Interaction of two spherical elastic waves in a nonlinear
five-constant
medium
IgorA. Beresnev
a)
Instituteof Physics
of theEarth, Russian,4cademyof Sciences,
BolshayaGruzinskaya10, Moscow
123810, Russia
(Received15 September1992;acceptedfor publication19 July 1993)
The interactionof two sphericallongitudinalwavesproducedby nonceincident
sources
with the
generationof a transversedifference-frequency
wave is studiedtheoretically.The equationof
motionfrom five-constant
elasticitytheoryto the secondorder in the displacement
is usedas a
mathematicalmodel. The generalsolutionfor the difference-frequency
field outsidethe wave
interaction region is obtainedin the form of a scatteringintegral. An analytical solution for
planeintersectingbeamsis derivedas a limit caseof smallinteractionvolumes.For the general
case,the solutionis analyzednumerically,andthe effectof the sphericityof interactingwavesis
comparedwith the resultsof plane-waveresonance
interaction.In the limit of largeinteraction
volumes,backscattering
at the differencefrequencyis formed.This effectsuggests
the useof this
typeof nonlinearinteractionfor remotesensingof the nonlinearpropertiesof the Earth'scrust.
PACS numbers:43.25.Lj
INTRODUCTION
Nonlinear interactions of noncollinear plane wave
soundbeamshave been studiedtheoreticallyin a number
f• and f2, respectively.
We alsoassumethat the interaction of thesewavestakesplacein a limited spatialregion
(Fig. 1). Elastic motion of the medium will be described
of a five-constant
elasticity
theoryJ
'•3'•4
of publications.
I-3 Selection
ruleshavebeenestablishedby theequations
that determined the conditions for resonance interactions
(the so-called"synchronism"
conditions),underwhichdirectedbeamsof sum and difference-frequency
waveswere
formed.Synchronism
conditions
definetherelationship
betweenfrequencies
of interactingwavesand compressional
For simplicity, the generationof only the differencefrequencywave (DFW) will be considered.
Let us write the equationof motionof a five-constant
theoryto the secondorder in the displacement:
o4u•)
• iis) /.
c•u
!z•. •au(•)
o- 1
and shear wave velocities for which the resonance interac-
tion is possible;the angleof a scatteredwave can be found
from theseconditionsfor given frequencyand velocity
ratios.
l These
selection
rulesforthescattering
ofsound
by
soundin solids(combinationscattering)havebeenvali-
datedexperimentally
in several
works.
4-7
Recentlythe useof combination
scatteringof seismic
waves to measure the earth's crust nonlinear coefficients
hasbeenproposed.
aThenonlinear
coefficient
ofrockmay
ou,
ou,
=(I•+• •Ox•,Ox,
Ox•,Oxt
/
!
ou,
x/ 0:%Out0%,Oui,
I
2
\c•uiOut //1
\ ! •uk Out
provideimportantinformationabout its mechanicalcon-
ditions,
porosity,
thepresence
of microcracks,
etefi
'9-12
However, seismicwavesexcitedby point-like surface-
basedsourcesspreadspherically,makingthe theoryof interactingplanebeamsof limited applicabilityto seismology. Spherical spreading also makes interactions
intrinsicallynonresonant.A theory of the interactionof
sphericallongitudinalwavesproducedby nonceincident
sourcesis developedin this paper.
O•utOuk\
O•ukOut
+Ox,
Ox•,
•-•x•
}+(a+2C)
ox-•-_-•,Ox/,
whereu?• isthe/thcomponent
ofthedisplacement
in the
scattereddifference-frequency
wave,ui is the /th componentof the primarywaves,K andp are second-order
elastic toodull, and A, B, and C are third-order moduli.
Following
Jones
andKobett's
• approach,
wewillsolve
Eq. (1) using a G-reen'sfunction method for an inhomo-
I. THEORY
We assumethat two pointsourcesO• and O2 of longitudinalelasticwavesare locatedin a homogeneous
and
isotropicelastic solid at some distancefrom each other.
