Ultrasonics 40 (2002) 927–933
www.elsevier.com/locate/ultras
Subsonic leaky Rayleigh waves at liquid–solid interfaces
V.G. Mozhaev a, M. Weihnacht
b
b,*
a
Faculty of Physics, Moscow State University, 117234 Moscow, Russia
Institut f€ur Festk€orper- und Werkstofforschung, Helmholtzstraße 20, 01069 Dresden, Germany
Abstract
The paper is devoted to the study of leaky Rayleigh waves at liquid–solid interfaces close to the border of the existence domain of
these modes. The real and complex roots of the secular equation are computed for interface waves at the boundary between water
and a binary isotropic alloy of gold and silver with continuously variable composition. The change of composition of the alloy
allows one to cross a critical velocity for the existence of leaky waves. It is shown that, contrary to popular opinion, the critical
velocity does not coincide with the phase velocity of bulk waves in liquid. The true threshold velocity is found to be smaller, the
correction being of about 1.45%. Attention is also drawn to the fact that using the real part of the complex phase velocity as a
velocity of leaky waves gives only approximate value. The most interesting feature of the waves under consideration is the presence
of energy leakage in the subsonic range of the phase velocities where, at first glance, any radiation by harmonic waves is not
permitted. A simple physical explanation of this radiation with due regard for inhomogeneity of radiated and radiating waves is
given. The controversial question of the existence of leaky Rayleigh waves at a water/ice interface is reexamined. It is shown that the
solution considered previously as a leaky wave is in fact the solution of the bulk-wave-reflection problem for inhomogeneous
waves. Ó 2002 Elsevier Science B.V. All rights reserved.
Keywords: Rayleigh waves; Subsonic leaky waves; Liquid–solid interface; Gold–silver alloy; Ice–water interface; Existence domain of leaky waves
1. Introduction
It is a widespread belief that leaky waves arise under
the conditions when the phase velocity of the waves
guided by any surface or interface exceeds the phase
velocity of the slowest bulk waves in one of the media
adjacent to this boundary [1]. From this point of view
the minimum bulk wave velocity is expected to be the
threshold value for the existence of leaky waves. However, our previous computational studies of the borders
of the existence domains for leaky surface acoustic
waves (SAW) in crystals revealed the cases when the
leaky wave branches extend into the subsonic regions of
the phase velocities, that is, below the values specified by
the slowest (so-called limiting) bulk waves [2,3]. The
analysis of these solutions in the subsonic regions shows
that they may be interpreted as subsonic leaky waves.
The present paper is motivated by the desire (i) to study
this phenomenon in a simpler situation without the
complication introduced by the presence of elastic an*
Corresponding author. Tel.: +49-351-4659-330; fax: +49-351-4659440/313.
E-mail address: [email protected] (M. Weihnacht).
isotropy of crystals, (ii) to gain insight into the occurrence of this phenomenon for leaky acoustic waves of
other types. As an object of the study, we have chosen a
very well-known and relatively simple example of leaky
Rayleigh waves at the boundary between an isotropic
solid and a fluid medium [1,4,5]. It is worth noting that
leaky waves of such a type are much more frequent in
their occurrence under natural environmental conditions than non-leaky ones since omnipresent air loading
on solid surfaces gives rise to radiating energy by surface
acoustic waves into the outer space. We study these
leaky waves under conditions close to those at the border of their existence domain.
2. Statement of the problem and secular equation
We consider the two-dimensional problem of surface acoustic wave propagation along the plane interface
between a homogeneous isotropic elastic solid and a nonviscous isotropic liquid. The particle-displacement field
of the wave is given by the following general form
ui ¼ u0i ðzÞ expðikx ixtÞ;
0041-624X/02/$ - see front matter Ó 2002 Elsevier Science B.V. All rights reserved.
