interaction of elastic waves in an isotropic solid

THE JOURNAL OF THE ACOUSTICAL SOCIETY OF AMERICA VOLUME 35, NUMBER 1 JANUARY 1963

Interaction of Elastic Waves in an Isotropic Solid

GERALD LEE JONES* AND DONALD R. KOBETT

Midwest Research Institute, Kansas City 10, Missouri (Received 5 January 1962)

Nonlinear elasticity theory is used to investigate the scattering of two intersecting, plane, elastic waves in a homogeneous, isotropic medium. A criterion for the occurrence of a strong scattered wave is derived. The criterion is formulated as a relationship between the first-order elastic constants of the material, the angle between the intersecting wave vectors, and the ratio of wave frequencies. The exact formulation also depends on the type of intersecting waves, i.e., longitudinal or transverse. The amplitude of the scattered wave is found to be proportional to the volume of interaction and dependent on the third-order elastic constants of the material. Numerical results are given for wave scattering in polystyrene.

INTRODUCTION

N the linear theory of elasticity two elastic waves do not interact. The equations of motion are linear and therefore the principle of superposition holds. Any solution of the equations of motion can be written as a linear combination of monochromatic waves. The linear

theory of elasticity results from assuming the elastic energy to be quadratic in the particle displacements. If terms cubic in the particle displacements are included in the elastic energy, the equations of motion become

nonlinear? This nonlinearity gives rise to an effective interaction between two plane elastic waves which can produce scattering. The scattering of two collimated, monochromatic, plane waves in an infinite isotropic solid is considered in the present paper.

I. THEORY

When terms which are cubic in the particle displacements are included in the elastic energy, the resultant nonlinear equations of motion for an isotropic solid are •

Ox•Oxk Oxz OxzOx• •x•/

( • A •(02uz Out q- K-{--nt---nt-B 3 4 / Ox-7Ox•Ox•

+(B+ OxiOx• • / 2C)\ox-•x• •x•/'

where o0 is the density of the undeformed solid, u• is the ith component of the particle displacement, K is the compression modulus, and u is the shear modulus. A, B, and C are third-order elastic constants, i.e., they are the coefficients of cubic-strain terms in the elastic

energy. Subscripts appearing twice in a single term indicate summation over the values 1, 2, 3. The terms x•, x2, and xa are rectangular coordinates.

* Present address' at State University of Iowa, Iowa City, Iowa.

The left side of Eq. (1) is linear in u while the right side is quadratic. In practical applications the displacement vector u is small and therefore the right side of Eq. (1) will be small compared to the left. To solve Eq. (1) we set u=uøq-u 8 where u ø is the solution when the right side of Eq. (1) is zero, and u 8 is a presumably small correction arising from the right-hand side. Being a solution to the linear equation, u ø consists of

x Z. A. Goldberg, Soviet Phys.--Acoustics 6, 306 (1961).

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6 G. L. JONES AND D. R. KOBETT

a superposition of monochromatic waves. Since we are interested in the mutual scattering of two waves we take

uø=Ao cos (o•lt-kl.r)+Bo cos (o•2t-k•..r). (2)

In Eq. (2) the amplitude vector may be chosen either parallel or perpendicular to the wave vector k, that is, the primary waves are either transversely or longitudi- nally polarized. For a transverse wave o•=Ctk, where Cs = (t•/oo) «. For a longitudinal wave o•=Ctk, where C•=E(K+•)/•o•.

Now to solve Eq. (1) we substitute u= uø+u s on the left side and u=u ø on the right side. This should be a good 'approximation if u s is small compared to u ø. Since u ø is a solution of the linear equation it disappears from the left side and we obtain

Ot 2OxkOxk •/OXDX, p•' (3) where the vector p is determined by putting u ø in the right side of Eq. (1). If we use Eq. (2) for u ø, p will involve a sum of products of two monochromatic waves. Some of the terms will represent the interaction of a primary wave with itself. This interaction has been treated before • and we shall not include it here.