Theyradiatecontinuous
sinusoidal
waveswithfrequencies
a)Present
address:
Institute
ofEarthSciences,
Academia
Sinica,
P.O.Box
1-55, Nankang,Taipei, Taiwan ! 1529,R.O.C.
geneous
waveequation.The right-handsideof Eq. ( 1) is a
knownfunctionof primarywaves.Usingthe designations
of Fig. 1, the primarywavefield canbe represented
as a
vector sum of spherical longitudinalwaves with the
sourcesat pointsOl and 02:
Fo
[Rl
ei(klRi-2•rflt)
q_R•
l'
ui=2[
2•>i(k•R2-2•r
(2)
3400 J.Acoust.
Sec.Am.94(6),Decemb.
er1993 0001-4966/93/94(6)/3400/5/$6.00
¸ 1993Acoustical
Societ•
ofAmerica
3400
02
M(x,y,z)
resonanceinteractionof planewaveswith the interaction
of sphericalwaves,to infer how sphericityaffectsthe scattered field.
Using the expression
(S)tr
21r
f) f•,Gij(ri,r
•,2rrf)pj(r;
,2rrf)do,
(5)
Ui
•, i•
whereGq(ri,r• ,2rrf) isthe"transverse"
pa• of a Green's
tensor,
pd(r• ,2•f) istheFourier
transfore
ofthe•. (3),
Y
]
and the integrationis cardedout over the interactionvolume V, we obt•n the frequency-domain
solutionfor ith
componentof the scatteredtransversewave field at the
......
pointM outside
theinteraction
region(thesuperscript
is omittedbelow). Applying an inverseFourier transfore,
FIG. 1. Geometryof theproblem.Here,OI andO2aresources
of primary
waves,k• andk2 aretheir wavevectors,M is an observation
point,M' is
a pointbelonging
to theinteraction
region,O isorigin,V istheinteraction
region;R, R•, R2, r, r' are radiusvectors.Directionsof the axesof a
Cartesiancoordinatesystemare alsoshown.
we obtain the final solution in the time domain. For the x
com•nent of the scatteredfield, the solutionis
cos
20+n[
[k2_k,•
u•tri,t)
=kik2zF•
3• f•K•1 m
R,R2
[k
R2
whereR1,2
= [R•.2I,F0 istheforceamplitude
produced
by
(1-KaRa
X K_R sin•+cos•
the sources,/•is the shearmodulus,and the subscripti in
the right-handsidemeansthe projectionof a fieldon the i
axis.We usein Eq. (2) the expression
for the far field of
1
-•[R(k2-k•)]
piston
sources
ina solid
•5tomaketheformulas
applicable
to seismicexplorationproblems.We assume,therefore,
that the interactionregionis locatedin the far fieldof the
X[ •
sources,
i.e.,theconditions
kl.2Rl,2•laresatisfied.
After
wesubstitute
expression
(2) intotheright-handsideof Eq.
( 1), andkeeponlytermswith the difference
frequency,
the
right-handsidebecomes
pi= (F•/4rrzl.
t2)Iisin[2rr(fl--f2)t-- (klRi-k2R2)],
sin•
(6)
whereK_ is the D•
wavenumber,m=K+7•/3+A
+2B,
n=K--2•/3+ 2B+ 2C,
kL2=kh:RL:/Rh•;
ß =--k•R•+k2R•--2•(f•--f2)(R/ct--t),
and ct is the
shearwave velocity.
(3)
where the amplitudefactor Ii has the followingcomponents:
I klk2[
WAVE
INTERACTION
In the generalcase,the integral (6) must be analyzed
numerically.However, in the casewhere the dimensionsof
k2 kl]
the interactionregionV aresmallenoughcomparedto the
distancefrom the sources
to the interactionregion,sothat
the curvatureof the primary sphericalfrontscan be neglectedwithin it, an analytic solutioncan be found. In
other words,this is the caseof plane-waveinteraction.
AssumeV to be sufficientlysmall.Then slowlychanging amplitudefactorsin the integrandof Eq. (6) can be
taken out of the integral. Also, square roots like
Ix= 2RiR2[rXl-L)•2-Xl
•lJ['"],
Iy=Yl
kj\['"1,
2 klk2/k2
I•= 2R•Rz•,Rz
R•J
['"1;
['"1: [(K++Z+2a)cos 2e
+ (K--• + 2B+ 2C)I;
II. PLANE
R:[(x--x')2+(y--y')2+(z--z')2] 1/2,etc.canbedevel(4)
x•, )h, and z• are coordinatesof the vector1t• and L is the
distance between the sources.
