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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: erlend.viggen@ntnu.no
Abstract
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·r−ωt). (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
ˆ
k=k/|k|. Their phase (k·r−ωt), and therefore also their pressure value p(r,t), is
constant over any plane perpendicular to ˆ
k.
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+iki∈C3, where kr∈R3,ki∈R3. (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·r−ωt)e−ki·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
ˆ
krdirection and still has planes of constant phase perpendicular to ˆ
kr, the pressure value
has an exponential decay in the ˆ
kidirection.
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 kr⊥ki. 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
c=ω
|kr|=c0
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 x–zplane on which an
attenuated surface vibration propagates with a complex wavenumber
kx=kxr+ikxi∈C, where kxr∈R≥0,kxi∈R≥0, (5)
giving a vibrational surface velocity
v(x,y=0, t) = v0ei(kxx−ωt)=v0ei(kxrx−ωt)e−kxix. (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]:
kxi=Iy0(kxr,kxi)
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.
2
cS0 <cP0 <cv
ˆ
kP
ˆ
kS
cS0 <cv<cP0
ˆ
kPr
ˆ
kS
kPi
cv<cS0 <cP0
ˆ
kPr
ˆ
kSr
kPi
kSi
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-
3
and S-waves’ particle velocities vPand vS, these wave equations are
1
c2
P0
∂2vP
∂t2− ∇2vP=0, with cP0 =sλ+2µ
ρand ∇ × vP=0, (8a)
1
c2
S0
∂2vS
∂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
v∗="v∗x0
v∗y0#ei(k∗·r−ωt)="v∗x0
v∗y0#ei(k∗r·r−ωt)e−k∗i·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 x–yplane 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., k∗x=kx. The wavenumbers’ ycomponents k∗y
can be calculated from the relation (ω/c∗0)2=k2
∗=k2
x+k2
∗y,
k∗y=k∗s1−kx
k∗2
=k∗v
u
u
t 1+kxi
k∗2
−kxr
k∗2!−i2kxrkxi
k2
∗
=k∗yr+ik∗yi. (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
k∗ris the radiation direction, we can use it to calculate the radiation angle. This
angle can be expressed in two ways:
θ∗=arctan kxr
k∗yr,α∗=arctan k∗yr
kxr, (11)
where θ∗is the radiation angle with respect to the surface normal, and α∗is the radiation
angle with respect to the surface.
The amplitudes v∗0of 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
vPx0
vPy0
vSx0
vSy0
=
vx0
vy0
0
0
(12)
that can be easily inverted with the help of a computer algebra system, giving
vP0 ="kx
kPy#kxvx0+kSyvy0
k2
x+kPykSy
,vS0 ="−kSy
kx#−kPyvx0+kxvy0
k2
x+kPykSy
. (13)
4
Thus, the P and S velocity fields v∗are fully determined. From them, we can calculate
the P and S strain tensors as
S∗ij =1
2 ∂u∗i
∂xj
+∂u∗j
∂xi!=−1
2iω ∂v∗i
∂xj
+∂v∗j
∂xi!=−1
2ωk∗jv∗i+k∗iv∗j. (14)
The time-harmonic nature of the wavefields means that the displacement u∗is related to
the velocity v∗as 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
σ∗ij =λδij S∗kk +2µS∗i j
(8)
=ρc2
P0 −2ρc2
S0δijS∗kk +2ρc2
S0S∗i 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
I∗i=Re −σ∗ijv∗j
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
2
=IPi+ISi+Re −σPi j vSj
2+Re −σSijvPj
2.
(18)
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
5
x
y
X
Yk∗r
k∗i
α∗
θ∗
X
Y
K∗r
K∗i
Figure 2: Real and imaginary components of the wavenumber k∗of a wave described in the x–
ycoordinate system and of the wavenumber K∗described in the wave-aligned X–Ycoordinate
system.
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
0≤α∗≤2π/4 in the x–ycoordinate system.
