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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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