PII: S 0 0 4 1 - 6 2 4 X ( 0 2 ) 0 0 2 3 3 - 0
ð1Þ
928
V.G. Mozhaev, M. Weihnacht / Ultrasonics 40 (2002) 927–933
where x is the propagation direction, z the normal to the
solid surface, k the wave number, x the angular frequency. The secular equation of the problem is found
from the equations of motion for these media and
boundary conditions [4]. It may be written in the form
DR Q ðqf =qÞqkt4 ¼ 0;
2
ð2Þ
2 2
where DR ¼ 4k qs ðk þ s Þ , s ¼ k q ¼ k2
2
2
2
2
kl , Q ¼ k kf , kt and kl are the wavenumbers of
transverse and longitudinal bulk waves in the solid, q is
the mass density of the solid. Variables corresponding to
fluid (liquid or gaseous) loading are denoted by subscript f. Equating DR to zero yields the well-known
secular equation for Rayleigh waves at a free surface of
solids if the ‘‘decay’’ constants in the solid, s and q, have
the same sign. According to the analysis by Ansell [6],
Eq. (2) produces eight-sheets Riemann surface, that
is, Eq. (2) possesses eight solutions. These solutions
correspond to all possible combinations of signs of constants s, q, and Q describing the decay or growth of the
wave amplitude with depth.
Let us first assume that the threshold velocity for
leaky waves is indeed equal to the phase velocity of bulk
waves in liquid. At the threshold velocity, radiation by
leaky waves tends to zero and the propagation constant
becomes real, that is, according to the above assumption
k ¼ kf :
2
2
2
kt2 ,
2
ð3Þ
Substituting Eq. (3) into Eq. (2) shows that Eq. (2) is
fulfilled only in the case of q ¼ 0, that is, k ¼ kl . But this
value of the propagation constant corresponds to the
fastest bulk wave rather than leaky Rayleigh waves.
Fig. 1. Structure of the wave field of leaky Rayleigh waves at a liquid–
solid interface.
Thus, the analysis of secular Eq. (2) shows that the
phase velocity of bulk waves in liquid is not accurate
threshold value for leaky Rayleigh waves. This very
simple estimation has served as a starting point of our
further numerical calculations.
The slowest real solution to Eq. (2) corresponds to
Stoneley waves called also in the case of a liquid–solid
interface as Scholte or Gogoladze waves [7] (see also the
bibliography in the book by Viktorov [4]). These waves
exist for arbitrary combinations of fluids and solids.
Under certain conditions studied by Brower et al. [5],
one of the complex roots of Eq. (2) corresponds to a
leaky Rayleigh wave. The structure of the wave field of
this mode is illustrated in Fig. 1. There are two features
of this field related to each other and important for
further discussion: (i) the inhomogeneity, along the
phase front, of bulk waves radiating into the liquid by
Rayleigh waves (ii) decaying of Rayleigh waves along
the propagation direction.
3. Interface between water and binary alloy
The case of a light gas loading on a solid substrate is
probably the most illustrative example which gives an
explanation of the appearance of leaky Rayleigh waves.
However, increasing the fluid load can change dramatically the character of the solution resulting, in particular, in the disappearance of leaky waves. Therefore,
one of the requirements to select the proper pair of
liquid and solid materials for the purpose of the present
study, is the requirement that the outer loading, which
may be estimated, for example, by the use of acoustic
impedances, should be rather small with respect to its
influence on the solid substrate. On the other hand, the
phase velocities of bulk waves in a liquid and Rayleigh
waves in a solid should be close to each other to study
the border of the existence domain for leaky waves.
These two requirements are compatible if to consider a
liquid/solid pair with a strong density contrast. Besides,
it is desirable to change continuously the ratio between
the wave velocities of a liquid and a solid that allows one
to cross the critical velocity of leaky waves. To realize
the last condition, a binary alloy with variable composition may be used as a solid substrate. In this case, the
Rayleigh wave velocity in one of the alloy components
should be higher than the velocity of sound in the liquid,
and quite the reverse for the other component. We have
found that all the described requirements are fulfilled for
the interface of water and the isotropic gold-silver alloy,
Au1x Agx , with variable content x.