Therefore, we take a similar approach to Poirée [2], defining a X–Ycoordinate 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 K∗is
K∗X=K∗Xr+i0, K∗Y=0±i|K∗Yi|, where K∗Xr∈R≥0,K∗Yi∈R. (19)
Here, ±is the sign of the angle between k∗rand k∗i, and specifies whether the decay
occurs in the +Yor −Ydirection. We can express the inhomogeneity magnitude |K∗Yi|
more simply as |K∗Y|.
Expressing the wave in the X–Ycoordinate 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 x–ycoordinate 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 x–ycoordinate system with the desired orientation. In the former
case, we can determine the X–Ywavenumber vector K∗from the original x–ywavenum-
ber vector k∗as
K∗=R−1(α∗)k∗, (20)
where R(α∗)is the rotation matrix used to transform from the x–ysystem to the X–Y
system, and R−1(α∗)the rotation matrix that performs the opposite transformation:
R(α∗) = "cos α∗−sin α∗
sin α∗cos α∗#,R−1(α∗) = RT(α∗) = "cos α∗sin α∗
−sin α∗cos α∗#. (21)
The inhomogeneous wave’s speed c∗is always less or equal to the matching speed of
sound c∗0:
c∗=ω
K∗X
=ω
pK2
∗+|K∗Y|2,c∗0=ω
K∗. (22)
From these two equations, we can derive a useful identity that relates the inhomogeneity
magnitude |K∗Y|and the resulting change in wave speed c∗:
|K∗Y|2
ω2=1
c2
∗
−1
c2
∗0
=1
c2
∗1−c2
∗
c2
∗0. (23)
6
3.1 Aligned velocity fields
From (9), we already have an expression for the velocity field that can be easily expressed
in the X–Ycoordinate system as
v∗="v∗X0
v∗Y0#ei(K∗XX−ωt)e∓|K∗Y|Y. (24)
If v′
∗0is the velocity amplitude vector of an original wave in the x–ycoordinate sys-
tem, we can straightforwardly determine the velocity amplitude vector in (24) as v∗0=
R−1(α∗)v′
∗0.
However, if we are constructing this wave from the ground up, it is useful to consider
the relationship between the Xand Ycomponents of v∗0. 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=0⇒KPXvPY−KPYvPX=0⇒vPY=±i|KPY|
KPX
vPX, (25a)
∇ · vS=iKS·vS=0⇒KSXvSX+KSYvSY=0⇒vSX=∓i|KSY|
KSX
vSY.(25b)
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 |K∗Y|/K∗X. 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 S∗XX ,S∗XY =S∗YX , and S∗YY, as well as S∗KK =S∗X X +S∗YY .
Expressing the P-wave strain components in terms of vPX, which we chose as a scalar
P-wave amplitude, we find
SPXX =−KPX
ωvPX=−vPX
cP
, (26a)
SPXY =−1
2±i|KPY|
ωvPX+KPX
ωvPY(25a)
=∓i|KPY|
ωvPX, (26b)
SPYY =∓i|KPY|
ωvPY
(25a)
=|KPY|2
ωKPX
vPX
(23)
=1−c2
P
c2
P0 vPX
cP
, (26c)
SPKK =SPXX +SPYY =−cP
cP0
vPX
cP0
. (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
7
be
SSXX =−KSX
ωvSX=±i|KSY|
ωvSY, (27a)
SSXY =−1
2±i|KSY|
ωvSX+KSX
ωvSY(25b)
=−1
2|KSY|2
ωKSX
+1
cSvSY
(23)
=− 2−c2
S
c2
S0 !vSY
2cS
,
(27b)
SSYY =∓i|KSY|
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 σ∗XX,σ∗XY =
σ∗YX, and σ∗YY directly from (15).