Applying a Fourier transformto Eq. ( 1) with a right-
opedin a series,and we can keeponly linear terms with
respectto r'/r. The quantitativecriterionfor applicability
of thesemanipulations
is givenby
r' • (/[R•TM
) •/2,
(7)
hand side defined by F__qs.
(3) and (4), we find a vector
where/1,
is theminimum
primarywavelength
andR• inis
Helmholtz equation,for which the exactform of a Green's
the minimum distancefrom the sourcesto the pointsin the
(tensor)function
isknown)UsingthisGreen's
function, interaction region.
verse scattered waves. We will confine ourselves to the
The abovemanipulationssignificantlysimplifythe integrandin Eq. (6). Assumingthat the interactionregion
transversescattered wave solution, since in the plane wave
has the form of a parallelepiped,that the origin and obser-
we canobtainthe solutionsfor bothlongitudinalandtrans-
case,onlythe transverse
difference-frequency
wavesatisfies vationpoint are both locatedin the (x,z) plane,and that
theresonance
conditions,
•-3whentwolongitudinal
waves the observationpointis in the far fieldof DFW (K_R >>1),
we can rewrite the solution (6) in the followingform:
interact.In thefollowing,wewill comparetheresultsof a
3401 J. Acoust.Soc. Am., Vol. 94, No. 6, December1993
Igor A. Beresnev:Interactionof two sphericalelasticwaves 3401
Fgflf2V
rnq-n
cos
/ k2
kI 1,cosq+e'
ux(r,
cp,t)_-8•rbt2pct2c•
RoRo(2
)20
sin•v(R--•0
--•00--•(•
sin•v)
X
sin[ (a/A_) •r cosq•q-(a/2)/J' ] sin[ (c/A_ ) •r sinq q-(c/2) e' ]
[ (a/A_)•r cos•+ (a/2)•5'] [ (c/A_),r sinq•+ (c/2)e']
sin[ -- kiRo+ k2R?-- 2rr(fl -- f2) (r/ct--t) ],
-
%
(8)
el \
where
R0andR(0
2)arethedistances
fromthesources
tothe
originlocatedat the centerof parallelepiped,
x0 andZoare
the coordinates
of the vectorRo, a and c are the parallelepipedsides,parallelto thex andz axes,respectively;
c/is
the longitudinal
wavevelocity,p is the density,A_ is the
DFW wavelength,
q is the anglebetweenthe vectorr and
the z axis,and 0 is the anglebetweenthe vectorsRe and
such that the center of the interactionregion is a point
where synchronismconditionsfor primary wave vectors
are satisfiedexactly. This means that the angle between
vectorsconnectingpointsO• and O, and O2 and O in Fig.
2 isdefined
bytheexpression
•
cos0=l/q22+
[ (q22 1) (•+ 1)/2q:•],
(9)
whereql =f2/f! and q2=ct/ct.
The displacementin Eq. (8) is the differenceStrictlyspeaking,the interactionis out of synchronism
frequencywavedisplacement
in the x direction,generated at everypoint exceptfor the centerof the region,because
by two planeintersecting
longitudinal
soundbeams.
I-3
Note that the amplitudeof the scatteredwave is proportionalto the primaryfrequencies,
the squaredamplitudeof
the force developedby the sources,the volumeof the interactionregion,and the nonlinearelasticconstantsof the
material. The angular factor in Eq. (8) dependingon q
describesthe directivity of the scatteredwave in the (x,z)
plane.A characteristic
featureof the scatteredwavedirectivity pattern is that the DFW radiationis summedup
coherentlyin the directiondefinedby the synchronism
conditions.Coherentsummationof the elementarysecondary sourcescausesthe proportionalityof the DFW amplitude to V. It can be shownby straightforwardcalculation
that the directionof a main lobe of the directivitypattern
givenby the angularfactorin Eq. (8) coincideswith the
directiondeterminedby synchronismconditions.