This binary alloy is distinguished by the following
relatively rare feature. Its phase diagram is practically
linear with respect to weight percent silver, w, or weight
percent gold, 1 w, in the whole range of composition,
0 6 x 6 1 [8]. This property is related to high homoge-
V.G. Mozhaev, M. Weihnacht / Ultrasonics 40 (2002) 927–933
929
neity of these solid solutions, with no formation of ordered structures, that allows us to find the material
constants of the alloy by a simple linear interpolation
between the values for pure gold and silver
Au
c11 ¼ wcAg
11 þ ð1 wÞc11 ;
ð4Þ
Au
c44 ¼ wcAg
44 þ ð1 wÞc44 ;
ð5Þ
q ¼ wqAg þ ð1 wÞqAu :
ð6Þ
Numerical calculations have been carried out for
the case under discussion using the following values of
Ag
Ag
material constants: cAg
¼ 10490
11 ¼ 13:95, c44 ¼ 2:7, q
3
Au
Au
Au
kg/m for silver [9], c11 ¼ 20:7, c44 ¼ 2:85, q ¼ 19300
kg/m3 for gold [9], and qf ¼ 1000 kg/m3 , vf ¼ 1457 m/s
for water. The elastic constants c11 and c44 are given in
units of 1010 N/m2 .
The real and complex roots of the secular equation
(Eq. (2)) computed as functions of weight percent silver
are shown in Figs. 2 and 3. Fig. 2 includes also the curve
for the angle of wave radiation into water (glancing
angle). Taking into account that the complex roots of Eq.
(2) occur as complex conjugate pairs, one can see that
Fig. 2 includes the data for all eight roots of the secular
equation. This full information on the roots may be
useful to understand an evolution and a mutual transformation of the wave solutions of different types, as
well as to interpret rather complicated wave spectra in
the layered structures. It is evident from Fig. 2 that the
value of the phase velocity of bulk waves in the liquid is
not a branching point for the leaky wave curve, since the
imaginary part of the propagation constant for leaky
mode does not tend to zero at this point. So, there is no
contact of different Riemann sheets and no possibility to
go from one Riemann sheet to another at this point.
Instead, a prolongation of the leaky-wave branch is
observed in a subsonic range of the phase velocities.
Thus, contrary to popular opinion, the critical velocity
for leaky waves does not coincide with the phase velocity of bulk waves in liquid. The true threshold velocity was found to be smaller, the correction being of
about 1.45%.
At the border of the existence region, the attenuation
of leaky waves along the surface, as well as the angle of
their radiation into the liquid, tends to zero, that is, the
constant of propagation becomes pure real. At this point,
the leaky waves degenerate into an unphysical (UP)
solution, with a contact of the leaky wave branch with a
turning point of UP branch. The unphysical solutions
below a point marked in the lower part of Fig. 3 have
the same transverse structure of the wave field like
common leaky Rayleigh waves (the decreasing wave
amplitude with depth in the solid and the inverse behavior in the liquid). Above the last mentioned point,
there is a change of sign of the ‘‘decay’’ constant s
that makes possible an intersection of the branch of
Fig. 2. Real and complex solutions of the secular equation for interface
waves including unphysical real solutions (UP) at the boundary of
water and alloy of silver and gold with variable composition. For highvelocity complex solutions of Brewster-reflection (BR) type, the reduced-scale curves are given in the upper part of the figure. BAW
velocities and SAW velocity for free surface are also included. A scale
for the attenuation of the branches BR2 and BR3 is shown in the righthand vertical axis. The lower part of the figure gives the angle between
the surface and the energy flux radiated into the liquid.
unphysical solutions with a velocity curve for SAW at
the free surface. The energy fluxes in both media for
the unphysical solutions are parallel to the interface.
This property gives no way of limiting the exponential
growth of the wave field for these solutions with distance
from the interface in the liquid by assuming that the
waves are excited by a source of finite sizes. That is the
reason why such solutions are referred to as unphysical.