For the P-wave, we get
σPXX = −ρcP+2ρcPc2
S0
c2
P0 !vPX−2ρc2
S0
cP
vPX=− 1+2c2
S0
c2
P1−c2
P
c2
P0 !ρcPvPX, (28a)
σPXY =∓2i|KPY|
ωρc2
S0vPX, (28b)
σPYY = −ρcP+2ρcPc2
S0
c2
P0 !vPX+2ρc2
S0 1
cP
−cP
c2
P0 vPX=− 1−2c2
S0
c2
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|
ωρc2
S0vSY, (29a)
σSXY =− 2c2
S0
c2
S
−1!ρcSvSY, (29b)
σSYY =∓2i|KSY|
ωρc2
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 v∗and 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:
I∗I=−1
2Re {σ∗IXv∗X+σ∗IYv∗Y}. (30)
8
(Bear in mind that the stress tensors are symmetric, so that σ∗YX =σ∗XY.) As σPIJ is
expressed through vPXand σSI J is expressed through vSY, we will end up with expressions
on the form
vPXvPX=|vPX|2(24)
=|vPX0|2e∓2|KPY|Y,vSYvSY=|vSY|2(24)
=|vSY0|2e∓2|KSY|Y. (31)
To calculate the P-wave intensity, we require the terms
σPXXvPX=− 1+2c2
S0
c2
P1−c2
P
c2
P0 !ρcP|vPX|2, (32a)
σPXYvPY
(25a)
=∓2i|KPY|
ωρc2
S0vPX∓i|KPY|
KPX
vPX(23)
=−2c2
S0
c2
P1−c2
P
c2
P0 ρcP|vPX|2, (32b)
σPXYvPX=∓2i|KPY|
ωρc2
S0|vPX|2, (32c)
σPYYvPY
(25a)
= −"1−2c2
S0
c2
P#ρcP!∓i|KPY|
KPX|vPX|2. (32d)
Together, they lead to a P-wave intensity
IPX= 1+4c2
S0
c2
P1−c2
P
c2
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
σSXXvSX
(25b)
=±2i|KSY|
ωρc2
S0vSY±i|KSY|
KSX
vSY(23)
=−2 c2
S0
c2
S
−1!ρcS|vSY|2, (34a)
σSXYvSY=− 2c2
S0
c2
S
−1!ρcS|vSY|2, (34b)
σSXYvSX
(25b)
= −"2c2
S0
c2
S
−1#ρcS!±i|KSY|
KSX|vSY|2, (34c)
σSYYvSY=∓2i|KSY|
ωρc2
S0|vSY|2, (34d)
which lead to an S-wave intensity
ISX= 4c2
S0
c2
S
−3!ρcS
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,cP→cS, 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
9
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
is
I∗=|G∗0|2e−2k∗i·rω
2ρω2k∗r+4[k∗r×k∗i]×[µik∗r−µrk∗i]
lossless
= 1+4c2
S0
c2
∗1−c2
∗
c2
∗0!ρω3
2|G∗0|2e−2k∗i·rk∗r.
(36)
In the last equality, we assumed the solid to be lossless, so that µi=0, µ=µr=ρc2
S0, and
k∗r⊥k∗i. 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 (k∗→K∗), 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 X–Ycoordinate system, which is aligned with the wave. We
can re-express these quantities in the x–ycoordinate system using the rotation matrix
from (21), so that
v′
∗=Rv∗,S′
∗=RS∗R−1,σ′
∗=Rσ∗R−1,I′
∗=RI∗. (38)
Here, the primed quantities are expressed in the x–ysystem, while the unprimed quanti-
ties are expressed in the X–Ysystem.
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
x–yplane 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:
10
0 0.5 1 1.5 2
0
0.25
0.5
0.75
1
Norm. vibration speed cv/cP0
Radiated intensity (arb. units)
kxi/kP=0
0 0.5 1 1.5 2
Norm. vibration speed cv/cP0
kxi/kP=0.02
IPy0ISy0IPy0+ISy0Iy0P-coincidence S-coincidence
0 0.5 1 1.5 2
Norm. vibration speed cv/cP0
kxi/kP=0.10
01234
Norm. vibration speed cv/cS0
01 2 34
Norm. vibration speed cv/cS0
01234
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 X–Ycoordinate system by the analytical expression
in (33) and (35) and use (38) to express them in the x–ysystem.
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.
11
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
Iy0̸=IPy0+ISy0.
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.
Acknowledgements
This work was supported by the Research Council of Norway under grant no. 237887.
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