Including only those terms representing interactions between the primary waves we find for p after some tedious manipulation

p(r,t) = I + sin{ (co1-t- co :•) /-- (kl-+-ks. r} + I- sin{ (Wl--Ws)l-- (kl-ks).r},

where

(4)

I :•= -« (u+-}A){ (Ao. Bo)(ks. ks)kl-q- (Ao. Bo)(kl. kl)k2 + (ao. kl) (ks ks)Ao+ (Ao. ks (kl. kl)ao + 2 (Ao. ks (kl. ks)Bo+ 2 (Bo. k;) (k•. k•)Ao}

« (K+ «t•+-}A + B){Ao. ao)(kl. ks)ks q- (Ao. Bo) (kl. ks)k1+ (Bo. ks (kl. ks)Ao

q- (Ao. k•)(k•. ks)Bo} « (-}A q- B){ (Ao. ks) (ao-k2)kl+ (Ao-kl) (ao. kl)k2

+ (Ao. ks)(Bo-kl)ks+ (Ao. ks)(Bo. k•)kl} « (B+ 2C){ (Ao. kl)(ao.ks)ks+ (Ao. kl)(ao. ks)k1}.

Point of

ß Observation

Fro. 1. Vector arrangement.

Since p is a known function of r and t, Eq. (3) is now a linear inhomogeneous equation for the scattered wave u s, with p acting as a source term. Instead of writing Eq. (3) in component form we prefer to use vector notation and write

(OSus/OF) (r,t)- C•SV { V. u s (r,t) } +CtSV x V x u s (r,t) = 4rq(r,t) (5)

where 4rq=p/p0. This is the standard form for the inhomogeneous vector wave equation? From now on we shall be interested only in the scattered wave, so we shall drop the superscript s until further notice. We introduce the time Fourier transform pairs

u (r,o•) = f_+• eiø•tu (r,t)dt, u(r,t)=--

q (r,o•) = f_+: ei•tq (r,t)dt, (6)

1 f_+: e-iø'tq (r,o•)&o. q (r,t) = •-•r From Eq. (5) the equation for the Fourier transform is

-- o•Su (r,o•) -- C tsV { V' u (r,o•) } + C tsV x V x u (r,o•) = 4;rq (r,co). (7)

From Eq. (4) and Eq. (6) one finds that I +

q (r,o•) = •{ exp[--- i (k•+ks. r-lb (o• + 4ip0

-- exp[-i (k1-1- k2). r-lb (o•- o•t-- o•s }

+•{ exp[--- i (k t- k2). r-]• (o• +o•x-- o•,) 4ip0

-- exp[-i (k l- ks. r•i (o•- o• +o•s } (8)

where •i(a)=0 for a•0, •i(a)= o• for a=0, and

Now Eq. (8) is not quite right. We are interested in the case where the primary beams are well collimated. The interaction term q, which is a product of the amplitudes of the primary beams, will be zero unless we are in the region where the primary beams intersect. We shall denote both the region of intersection and its volume by V. Then Eq. (8) is valid only inside V and q is zero outside V.

If one assumes that u(r,o•) decreases at least as fast as 1/r for large r, then it can be shown sthat the solution for Eq. (7) in the infinite region can be written as

•' P.M. Morse and H. Feshbach, Methods of Theoretical Physics, Part II (McGraw-Hill Book Company, Inc., New York, 1953), Chap. 13.


WAVE INTERACTION IN A SOLID 7

u[r,w)= j l•(r,r',w)q(r',w)dV (9) v ,

where I• is a tensor or dyadic operator. For the infinite region, I• can be written •'

= + (lO)

where R= r- r' (Fig. 1) and R= I R[ and ! is the unit dyadic.