930
V.G. Mozhaev, M. Weihnacht / Ultrasonics 40 (2002) 927–933
locity as a velocity of leaky waves gives only an approximate value for the true phase velocity. The true
phase velocity v is determined by both real and imaginary parts of the complex phase velocity v0 ¼ vr þ ivi
v ¼ vr ð1 þ v2i =v2r Þ:
ð7Þ
4. Why subsonic leakage is admissible
The most interesting feature of the waves under consideration is the presence of energy leakage in the subsonic range of the phase velocities where, at first glance,
any radiation by harmonic waves is not permitted. Note
that the structure of the wave field of subsonic leaky
waves is the same like for common leaky waves, that is,
these are surface waves with amplitude decreasing into
the depth in the solid, while the wave amplitude in the
liquid is growing with distance from the interface. Thus
the single point to be explained in this problem is the
occurrence of radiation in the subsonic range of the
phase velocities. The simplest explanation is based on
the analysis of the phase matching condition
cos h ¼ vb =vR ;
Fig. 3. Fragments of Fig. 2 including the points of degeneracy on an
enlarged scale.
At upper turning point of the unphysical branch, there is
a contact of this branch with a branch of complex solutions denoted by the abbreviation BR1. This complex
branch corresponds to a wave solution with the same
transverse structure as that of leaky Rayleigh waves.
With one exception of a small range of alloy compositions near the mentioned upper turning point, the phase
velocity of BR1 solution is greater than that of shear
bulk acoustic waves (BAW) in the solid. This allows us
to interpret this solution as a mode of predominantly
Brewster-reflection (BR) type for the inhomogeneous
bulk waves, although in the small region before the intersection of this branch and a velocity curve for shear
bulk waves in the solid, there is a similarity of this mode
to the subsonic leaky waves.
Since the correction found to the threshold velocity is
not so great, a special attention should be given in this
problem to the accurate definition of the phase velocity.
It is easy to show from Eq. (1) that the use in some
papers [5,10–12] of the real part of complex phase ve-
ð8Þ
where vb is the phase velocity of inhomogeneous bulk
waves radiated into the liquid, vR is the phase velocity of
leaky Rayleigh waves, h is the angle between the propagation directions of leaky Rayleigh waves and waves
radiated into the liquid. The threshold velocity for leaky
Rayleigh waves was determined previously [5] on the
basis of the phase matching condition by the use of vb
to be equal to the phase velocity of homogeneous bulk
waves in a liquid ignoring the fact that this is only
an approximation. In this approximation, the phase
matching condition admits the existence of leaky waves
only in a supersonic range of the phase velocities [5]. In
principle, this conclusion might be accurate and correct
if the attenuation of leaky waves along the propagation
direction tends to zero simultaneously as their phase
velocity approaches the phase velocity of bulk waves in
liquid. However, it is not the case in the problem under
study. The solution of this problem in the self-consistent
formulation involves taking account of inhomogeneity
of the radiated waves (this inhomogeneity in its turn is a
consequence of radiation), but inhomogeneous waves
are slower than homogeneous ones. Therefore, for small
angles of radiation (angles h) inhomogeneous radiated
waves can propagate along the surface slower than homogeneous ones that reduces the true threshold velocity
for the existence of leaky waves. The direct calculations
of angle of the wave radiation into the liquid (the lower
part of Fig. 3) demonstrate non-zero values of this angle
in the subsonic region.
The Fourier analysis of the leaky wave field distribution on the half-line (x P 0) along the interface
V.G. Mozhaev, M. Weihnacht / Ultrasonics 40 (2002) 927–933
gives an additional explanation for possible leakage in
the subsonic range of the phase velocities. Due to the
decay of the field, the spatial spectrum of this distribution is a peak of finite width and height rather than a
delta function. Therefore, when the maximum of the
peak is in the subsonic region forbidden for the radiation of homogeneous waves, one of the edges of the
peak will be still in the region permitted for the radiation
(Fig. 4).
