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Inhomogeneous P- and S-wavefields radiated into isotropic elastic solids


Abstract and Figures

A vibrating surface in contact with a solid material will generate P- and S-waves in the solid. When the surface vibration is spatially attenuated, we must take into account that the generated waves are always inhomogeneous. In an isotropic elastic solid, such inhomogeneous waves are attenuated perpendicularly to their direction of propagation. When the surface vibration's phase speed is lower than the P- and/or S-waves' speed of sound, the inhomogeneity affects the radiation of P- and S-waves in a major but relatively poorly understood way. For a better understanding, finding the total radiated intensity of the two inhomogeneous waves is key. Our work takes a step towards such an understanding by deriving analytical expressions for the velocity, strain, stress, and intensity fields of arbitrarily inhomogeneous P- and S-waves. Furthermore, we investigate whether the total radiated intensity can be found as the sum of the intensities of the individual P- and S-waves. We find that this is only possible when the surface vibration is unattenuated; for attenuated vibrations, the total radiated intensity should be calculated numerically.
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Proceedings of the 45th Scandinavian Symposium on Physical Acoustics, Online, 31 Jan – 1 Feb 2022
Inhomogeneous P- and S-wavefields radiated into
isotropic elastic solids
Erlend Magnus Viggen1, Håvard Kjellmo Arnestad2
1Centre for Innovative Ultrasound Solutions, Department of Circulation and Medical Imaging,
Norwegian University of Science and Technology
2Department of Informatics, University of Oslo
Contact email:
A vibrating surface in contact with a solid material will generate P- and S-waves
in the solid. When the surface vibration is spatially attenuated, we must take into
account that the generated waves are always inhomogeneous. In an isotropic elas-
tic solid, such inhomogeneous waves are attenuated perpendicularly to their direc-
tion of propagation. When the surface vibration’s phase speed is lower than the P-
and/or S-waves’ speed of sound, the inhomogeneity affects the radiation of P- and
S-waves in a major but relatively poorly understood way. For a better understand-
ing, finding the total radiated intensity of the two inhomogeneous waves is key. Our
work takes a step towards such an understanding by deriving analytical expressions
for the velocity, strain, stress, and intensity fields of arbitrarily inhomogeneous P-
and S-waves. Furthermore, we investigate whether the total radiated intensity can
be found as the sum of the intensities of the individual P- and S-waves. We find
that this is only possible when the surface vibration is unattenuated; for attenuated
vibrations, the total radiated intensity should be calculated numerically.
1 Introduction
The most basic type of wave in acoustics is a homogeneous plane wave. For a pressure
wave in a fluid, it can be expressed as
p(r,t) = p0ei(k·rωt). (1)
Here, kis the real-valued wavenumber vector, r= (x,y,z)is the spatial coordinate, ωis
angular frequency, tis time, and the subscripted zero indicates amplitude — the value at
r=0,t=0. These waves propagate in a direction given by the wavenumber direction
k=k/|k|. Their phase (k·rωt), and therefore also their pressure value p(r,t), is
constant over any plane perpendicular to ˆ
Inhomogeneous plane waves is a lesser-known [1] type of plane wave. In such a plane
wave, the wavenumber vector kis no longer real-valued, but complex:
k=kr+ikiC3, where krR3,kiR3. (2)
Here, the subscripted r and i denote the real part and the imaginary part, respectively.
Inserted into (1), this leads to
p(r,t) = p0ei(kr·rωt)eki·r. (3)
Open Access
1 ISBN 978-82-8123-022-4
Hence, the real wavenumber component krrepresents a propagation vector, while the
imaginary component kirepresents an attenuation vector: While the wave moves in the
krdirection and still has planes of constant phase perpendicular to ˆ
kr, the pressure value
has an exponential decay in the ˆ
The attenuation vector kiis often split into two parts [1]: A damping vector parallel
with ˆ
kr, representing the effect of losses in the medium, and an inhomogeneity vector
perpendicular to ˆ
kr, representing an exponential decay along the wavefronts. For a loss-
less medium there is no damping vector, and krki. This can also be proven by insert-
ing (3) into a lossless pressure wave equation, which at the same time can tell us that
inhomogeneous waves propagate at a speed
q1+(|ki|/k)2. (4)
Here, k=|k|=ω/c0is the wavenumber magnitude, and c0is the medium’s speed of
sound, which is the speed at which homogeneous waves propagate.
