Lesage, S. Catheline, and G. Cloutier, Appl.
Phys. Lett. 109, 221901 (2016).
12
C. Desmet, V. Gusev, W. Lauriks, C. Glorieux, and J. Thoen, Appl. Phys. Lett.
68, 2939 (1996).
13
N. Hosoya, A. Yoshinaga, A. Kanda, and I. Kajiwara, Int. J. Mech. Sci. 140,
486 (2018).
14
C. T. Schröder and W. R. Scott, J. Acoust. Soc. Am. 110, 2867 (2001).
15
M. K. Teshaev, I. I. Safarov, N. U. Kuldashov, M. R. Ishmamatov, and
T. R. Ruziev, J. Vib. Eng. Technol. 8, 579 (2020).
16
Q. Han, M. Qian, and H. Wang, J. Appl. Phys. 100, 093101 (2006).
17
J. Zhu, J. S. Popovics, and F. Schubert, J. Acoust. Soc. Am. 116, 2101 (2004).
18
F. Padilla, M. de Billy, and G. Quentin, J. Acoust. Soc. Am. 106, 666 (1999).
19
C. Glorieux, K. Van de Rostyne, K. Nelson, W. Gao, W. Lauriks, and J. Thoen,
J. Acoust. Soc. Am. 110, 1299 (2001).
20
R. A. Phinney, Bull. Seismol. Soc. Am. 51, 527 (1961).
21
A. Le Goff, P. Cobelli, and G. Lagubeau, Phys. Rev. Lett. 110, 1 (2013).
22
J. J. Pitre, M. A. Kirby, L. Gao, D. S. Li, T. Shen, R. K. Wang, M. O’Donnell,
and I. Pelivanov, Appl. Phys. Lett. 115, 083701 (2019).
23
L. Gao, J. Xia, and Y. Pan, Geophys. J. Int. 199, 1452 (2014).
24
M. Roth and K. Holliger, Geophys. J. Int. 140, F5 (2000).
25
S. Das, “Supershear earthquake ruptures–Theory, methods, laboratory experi-
ments and fault superhighways: An update,”in Perspectives on European
Earthquake Engineering and Seismology, Geotechnical, Geological and
Earthquake Engineering Vol. 39, edited by A. Ansal (Springer, Cham, 2015), pp.
1–20.
26
N. Allouche and G. Drijkoningen, Geophys. Prospect. 64, 543 (2016).
27
Z. Cai, S. Liu, and C. Zhang, AIP Conf. Proc. 1806, 050015 (2017).
28
V. Gusev and P. Hess, Nondestruct. Test. Eval. 11, 313 (1994).
29
C. Desmet, V. Gusev, W. Lauriks, C. Glorieux, and J. Thoen, Opt. Lett. 22,69
(1997).
30
H. Coufal, K. Meyer, R. K. Grygier, P. Hess, and A. Neubrand, J. Acoust. Soc.
Am. 95, 1158 (1994).
31
A. Neubrand and P. Hess, J. Appl. Phys. 71, 227 (1992).
32
A. Dräbenstedt, AIP Conf. Proc. 1600, 263 (2014).
33
B. Pouet, A. Wartelle, and S. Breugnot, Nondestruct. Test. Eval. 26, 253
(2011).
34
J. Xiong, X. Xu, C. Glorieux, O. Matsuda, and L. Cheng, Rev. Sci. Instrum. 86,
053107 (2015).
35
A. A. Kamshilin, R. V. Romashko, and Y. N. Kulchin, J. Appl. Phys. 105,
031101 (2009).
36
T. A. Carolan, R. L. Reuben, J. S. Barton, and J. D. C. Jones, Appl. Opt. 36,
380 (1997).
37
I. Pelivanov, T. Buma, J. Xia, C.-W. Wei, and M. O’Donnell, J. Appl. Phys.
115, 113105 (2014).
38
I. Pelivanov, T. Buma, J. Xia, C.-W. Wei, and M. O’Donnell, Photoacoustics 2,
63 (2014).
39
J. Spytek, A. Ziaja-Sujdak, K. Dziedziech, L. Pieczonka, I. Pelivanov, and
L. Ambrozinski, NDT E Int. 112, 102249 (2020).
40
P. Pyzik, A. Ziaja-Sujdak, J. Spytek, M. O’Donnell, I. Pelivanov, and
L. Ambrozinski, Photoacoustics 21, 100226 (2021).
41
T. T. P. Nguyen, R. Tanabe, and Y. Ito, Appl. Phys. Lett. 102, 124103
(2013).
42
T. T. P. Nguyen, R. Tanabe-Yamagishi, and Y. Ito, Appl. Surf. Sci. 470, 250
(2019).
43
T. T. P. Nguyen, R. Tanabe-Yamagishi, and Y. Ito, Appl. Phys. Express 14,
042005 (2021).
44
Y. Zhang, C. Yang, H. Qiang, and P. Zhong, Phys. Rev. Res. 1, 33068 (2019).
45
J. H. Ansell, Pure Appl. Geophys. 94, 172 (1972).
46
M. Hayes and R. S. Rivlin, Z. Angew. Math. Phys. 13, 80 (1962).
47
L. Tsang, J. Acoust. Soc. Am. 63, 1302 (1978).
48
V. G. Mozhaev and M. Weihnacht, Ultrasonics 40, 927 (2002).
49
T. T. P. Nguyen, R. Tanabe, and Y. Ito, Appl. Phys. A 116, 1109 (2014).
50
T. T. P. Nguyen, R. Tanabe, and Y. Ito, Opt. Laser Technol. 100, 21 (2018).
51
J. A. Martin, D. G. Schmitz, A. C. Ehlers, M. S. Allen, and D. G. Thelen,
J. Biomech. 90, 9 (2019).
52
K. Shukla, J. M. Carcione, J. S. Hesthaven, and E. L’heureux, J. Acoust. Soc.
Am. 147, 3136 (2020).
53
J. Lubbers and R. Graaff, Ultrasound Med. Biol. 24, 1065 (1998).
54
J. Ivanov, C. B. Park, R. D. Miller, J. Xia, J. A. Hunter, R. L. Good, and
R. A. Burns, SEG Technical Program Expanded Abstracts 2000 (Society of
Exploration Geophysicists, 2000), Abstract No. 1307-1310.
55
N. Allouche, G. G. Drijkoningen, W. Versteeg, and R. Ghose, Geophysics 76,
T1 (2011).
56
C. Bakre, P. Rajagopal, and K. Balasubramaniam, AIP Conf. Proc. 1806,
020004 (2017).
Journal of
Applied Physics ARTICLE scitation.org/journal/jap
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