A very unusual property of the subsonic leakage is
that it is admissible only due to the presence of non-zero
imaginary part of the wavenumber of leaky waves, while
the value of the associated real part would not allow any
radiation in the absence of the imaginary part. Hence
such well-known experimental methods like the observations of Schoch displacement of acoustic beams
reflected from a liquid–solid interface and the measurements of surface Brillouin light scattering are not
applicable, at least in their direct form, to detect subsonic leaky waves. This is due to the fact that acoustic
and optical beams of homogeneous-wave type commonly used in these experiments do not provide the
proper value of the imaginary part of the wavenumbers. Nevertheless, the use of the beams of inhomogeneous-wave type produced, for example, with the aid of
strongly absorbing prisms could probably create the
necessary conditions to observe subsonic leaky waves in
these experiments. The transmission of Rayleigh waves
in a plate partially emmersed into a liquid is another
possible experiment to observe the subsonic leakage. In
this case, Rayleigh waves incident from the free side of
the plate onto the plate surface contacted with the liquid
could be transformed in part into the subsonic leaky
Fig. 4. Spacial spectrum of leaky wave field distribution on the halfline along the interface.
931
waves. The observation of the last waves may be done
either by measuring their absorption, or by visualization
of acoustic field radiated into the liquid, or by measuring the radiation pattern in the liquid. Note that the use
of liquid solutions with a variable concentration may be
an additional way (perhaps simpler for an experiment
than changing the solid substrate properties) to provide
the necessary conditions for the existence of subsonic
leaky waves.
5. Interface between water and ice
Controversial results are reported in the literature
on leaky Rayleigh waves at a water/ice interface. This
leaky mode does not exist, according to calculations by
Brower et al. [5]. On the other hand, Chamuel reported
about observations of such a wave in a laboratory experiment [13]. A more recent numerical study by Weng
and Yew [14] revealed a ‘‘leaky wave’’ which is faster
than the bulk shear waves in ice. This, according to the
opinion of the authors [14], implies that such a leaky
wave is radiating energy into both the solid as well as
liquid medium. The radiation of energy into solid implies, in turn, that one of the partial waves in the solid
should have increasing amplitude with distance from the
interface. But this is a quite different type of the wave
solution in comparison with Rayleigh waves, because
for Rayleigh waves all the partial waves are decaying
into the depth. Thus, one cannot conclude, on the basis
of results reported in the literature, what happened is
with Rayleigh waves under the influence of water loading on ice surface.
To clarify this question and also to try to reveal (or to
exclude) the possibility of the existence of subsonic leaky
Rayleigh waves in this case, we have performed the
computational study of the evolution of the Rayleigh
wave solution with various loading on an ice substrate.
As limiting states, air and water loadings are considered.
As previously, we use a linear interpolation of material
parameters of the fluid between these two limiting states.
The material parameters of ice and water are taken to be
the same as in Ref. [14]: q ¼ 940 kg/m3 , Young’s modulus E ¼ 1 1010 Pa, Poisson’s ratio m ¼ 0:333 for ice,
and qf ¼ 1026 kg/m3 , vf ¼ 1440 m/s for water. The results of calculations of the phase velocity and attenuation for quasi-Rayleigh wave (curve 1) and the wave
solutions of Brewster type (curves 2 and 3) are shown in
Fig. 5. The transition parameter used in Fig. 5 is determined by the same manner as weight percent silver in
Eqs. (4)–(6). One can see that the transition from air to
water causes the Rayleigh wave velocity to increase.
Thus, subsonic leaky waves do not occur in this case.
Below a certain value of the transition parameter (approximately below 0.4), the phase velocity becomes
higher than that of shear bulk waves shown in Fig. 5 by
932
V.G. Mozhaev, M. Weihnacht / Ultrasonics 40 (2002) 927–933
Fig. 5. Evolution of solutions for leaky Rayleigh waves (curve 1) and
Brewster-angle-reflections of inhomogeneous incident waves (curves 2
and 3) with various loading on ice substrate.
the dashed line. The final phase velocity for water
loading coincides with the value obtained by Weng and
Yew [14]. However, the calculations of the wave field
distribution in depth (Fig. 6) leads us to the opposite
conclusion about the type of the solution. The decaying
character of the wave fields in depth remains for ice
during the transition from air to water loading. This
property of the solution, in parallel with the fact that
its phase velocity is higher than the shear-bulk-wave
one, implies that the solution includes shear bulk waves
incident from ice upon the ice/water interface. Hence,
this solution should be classified as a solution of bulkwave-reflection problem rather than a leaky wave. It is a
rather interesting fact that such an unusual type of
transformation is admissible for Rayleigh waves.