The main motivation for working with inhomogeneous waves is their prevalence
when dealing with lossy materials such as thermoviscous liquids [2] and viscoelastic
solids [1, 3]. Our work, however, is motivated by understanding how leaky Lamb and
Rayleigh waves moving at a speed cvcan radiate energy into an adjacent medium in the
subsonic domain, where cv<c0. While several works demonstrate this phenomenon [4–
7], the literature has widely regarded it as impossible; many references state that this can
only occur in the supersonic domain, where cv>c0[8–13].
However, previous and current work on subsonic radiation [5, 14, 15] has found that
properly considering the inhomogeneity of radiated waves is the key to understanding
how subsonic radiation is possible. This inhomogeneity occurs even if all materials are
lossless — it is a consequence of the attenuation of the leaky Lamb or Rayleigh wave,
which occurs due to the loss of power into the radiated wave.
The problem can be simplified by considering a surface in the xzplane on which an
attenuated surface vibration propagates with a complex wavenumber
kx=kxr+ikxiC, where kxrR0,kxiR0, (5)
giving a vibrational surface velocity
v(x,y=0, t) = v0ei(kxxωt)=v0ei(kxrxωt)ekxix. (6)
Whenever kxi>0, an inhomogeneous (ki̸=0) but non-evanescent (kyr>0) wave is
radiated into the fluid [14, 15].
The power associated with a surface vibration can be quantified by its power flow
amplitude Px0. For a Lamb wave in a plate, for example, the power flow is the integral
over the plate’s cross-section of the intensity component parallel to the plate. The power
radiated into the fluid can be equated with a loss in power flow per unit length to get an
implicit equation for the vibrational attenuation [14, 15]:
2Px0(kxr,kxi). (7)
Here, Iy0is the intensity radiated into the fluid per unit length along the surface. Both the
radiated intensity and the power flow can be functions of both kxrand kxi.
cS0 <cP0 <cv
cS0 <cv<cP0
cv<cS0 <cP0
Figure 1: Three domains of P-wave (orange) and S-wave (green) radiation into a solid from an
unattenuated surface vibration. Left: P-supersonic and S-supersonic, where both types of waves
radiate into the solid. Middle: P-subsonic and S-supersonic, where the P-wave becomes evanes-
cent. Right: P-subsonic and S-subsonic, where both waves are evanescent. Unit vectors are
indicated by hats and drawn scaled to one wavelength.
For radiation into a fluid, the intensity is relatively straightforward to calculate. We
show in an upcoming article [15] (partly summarised in an extended abstract [14]) that (7)
predicts subsonic radiation into a fluid in a qualitatively correct way, as well as validate
it against leaky Lamb waves.
While this model was derived for radiation into a fluid, it can be straightforwardly
repurposed for radiation into a solid by calculating Iy0for solids. However, radiation into
solids is much less straightforward than radiation into fluids. While fluids only support
longitudinal pressure waves, solids support two types of waves: Longitudinal P-waves,
with a speed of sound cP0, and transversal S-waves, with a speed of sound cS0 <cP0.
Hence, for a surface vibration radiating into a solid, two inhomogeneous wavefields must
be determined in order to calculate the total radiated intensity Iy0.
Furthermore, in a fluid there is only one coincidence point cv=c0that separates two
domains of interest: The supersonic domain (cv>c0) and the subsonic domain (cv<c0).
Solids, with their two types of waves, have two coincidence points — one for P-waves
(cv=cP0) and one for S-waves (cv=cS0 ). Hence, solids have three domains of interest,
also shown in Figure 1:
P-supersonic and S-supersonic, where cS0 <cP0 <cv,
P-subsonic but S-supersonic, where cS0 <cv<cP0, and
P-subsonic and S-subsonic, where cv<cS0 <cP0.
In this article, we investigate how to determine the total intensity radiated from a sur-
face vibration into the simplest type of solid, which is isotropic (behaving independently
of orientation) and elastic (lossless). In Section 2, we determine the radiated waves’ veloc-
ity fields and sketch how to determine the strain, stress, and intensity fields. In Section 3,
we derive analytical expressions for these fields in a coordinate system aligned with the
wave. In Section 4, we investigate in which cases the total radiated intensity Iy0can be
calculated as the sum IPy0+ISy0of the P- and S-wave intensities. Section 5 summarises
our findings and concludes.