III. THE GENERAL
INTERACTION
When the maximum
of the sphericityof interactingfronts. In the caseof small
V, the interactioncan be consideredapproximatelyresonant, but for large V it is no longer resonant.In our calculations,we consecutivelyincreasedthe volume of interaction, beginningwith small V satisfyingthe condition
(7), and increasingV to sizessignificantlyexceedingthis
limit, as shownin Fig. 2, so that the influenceof sphericity
on the resonanceinteractioncouldbe studied.Amplitudes
and directivitypatternsof the DFW werecalculatedwithin
a circle of radius 1000 m.
Figure 3 shows the calculated directivity patterns
u•(cp)/u•m•'(q)ofthe DFW in the (x,z) plane. Figure
01
L
02
CASE
dimension of the interaction
re-
gion does not satisfycondition (7), the curvature of the
primary frontscannotbe neglected,and the integralin Eq.
(6) must be analyzed numerically.
In the numericalexamplegiven below, we use values
of the parametersin Eq. (6) that are typical for explora-
tion seismologyexperiment: L=600
f2=70
m, fl=50
Hz,
Hz, F0=500 kN, cl=3000 m/s, ct= 1500 m/s,
p=2000kg/m3;andthe values
for m/pc•andn/Pc•,
which are the analogsof the nonlinearparameterfor this
typeof interaction,
areequalto 103
. Notethatnonlinear
parameters
forearthmaterials
canhavevalues
upto 104
(12).To simplify
thenumerical
integration,
theinteraction
regionwas taken to be a cubewith its centerlocatedsymmetrically betweenthe sources(Fig. 2). The depth of the
center d was taken to be 1000 m.
The geometryusedin a numericalexampleis selected
3402 d. Acoust.Sec.Am.,VoL94, No.6, December1993
FIG. 2. Geometryusedin numericalexample.Here, a is sideof the cube,
d is depthof a centerof the cube,1-3 showconsecutively
increasing
size
of interaction
volume,4 isa circleonwhichtheamplitudes
anddirectivity
patternsof DFW are calculated.
IgorA. Beresnev:
Interaction
of twospherical
elasticwaves 3402
(b)
(c)
10-10
10-12
(d)
(e)
(f)
10-14
FIG. 3. Directivitypatternsof the difference-frequency
wavecalculated
for differentsizesof interaction
volume:(a)•a= ( 1/3)•._,
(c)--(16/3)•_, (d)--(22/3)Z_, (e)--(28/3),•_, (f)--(56/3)•._,
wherea is thecube•ideand•_ is theDFW wavelength.
10-16
-
tc• I
105
I
107
I
I
109
Volume
of Interaction
(m3)
3(a)-(f) corresponds
to increasing
the sizeof the interacof thedifference-frequency
wave
tion volume.The maximumdimension
in Fig. 3(a) and FIG. 4. Amplitudeof thex component
versus
thevolume
ofinteraction
(a) in thesynchro(b) satisfies
thecriterionof Eq. (7), sothattheplane-wave particledisplacement
nismdirectionand (b) the directionof 310e,corresponding
to a maxiapproximation
can be applied,and numericallycalculated mum radiationfor largeinteractionvolumes.
directivitypatternscoincidewith the resultsof calculation
usingasymptotic
formula(8). Figure3(a) hasa quasiomThereis a question
abouttheeffectof theshapeof the
nidirectionaldirectivitypattern,becausethe interaction
volumeon the directivityof the scatteredDFW wave.The
volumeis sosmallthatit canbeconsidered
a pointsource
calculations
so far are limitedto cubes.We cansuppose
(cube side a----/•_/3). Directional radiation is formed in
from qualitativeconsiderations
that for volumesthat are
Fig. 3(b), wherethe main lobeis orientedin the direction
smallcompared
withtheDFW wavelength
theeffectof the
of synchronism
(a = 23._). Figure3(c)-(f) corresponds
to
shapeis negligible,
because
the regionof interactioncanbe
the interaction
volumesthat no longersatisfythe criterion
regardedas the point-like source.On the other hand, for
(7), andthe form of the direetivitypatternsis affectedby
largevolumeshavingmanywavelengths
on its lineardithe sphericityof interactingwaves. In Fig. 3(c)
mension
thecharacteristic
featuresof the radiationpattern
[a=(16/3})t_] the angularwidth of the main lobedoes
are formedin the bulk of the volume,and the presence
of
not decrease,
comparedwith Fig. 3(b), as it shouldbe for
the comersof a cubecanaffectonly minordetailsof the
theplanewavecase,andFig. 3(d) [a= (22/3}J._] already directivitypattern.However,an accuratestatementcould
showsa significantdeviationof the radiationdirection be made after numerical examination.