In conclusion, the new surprising phenomenon of the
existence of subsonic leaky Rayleigh waves at interfaces
of liquid and isotropic solid (water and gold–silver
alloy) is found in the computational study of the problem. This result gives a new insight into the properties
of leaky waves and corrects the previously accepted
Fig. 6. Variation of displacement components with depth for leaky
Rayleigh waves at an air/ice interface and for pseudoleaky waves at a
water/ice interface.
notion of a threshold condition for the existence of
leaky Rayleigh waves at such a boundary. The study of
the case of a water/ice interface does not reveal the
existence of subsonic or any other leaky waves, but
instead shows that the solution previously considered as
a leaky wave is in fact an erroneous interpretation of
bulk-wave-reflection solution for inhomogeneous incident waves. Thus, because of contradictions of available theoretical and experimental results, the problem
of leaky waves at a water/ice interface still remains to
be solved.
Acknowledgements
The authors are grateful to Dr. R. Wobst for his
programming work and to Prof. A.G. Every for critical
reading of the manuscript. The work of one of us
(VGM) is supported in part by S€achsiches Ministerium
V.G. Mozhaev, M. Weihnacht / Ultrasonics 40 (2002) 927–933
f€
ur Wissenschaft und Kunst and by Russian Foundation
for Basic Research (grant 01-02-17411).
References
[1] L.M. Brekhovskikh, O.A. Godin, Acoustics of Layered Media: I.
Plane and Quasi-Plane Waves, Springer-Verlag, Berlin, 1990, p. 111.
[2] V.G. Mozhaev, F. Bosia, M. Weihnacht, Types of leaky SAW
degeneracy in crystals, in: 1998 IEEE Ultrasonics Symposium
Proceedings, IEEE, New York, 1998, p. 143.
[3] F. Bosia, V.G. Mozhaev, M. Weihnacht, Subsonic leaky surface
waves in crystals, in press.
[4] I.A. Viktorov, Rayleigh and Lamb waves, Plenum Press, New
York, 1967.
[5] N.G. Brower, D.E. Himberger, W.G. Mayer, Restrictions on the
existence of leaky Rayleigh waves, IEEE Trans. Sonics Ultrason.
26 (1979) 306.
[6] J.H. Ansell, The roots of the Stoneley wave equation for solid–
liquid interfaces, Pure Appl. Geophys. 94 (1972) 172.
933
[7] A.M.B. Braga, G. Herrmann, Scholte-Gogoladze-like waves at
the interface between a fluid and an anisotropic layered composite,
J. Acoust. Soc. Am. 84 (1988) S207.
[8] Binary Alloy Phase Diagrams, 2nd ed., vol. 1, in: T.B. Massalski
et al. (Eds.), ASM International, Materials Park, Ohio, 1990.
[9] B.A. Auld, Acoustic Fields and Waves in Solids, vol. I, second ed.,
R.E. Krieger Publishing Company, Malabar, FL, 1990.
[10] J.J. Campbell, W.R. Jones, Propagation of surface waves at the
boundary between a piezoelectric crystal and a fluid medium,
IEEE Trans. Sonics Ultrason. 17 (1970) 71.
[11] P.A. Doyle, C.M. Scala, Toward laser–ultrasonic characterization
of an orthotropic–isotropic interface, J. Acoust. Soc. Am. 93
(1993) 1385.
[12] K. Toda, K. Motegi, Coupling of velocity dispersion curves of
leaky Lamb waves on a fluid-loaded plate, J. Acoust. Soc. Am.
107 (2000) 1045.
[13] J.R. Chamuel, On the controversial leaky Rayleigh wave at a
water/ice interface, J. Acoust. Soc. Am. 80 (1986) S54.
[14] X. Weng, C.H. Yew, The leaky Rayleigh wave and Scholte wave
at the interface between water and porous sea ice, J. Acoust. Soc.
Am. 87 (1990) 2481.