2 Radiated P- and S-wavefields
From the momentum conservation equation of an elastic and isotropic solid, separate
wave equations for P- and S-waves can be derived [16–18]. Expressed in terms of the P-
and S-waves’ particle velocities vPand vS, these wave equations are
t2− ∇2vP=0, with cP0 =sλ+2µ
ρand ∇ × vP=0, (8a)
t2− ∇2vS=0 with cS0 =rµ
ρand ∇ · vS=0. (8b)
Here, the speeds of sound cP0 and cS0 are expressed through the density ρand Lame’s
first (λ) and second (µ) parameters, the latter better known as the shear modulus. Each
velocity field has a condition applied to it: vPis irrotational, and vSis divergence-free.
Both wave equations have inhomogeneous plane wave solutions like
vy0#ei(kr·rωt)eki·r, (9)
where the asterisks represent either P or S. (From here on, we treat the problem as two-
dimensional, as the wave propagates in the xyplane and is zinvariant.) If the waves are
radiated from a surface vibration like (6), the wavenumbers’ xcomponents must match
the surface vibration’s wavenumber, i.e., kx=kx. The wavenumbers’ ycomponents ky
can be calculated from the relation (ω/c0)2=k2
t 1+kxi
=kyr+ikyi. (10)
Here, we have chosen the positive sign for the square root, which corresponds to a radi-
ated wave. (The negative sign corresponds to an incoming wave.)
As ˆ
kris the radiation direction, we can use it to calculate the radiation angle. This
angle can be expressed in two ways:
θ=arctan kxr
kyr,α=arctan kyr
kxr, (11)
where θis the radiation angle with respect to the surface normal, and αis the radiation
angle with respect to the surface.
The amplitudes v0of the two waves are determined by matching their combined
velocities’ xand ycomponents with those of the surface vibration from (6) at the surface,
as well as relating the xand ycomponents of each wave type in (9) using the conditions
in (8). This leads to a set of equations
1 0 1 0
0 1 0 1
kPykx0 0
0 0 kxkSy
that can be easily inverted with the help of a computer algebra system, giving
vP0 ="kx
,vS0 ="kSy
. (13)
Thus, the P and S velocity fields vare fully determined. From them, we can calculate
the P and S strain tensors as
Sij =1
2 ui
2iω vi
2ωkjvi+kivj. (14)
The time-harmonic nature of the wavefields means that the displacement uis related to
the velocity vas v=u/t=iωu. We also use the Einstein summation convention,
where we use i,j, and kas generic Cartesian indices, and repeating one of these generic
indices in a term implies a summation over all possible values of that index.
From the P and S strain tensors, we can calculate the P and S stress tensors for an
isotropic elastic solid as
σij =λδij Skk +2µSi j
P0 2ρc2
S0δijSkk +2ρc2
S0Si j. (15)
Here, we express the stresses through the more acoustically relevant speeds of sound cP0
and cS0 instead of the more traditional Lamé’s first and second parameters λand µ.
Then, we can use the P and S velocity fields and stress tensors to calculate the time-
averaged intensities for P- and S-waves as
Ii=Re σijvj
2. (16)
The bar over the velocity denotes complex conjugation.
The fields of velocity, strain, and stress are all linear quantities. Hence, the corre-
sponding total fields can be found directly as the sum of the P and S components:
v=vP+vS,Sij =SPij +SSi j,σij =σPij +σSi j . (17)
Intensity, however, is a nonlinear quantity, as it is calculated through products of stress
and velocity. Hence, the total intensity becomes
Ii=Re ((σPij +σSij )(vPj+vSj)
2)=Re σPijvPj+σSij vSj+σPijvSj+σSijvPj
=IPi+ISi+Re σPi j vSj
2+Re σSijvPj
This would indicate that the total intensity Iicannot simply be calculated as the sum
IPi+ISi, due to the latter two cross-terms between the P and S wavefields. Even so,
Brekhovskikh and Godin proved for fluid-to-solid reflection-transmission problems that
a simple summation of the normal component of the P- and S-intensities actually cor-
rectly provides the normal component of the total intensity in the solid [10]. In their case,
the two cross-terms must necessarily sum to zero. In Section 4, we investigate whether it
also holds true for our case.