fromtheplane-wave
radiation
direction.
Newlobesappear
The existence
of a limitingdirectionof the DFW raas well. Further increasingthe interactionvolume, the
diation, which doesnot changeas interactionvolumeinDFW radiationconcentrates
in the directiondefinedapcreases,
canbe explainedby the sphericalspreading
of the
proximately by the angle 310ø [Fig. 3(e) and (f),
primary waves,whoseamplitudesdecreaseas 1/R. Remote
a=(28/3)J._ and (56/3)J._, respectively].
Calculations
for larger V showedthat this latter directionis stableand
doesnot change(increasingof interactionvolumewaslimited, of course,by the requirementthat it must have remained in the far field of sources).
Figure4 showsthe amplitudeof thex component
of
the DFW fieldversusthevolumeof interaction
in thesynchronism direction (a) and in the 310ø direction (b).
Curve (a) showsthat the amplitude in the synchronism
directiongrows linearly with volume, in accordancewith
zonesof the interactionregion do not affect the result of
the interaction,becauseamplitudesof the primariesbecome very small there. Thus, characteristicsof the DFW
field are determinedby a limited regionof space,where
intensities
of the secondary
sourcesare the largest.This
factshowsthat in the realseismic
experiments
the characteristicsof the recordednonlinearlyscatteredfield will be
dominatedby the integratednonlinearpropertiesof a spatially limited regionof the Earth'scrust.In theseexperiments the actual directivity of the primary vibratory
sourceswill alsocontributeto the confiningthe interaction
Eq. (8), while the approximationof plane interacting
wavesholds.Then amplitudegrowthstops.Amplitudeof
volume.
radiationin 310ø directionis two ordersof magnitude
weakerfor smallinteractionvolumes,but increases
rapidly IV. DISCUSSION AND CONCLUSIONS
when the interactionvolumesare large. This spatialdirection corresponds
to a stablesummationof quasicoherent
Figure 3(e) and (f) showsthat the main lobeof the
secondarysources,whenthe regionis largeand the influDFW radiationhasa "backward"direction,sincethe prienceof sphericityis significant.
mary sourcesare at the top of (x,z) plane. Thus, the in3403 J. Acoust.Soc. Am., VoL 94, No. 6, December1993
Igor A. Beresnev:Interactionof two sphericalelasticwaves 3403
teractionof sphericalwaveson largenonlinearinhomoge-
I G. L. Jonesand R. D. Kobett,"Interaction
of ElasticWavesin an
neities leads to a backscattered
IsotropicSolid,"J. Acoust.Soc.Am. 35, 5-10 (1963).
2j. D. Childtessand C. G. Hambrick,"InteractionsBetweenElastic
transverse difference-
frequencywave.The amplitudeof the backscattered
wave
reaches
thevalueof 10-1ø-10
-9 m (Fig.4), forthevalues
of parametersusedin the numericalexample.These conditions could be reconstructedin a real field experiment.
The aboveamplitudeof the DFW is equalto amplitudesof
weak microseismicnoise, implying that nonlinearly scattered radiation can be realisticallyrecorded.
Due to the fact that the interactionregionfor spherical
wavesis localizedin space,and that the DFW-radiation
amplitudeis proportionalto its nonlinearparameter,this
type of nonlinearinteractioncouldbe usedin remotesensing of nonlinearpropertiesof the Earth'scrust.If vibratory
sourcesof seismicwavesare employed,the depth of the
interactionregioncan be alteredby changingdistancebetween sources.The valuesof nonlinearparametersat different depths would provide valuable information about
the physicalstateof the in situ rock.