3 Axis-aligned inhomogeneous wavefields
In Section 2 we fully determined the velocity fields of the inhomogeneous P- and S-waves
radiated from a surface vibration, and provided the equations required to calculate the P-
and S-fields of strain, stress, and intensity. However, calculating analytical expressions
Figure 2: Real and imaginary components of the wavenumber kof a wave described in the x
ycoordinate system and of the wavenumber Kdescribed in the wave-aligned XYcoordinate
for the latter fields would be very cumbersome and would likely lead to cluttered ex-
pressions, considering that the radiated inhomogeneous waves can have any orientation
0α2π/4 in the xycoordinate system.
Therefore, we take a similar approach to Poirée [2], defining a XYcoordinate system
that is aligned with the P- or S-wave in question. As Figure 2 shows, the wave prop-
agates along the Xaxis and decays along the ±Yaxis. In this coordinate system, the
wavenumber Kis
KX=KXr+i0, KY=0±i|KYi|, where KXrR0,KYiR. (19)
Here, ±is the sign of the angle between krand ki, and specifies whether the decay
occurs in the +Yor Ydirection. We can express the inhomogeneity magnitude |KYi|
more simply as |KY|.
Expressing the wave in the XYcoordinate system is useful in at least two circum-
stances. First, it lets us take a wave, typically expressed in terms of its velocity as in (9),
from the xycoordinate system and analytically determine its other fields, such as strain,
stress, or intensity, in a more convenient coordinate system. Second, it lets us us construct
an arbitrarily inhomogeneous wave in a convenient axis-aligned coordinate system, and
then place it into the xycoordinate system with the desired orientation. In the former
case, we can determine the XYwavenumber vector Kfrom the original xywavenum-
ber vector kas
K=R1(α)k, (20)
where R(α)is the rotation matrix used to transform from the xysystem to the XY
system, and R1(α)the rotation matrix that performs the opposite transformation:
R(α) = "cos αsin α
sin αcos α#,R1(α) = RT(α) = "cos αsin α
sin αcos α#. (21)
The inhomogeneous wave’s speed cis always less or equal to the matching speed of
sound c0:
K. (22)
From these two equations, we can derive a useful identity that relates the inhomogeneity
magnitude |KY|and the resulting change in wave speed c:
0. (23)
3.1 Aligned velocity fields
From (9), we already have an expression for the velocity field that can be easily expressed
in the XYcoordinate system as
vY0#ei(KXXωt)e∓|KY|Y. (24)
If v
0is the velocity amplitude vector of an original wave in the xycoordinate sys-
tem, we can straightforwardly determine the velocity amplitude vector in (24) as v0=
However, if we are constructing this wave from the ground up, it is useful to consider
the relationship between the Xand Ycomponents of v0. The component relationships
for P- and S-waves can be found from the P- and S-wave conditions in (8), which give
∇ × vP=iKP×vP=0KPXvPYKPYvPX=0vPY=±i|KPY|
vPX, (25a)
∇ · vS=iKS·vS=0KSXvSX+KSYvSY=0vSX=i|KSY|
This shows the well-known fact that homogenous P-waves only move the medium in
the Xdirection, while homogeneous S-waves only move it in the Ydirection. As the
inhomogeneity grows, the medium’s motion becomes elliptical, with an inverse aspect
ratio |KY|/KX. Hence, it is natural to treat vPXand vSYas scalar wave amplitudes for
our P- and S-waves, respectively, and use (25) to calculate the other velocity components.
3.2 Aligned strain fields
Knowing the velocity field, we can use (14) to calculate analytical expressions for the
strain field components SXX ,SXY =SYX , and SYY, as well as SKK =SX X +SYY .
Expressing the P-wave strain components in terms of vPX, which we chose as a scalar
P-wave amplitude, we find
, (26a)
ωvPX, (26b)
P0 vPX
, (26c)
. (26d)
In a homogeneous P-wave, where |KPY|=0 and cP=cP0, only SPX X (and hence SPKK)
is nonzero. In an inhomogeneous P-wave, SPX X is expressed similarly, albeit through the
inhomogeneous P-wave speed cPinstead of the homogeneous P-wave speed cP0. The
inhomogeneity also leads SPXY and SPYY to become nonzero.
We similarly find the strain components of the S-wave, expressed in terms of vSY, to
ωvSY, (27a)
= 2c2
S0 !vSY
ωvSY, (27c)
SSKK =SSXX +SSYY =0. (27d)
For a homogeneous S-wave, only the shear strain SSXY is nonzero. In an inhomogeneous
wave, the normal strains SSX X and SSYY become nonzero as well. Still, these normal
strains do not lead to any change in volume, as the volumetric strain SSKK remains zero.
3.3 Aligned stress fields
The strain field components let us calculate the stress field components σXX,σXY =
σYX, and σYY directly from (15).
For the P-wave, we get
σPXX = ρcP+2ρcPc2
P0 !vPX2ρc2
vPX= 1+2c2
P0 !ρcPvPX, (28a)
σPXY =2i|KPY|
S0vPX, (28b)
σPYY = ρcP+2ρcPc2
P0 !vPX+2ρc2
S0 1
P0 vPX= 12c2
P!ρcPvPX. (28c)
The inhomogeneity causes changes to the normal stresses σPXX and σPYY that can be ex-
pressed in terms of the changes in P-wave speed cP. It also causes the shear stress σPXY
to become nonzero.
For the S-wave, calculating stress from (15) is more straightforward as SSKK =0:
σSXX =±2i|KSY|
S0vSY, (29a)
σSXY = 2c2
1!ρcSvSY, (29b)
σSYY =2i|KSY|
S0vSY. (29c)
The inhomogeneity causes a minor change to the shear stress σSXY, and cause the normal
stresses σSX X and σSYY to be nonzero. Even so, the wave does not cause a bulk stress σSKK,
as the two normal stresses cancel each other.
3.4 Aligned intensity fields
Now that we know both the velocity fields vand the stress fields σ, we can calculate
the time-averaged intensity of individual P- or S- waves from (16). Writing it out, we find
that we need to calculate two terms for the Ith component of each wave:
2Re {σIXvX+σIYvY}. (30)
(Bear in mind that the stress tensors are symmetric, so that σYX =σXY.) As σPIJ is
expressed through vPXand σSI J is expressed through vSY, we will end up with expressions
on the form
=|vSY0|2e2|KSY|Y. (31)
To calculate the P-wave intensity, we require the terms
σPXXvPX= 1+2c2
P0 !ρcP|vPX|2, (32a)
P0 ρcP|vPX|2, (32b)
S0|vPX|2, (32c)
= "12c2
KPX|vPX|2. (32d)
Together, they lead to a P-wave intensity
IPX= 1+4c2
P0 !ρcP
2|vPX|2,IPY=0. (33)
The last equality follows because σPXYvPXand σPYYvPYare purely imaginary, with no
real component. Therefore, energy only flows along the direction of wave propagation
for inhomogeneous P-waves, in the same way as for homogeneous ones.
For S-waves, we require the terms
=2 c2
1!ρcS|vSY|2, (34a)
σSXYvSY= 2c2
1!ρcS|vSY|2, (34b)
= "2c2
KSX|vSY|2, (34c)
S0|vSY|2, (34d)
which lead to an S-wave intensity
ISX= 4c2
2|vSY|2,ISY=0. (35)
As with the P-waves, the Ycomponent of the S-wave intensity is zero because the real
parts of σSXYvSXand σSYYvSYare zero, so that energy only flows in the Xdirection.
Closer inspection of the expressions for P- and S-wave intensities reveals that they
are related. By the substitutions |vPX|2→ |vSY|2,cPcS, and cP0 cS0, (33) can be
transformed into (35).
While we are not aware of any literature providing expressions for inhomogeneous
strain and stress that we can compare our corresponding expressions with, a 1985 article
by Borcherdt and Wennerberg provides a general expression for the intensity of inhomo-
geneous P- or S-waves in a lossy medium [3]. Their medium is characterised by a com-
plex shear modulus µ=µr+iµi, where the real part is tied to elasticity and the imaginary
part is tied to loss. Adapted to our notation, their expression for time-averaged intensity
= 1+4c2
In the last equality, we assumed the solid to be lossless, so that µi=0, µ=µr=ρc2
S0, and
krki. In the reference, |GP0|represents the scalar displacement potential amplitude,
and GS0 =GSz0ˆzrepresents the vector displacement potential amplitude. We can relate
these potential amplitudes to our velocity amplitudes as
vP0 =ωkP|GP0|and vS0 =ωkS×GS0. (37)
If we choose this wave to be aligned along the Xand Yaxes (kK), we find that
Borcherdt and Wennerberg’s general expression for intensity matches our (33) and (35)
when the solid is lossless.
3.5 Reorienting the wavefields
The expressions for velocity v, strain S, stress σ, and intensity Ifound in Sections 3.1–3.4
were all expressed in the XYcoordinate system, which is aligned with the wave. We
can re-express these quantities in the xycoordinate system using the rotation matrix
from (21), so that
=RI. (38)
Here, the primed quantities are expressed in the xysystem, while the unprimed quanti-
ties are expressed in the XYsystem.