Wavesin an IsotropicSolid," Phys.Rev. A 136, A411-A418 (1964).
3L.H. TaylorandF. R. Rollins,
Jr.,"Ultrasonic
StudyofThree-Phonon
Interactions. I. Theory," Phys. Rev. A 136, A591-A596 (1964).
4F. R. Rollins,Jr., "Interactionof UltrasonicWavesin SolidMedia,"
Appl. Phys. Lett. 2, 147-148 (1963).
5H.-F.Kun, L. K. Zarembo,
andV. A. Krasil'nikov,
"Experimental
Investigationof CombinationScatteringof Soundby Soundin Solids,"
Soy. Phys. JETP 21, 1073-1077 (1965).
6p.A. Johnson,
T. J. Shankland,
R. J. O'Connell,
andJ. N. Albright,
"Nonlinear Generationof ElasticWavesin CrystallineRock," J. Geophys.Res. 92 (BS), 3597-3602 (1987).
P. A. Johnson
andT. J. Shankland,
"Nonlinear
Generation
of Elastic
Waves in Granite and Sandstone: Continuous Wave and Travel Time
Observations,"$. Geophys.Res. 94(B12), 17 729-17 733 (1989).
sG. M. Shalashov,
"An Estimation
of thePossibility
of Nonlinear
Vibrational SeismicProspecting,"Izv. Acad. Sci. USSR, Fiz. Zemli
(Physics of the Solid Earth) 21, 234-235 (1985).
0I. A. Beresnev,
A. V. Nikolayev,
V. S.Solov'yev,
andG. M. Shalashov,
"NonlinearPhenomena
in SeismicSurveyingUsingPeriodicVibrosighals,"lzv. Acad. Sci.USSR, Fiz. Zemli (Physicsof the SolidEarth) 22,
804-811 (1986).
ACKNOWLEDGMENTS
mi. A. Beresnev
andA. V. Nikolaev,
"Experimental
Investigations
of
I would like to thank A. V. Nikolaev
from the Insti-
NonlinearSeismicEffects,"Phys.Earth Plan. lnt. 50, 83-87 (1988).
B. P. BonnerandB. J. Wanamaker,
"Acoustic
Nonlinearities
Produced
tute of Physicsof the Earth in Moscow, G. M. Shalashov
from the RadiophysicsResearch Institute in Nizhniy
Novgorod (formerly Gorky), Russia, B. Ya. Gurevich
from the VNIIGeosystemsInstitute in Moscow,and P. A.
•2L.A. Ostrovsky,
"WaveProcesses
in MediaWith StrongAcoustic
Johnsonfrom the Los AlamosNational Laboratory,USA,
for the discussions
and suggestions
that largelyhelpedin
•3L. D. Landau
andE.M. Lifshitz,
Theory
ofElasticity
(Pergamon,
New
the fulfillment
of this work.
Katherine
McCall
of Los
AlamosLaboratorycarefullyreadthe manuscriptandsuggestedsignificantimprovements.
I appreciatethe recommendationsof the reviewer of this paper. My special
thanksis to Ruth Bigiv for preparingthe illustrations.
3404 J.Acoust.
Soc.Am.,Vol.94,No.6, December
1993
by a SingleMacroscopic
Fracturein Granite,"in Reviewof P•ogress
in
Quantitative
Nondestructive
Evaluation,
editedby D. O. Thompson
and
D. E. Chimenti (Plenum, New York, 1991), pp. 1861-1867.
Nonlinearity," J. Acoust. Soc. Am. 90, 3332-3337 (1991).
York, 1986), pp. 104-107.
•4Z.A. Gol'dberg,
"Interaction
of PlaneLongitudinal
andTransverse
ElasticWaves,"Soy.Phys.Acoust.6, 306-310 (1960).
•SG.F. MillerandH. Pursey,
"OnthePartition
of Energy
Between
ElasticWavesin a Semi-InfiniteSolid," Proc. R. Soc.London,Ser. A
233, 55-69 (1955).
IgorA.Beresnev:
Interaction
oftwospherical
elastic
waves 3404