3.6 Summary of axis-aligned inhomogeneous wavefields
In this section, we have derived generic analytical expressions of the basic fields — ve-
locity, strain, stress, and intensity — of individual inhomogeneous P- and S-waves prop-
agating along the Xaxis and decaying along the ±Yaxis as shown in Figure 2. While
these fields are aligned to an artificial coordinate system, they can be transferred to the
xyplane with any orientation, as shown in (38).
Not only are the final expressions for intensity equivalent to those provided in [3], but
every expression provided in this section can be (and has been) verified through numeri-
cal calculation. With the velocity field specified as in (24), the strain, stress, and intensity
amplitudes can be computed numerically using (14), (15), and (16). These values can then
be compared against values calculated from the analytical expressions in Sections 3.2–3.4.
4 Total radiated intensity
At this point, we know three equivalent ways to calculate the intensity of the individual
P- and S-waves radiated from an attenuated surface vibration. Starting with the P and S
velocity fields from (9) and (13), we can:
0 0.5 1 1.5 2
Norm. vibration speed cv/cP0
Radiated intensity (arb. units)
0 0.5 1 1.5 2
Norm. vibration speed cv/cP0
IPy0ISy0IPy0+ISy0Iy0P-coincidence S-coincidence
0 0.5 1 1.5 2
Norm. vibration speed cv/cP0
Norm. vibration speed cv/cS0
01 2 34
Norm. vibration speed cv/cS0
Norm. vibration speed cv/cS0
Figure 3: Radiated intensity in arbitrary units against vibration speed for the P-wave, the S-wave,
the sum of individual P- and S-waves, and the total field, for three different attenuations of the
surface vibration. For either wave type, the waves are subsonic at vibration speeds below coinci-
dence and supersonic above.
1. Calculate the intensities in the XYcoordinate system by the analytical expression
in (33) and (35) and use (38) to express them in the xysystem.
2. Calculate the intensities numerically, using (14)–(16).
3. Calculate the intensities using Borcherdt and Wennerberg’s expression [3] in (36).
However, this only gives us the intensities of the individual P- and S-waves. What we
need for (7) is the total radiated intensity Iy0, which must be calculated from the combined
P- and S-wavefields as shown in (18). As we explained at the end of Section 2, (18)
shows that Iy0is not necessarily simply a sum of IPy0and ISy0, although Brekhovskikh
and Godin [10] show such a summation to be valid for the case of wave transmission
from a fluid into a solid.
To investigate whether this summation is valid for wave radiation into solids from
an attenuated surface vibration, we calculate and compare IPy0,ISy0, and Iy0for different
attenuations of the surface vibration. In all cases, we choose a purely normal surface
velocity of arbitrary amplitude, so that vx0=0, and vy0=1. Furthermore, we choose
arbitrary parameters ρ=1, cP0 =1, cS0 =0.5, and ω=2π.
Figure 3 compares the radiated intensities for three different vibrational attenuations
kxi. As the attenuation increases, the radiated intensities are smoothed, similarly to when
a surface vibration radiates into a fluid [14]. With a nonzero attenuation (kxi>0), the
intensities of the individual P- and S-waves diverge when we go into in the S-subsonic
domain. (Even when choosing different values of cS0/cP0 , this divergence only occurs
when cv<cS0.) Despite this divergence of the individual waves’ intensity, the total
intensity Iy0remains finite and small.
So, can we confirm Brekhovskikh and Godin’s result [10] that Iy0=IPy0+ISy0? For
unattenuated surface vibrations (kxi=0), this holds exactly true. For attenuated surface
vibrations (kxi>0), however, Iy0̸=IPy0+ISy0. The discrepancy is higher for higher
attenuations and becomes extreme as we go into the S-subsonic domain. As for the x-
component, Ix0̸=IPx0+ISx0in all cases.
Hence, Brekhovskikh and Godin only found their result because they were looking
at a reflection-transmission problem with incoming plane waves from a lossless medium.
In this case, the incoming wave amplitude is the same across the entire interface. For
incoming waves from a lossy medium, however, the incoming wave amplitude would be
exponentially attenuated along the interface, leading to an inhomogeneous transmitted
wave [1]. This would be a situation similar to our attenuated surface vibration, so that
Even so, how can Iy0=IPy0+ISy0even for kxi=0, when (18) shows that there are
cross-terms between the P and S wavefields? A closer numerical investigation shows
that all of these cross-terms are nonzero, but that σPyx0vSx0=σSyy0vPy0and σPyy0vSy0=
σSyx0vPx0. Thus, all the cross-terms in (18) end up cancelling. For kxi>0, these relations
no longer hold. However, a closer investigation of this phenomenon is outside the scope
of this article.
5 Conclusion
In this article, we have taken an important step towards understanding subsonic radi-
ation into a solid through the explanatory model underlying (7). We have investigated
how to correctly determine the required total intensity radiated into a solid by a vibrating
surface as specified in Section 2.
In Section 3, we derived analytical expressions for the velocity, strain, stress, and in-
tensity fields of arbitrarily inhomogeneous individual P- and S-waves in lossless isotropic
solids. However, we also found in Section 4 that the individual P- and S-wave intensi-
ties are not generally sufficient to find the total radiated intensity, due to the cross-terms
between the P- and S-wavefields shown in (18). Whenever the surface vibration is atten-
uated due to loss of vibrational energy, the total radiated intensity should be calculated
numerically instead. Fortunately, given the P and S velocity fields, it is quite straight-
forward, as well as exact to machine precision, to calculate the total radiated intensity
via the velocity, strain, and stress of the P- and S-waves. While the analytical wavefield
expressions in Section 3 are not as useful for our end goal as we had hoped, we have still
included them in the hope that they may be useful to other researchers.
We aim to follow up this investigation with a future article that explores the behaviour
of subsonic radiation into solids as described by (7) and the explanatory model underly-
ing it.
This work was supported by the Research Council of Norway under grant no. 237887.
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Full-text available
It is well-known that vibrating surfaces generate sound waves in adjacent fluids. According to the classical radiation model, the nature of these waves depends on whether the vibration’s phase speed cv is above (supersonic) or below (subsonic) the fluid sound speed cf. The transition between these two domains is known as coincidence. In the supersonic domain, the sound wave radiates into the fluid. In the subsonic domain, the classical model states that the wave becomes evanescent and clings to the surface. In the last 30 years, however, several articles on leaky guided waves have reported radiating waves in the subsonic domain, which is at odds with the classical model. In this article, we investigate an enhanced model for sound radiation near and below coincidence. Unlike the classical model, this model fully respects conservation of energy by balancing the radiated power with power lost from the guided wave underlying the vibration. The model takes into account that this power loss and the consequent attenuation of the surface vibration result in an inhomogeneous radiated sound wave — an effect that cannot be neglected near coincidence. We successfully validate the model against exact solutions for leaky A0 Lamb waves around coincidence. The model can also be used as a perturbation method to predict the attenuation of leaky A0 waves from the properties of free A0 waves, giving more accurate estimates than existing perturbation methods. We further investigate subsonic leaky A0 waves using the enhanced model. Thereby we, for example, explain the peculiar reappearance or persistence of the leaky A0 wave at lower frequencies, an effect brought to attention by previous theoretical studies.
Full-text available
Lamb waves are elastodynamic guided waves in plates and are used for non-destructive evaluation, sensors, and material characterization. These applications rely on the knowledge of the dispersion characteristics, i.e., the frequency-dependent wavenumbers. The interaction of a plate with an adjacent fluid leads to a nonlinear differential eigenvalue problem with a square root term describing exchange of energy with the surrounding medium, e.g., via acoustic radiation. In this contribution, a spectral collocation scheme is applied to discretize the differential eigenvalue problem. A change of variable is performed to obtain an equivalent polynomial eigenvalue problem of fourth order, which is linear in state-space and can reliably be solved using modern numerical methods. Traditionally, the leaky Lamb wave problem has been solved by finding the roots of the characteristic equations, a numerically ill-conditioned problem. In contrast to root-finding, the approach described in this paper is inherently able to find all modes and naturally handles complex wavenumbers. The full phase velocity dispersion diagram and attenuation curves are presented and are shown to be in excellent agreement with solutions of the characteristic equation as well as computations made with a perturbation method. The procedure is applicable to anisotropic, viscoelastic, inhomogeneous, and layered plates coupled to an inviscid fluid.
Full-text available
The physical characteristics for general plane-wave radiation fields in an arbitrary linear viscoelastic solid are derived. Expressions for the characteristics of inhomogeneous wave fields, derived in terms of those for homogeneous fields, are utilized to specify the characteristics and a set of reference curves for general P and S wave fields in arbitrary viscoelastic solids as a function of wave inhomogeneity and intrinsic material absorption. The expressions show that an increase in inhomogeneity of the wave fields cause the velocity to decrease, the fractional-energy loss (Q** minus **1) to increase, the deviation of maximum energy flow with respect to phase propagation to increase, and the elliptical particle motions for P and type-I S waves to approach circularity. Q** minus **1 for inhomogeneous type-I S waves is shown to be greater than that for type-II S waves, with the deviation first increasing then decreasing with inhomogeneity. The mean energy densities (kinetic, potential, and total), the mean rate of energy dissipation, the mean energy flux, and Q** minus **1 for inhomogeneous waves are shown to be greater than corresponding characteristics for homogeneous waves, with the deviations increasing as the inhomogeneity is increased for waves of fixed maximum displacement amplitude.
Full-text available
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.
Full-text available
This paper gives a historical survey of the development of the inhomogeneous wave theory, and its applications, in the field of ultrasonics. The references are listed predominantly chronologically and are as good as complete. Along the historical description, several scientific features of inhomogeneous waves are described. All topics of inhomogeneous wave research are taken into account, such as waves in viscoelastic solids and liquids, thermoviscous liquids and solids, and anisotropic viscoelastic materials. Also inhomogeneous waves having complex frequency are described. Furthermore, the formation of bounded beams by means of inhomogeneous waves is given and the diffraction of inhomogeneous waves on periodically corrugated surfaces. The experimental generation of inhomogeneous waves is considered as well.
Complex harmonic plane waves, which are characterized by a complex wave-vector and a complex frequency, may be divided into homogeneous plane waves (having parallel propagation and attenuation vectors) and nonhomogeneous or inhomogeneous plane waves. The last ones may be evanescent plane waves (having perpendicular propagation and attenuation vectors) or heterogeneous plane waves (the propagation and attenuation vectors being neither parallel nor perpendicular). All these waves may be either permanent ones or transient ones.
Ultrasonic guided waves in solid media have become a critically important subject in nondestructive testing and structural health monitoring, as new faster, more sensitive, and more economical ways of looking at materials and structures have become possible. This book will lead to fresh creative ideas for use in new inspection procedures. Although the mathematics is sometimes sophisticated, the book can also be read by managers without detailed understanding of the concepts as it can be read from a 'black box' point of view. Overall, the material presented on wave mechanics - in particular, guided wave mechanics - establishes a framework for the creative data collection and signal processing needed to solve many problems using ultrasonic nondestructive evaluation and structural health monitoring. The book can be used as a reference in ultrasonic nondestructive evaluation by professionals and as a textbook for seniors and graduate students. This work extends the coverage of Rose's earlier book Ultrasonic Waves in Solid Media.
In a recent Letter by Dickey et al., the velocity dispersion curve for the A0 mode for a fluid-loaded plate shows a splitting. However, this dispersion curve is incomplete and cannot reveal the reason for this anomaly. The nature of this anomaly has been investigated and the complete dispersion curve is presented. It is believed that this anomaly occurs because of the coupling of the A0, antisymmetrical zeroth mode, and the AS, antisymmetrical interface Scholte wave. The strength of this coupling depends on the ratio of the density of the loading fluid to the density of the plate material. The results of this numerical investigation for a solid plate loaded on the top and bottom surfaces with the same fluid and two different fluids are also presented.
Intended for use as both a textbook and a reference, "Fourier Acoustics" develops the theory of sound radiation uniquely from the viewpoint of Fourier Analysis. This powerful perspective of sound radiation provides the reader with a comprehensive and practical understanding which will enable him or her to diagnose and solve sound and vibration problems in the 21st Century. As a result of this perspective, "Fourier Acoustics" is able to present thoroughly and simply, for the first time in book form, the theory of nearfield acoustical holography, an important technique which has revolutionised the measurement of sound. Relying little on material outside the book, "Fourier Acoustics" will be invaluable as a graduate level text as well as a reference for researchers in academia and industry. It talks about the physics of wave propogation and sound vibration in homogeneous media. It deals with acoustics, such as radiation of sound, and radiation from vibrating surfaces; inverse problems, such as the theory of nearfield acoustical holography; and, mathematics of specialized functions, such as spherical harmonics.