We now choose r so large that

(w/C•)R>>I, (w/COR>>i (13)

for all r' in V. If in addition I r'[<<lr[ we have R"-•r -- •. r' (14)

In Eq. (9) the term arising from I• will be longitudinal, while the term arising from l•t will be transverse. In dyadic notation I• and l•t are given by •'

RR[3-i(w/Cz)R-

where • is a unit vector in the direction of r, i.e., •= r/r. With these assumptions we can simplify Eq. (11) and Eq. (12). Firstly, we keep only the leading terms in (w/C•)R and (w/C,)R. Secondly, we replace R by r--•. r' in the exponentials and R by r elsewhere. We obtain

exp•i(w/C•)R• X , (11) l•(r,r',w/Ct) R

=Pi exp --i--•.r' , (15) r C•

exp•i(co/Ct)R• X , (12)

R

(16)

Now using Eqs. (15) and (16) in Eq. (10) we can find I• and using I• in Eq. (9) with q given by Eq. (8) we can find u(r,w). Then from Eq. (6) we can get u(r,t). The result of this straightforward but tedious procedure is

u (r,t)-(I+'•) i sin{(W•+W2r kl-k2)ort-((.Ol+(.o2)(•ll-t)}dV d .... sin. •-kl+k• .r'--(wx--w•) --t dV

4•C•2p0 r k C•

) _ d sin -k•-k2 .r' (Wl+W2) •--t dV 4rC•2por Ct C•

) d .... sin •-k•+k2 .r'- (w•-w2) -t dV. 4•Ct2por Ct

(17)

The first and second terms in Eq. (17) are longitudinal waves with the sum frequency w•q-w•. and the difference frequency w•--w2, respectively. The third and fourth terms are transverse waves with the sum and difference

frequencies, respectively. The terms in the arguments of the sine function

involving ['(r/C)--t-] do not vary during the r' integration. Now look at the first term in Eq. (17). As we integrate over r' the integrand oscillates with frequency determined by the coefficient of r', i.e., [-(w•+wa)/C•]• -kl-k• in this case. In general, the result of this integration will depend on just how the waves fit into the region V. As we increase V the value of the integral

will oscillate between fixed limits, unless we can find a direction •8 for which

0. (18)

If we can find such an 28, the integrand becomes constant for 2-2, and the amplitude of the scattered wave in this direction becomes proportional to the volume of interaction V. By increasing the volume V one can indefinitely increase the amplitude of the scattered wave in the direction 2•. In any other direction the amplitude does not increase indefinitely with volume but oscillates. For the proper choice of experimental parameters this will lead to scattering


8 G. L. JONES AND D. R. KOBETT

TABLE I. Interaction cases which produce a scattered wave.

Primary waves Resonant wave Direction of

type and frequency scattered wave cos qo •

Longitudinal Two transverse

Transverse

Two longitudinal (•-•)

One longitudinal and Longitudinal one transverse e

One longitudinal and Longitudinal one transverse e

(•-•)

One longitudinal and Transverse one transverse e

(•-•)

kl + ks cSq - [ (d- 1) (d'+ 1)/2a-]

(kl- ks)/(col-co,.) 1/c"+[-(d'- 1)(aS+ 1)/2ad']

kid-ks c+[-a(d'- 1)/2c-]

Frequency limits b

c + [-a (1--d') /2c']

1--c l+c --<a<----

l+c 1--c

1--c l+c <a<--

l+c 1-c

2c 0<a<•

2c 0<a<--

l+c

1--c l+c

2 2

•, is the angle between kz and ks at resonance; a is the frequency ratio co,./co•; c is the velocity ratio Ct/C•. When a is within the limits shown, it is possible to choose an angle • that will give a scattered wave. The frequency of the longitudinal primary wave is ,oz.

which is sharply peaked in one direction and whose amplitude is proportional to the volume of interaction. We shall call this part of the wave the scattered wave as opposed to the rest of the outgoing wave which has the character of a diffracted waveß Equation (18) will be called the resonance condition. We have so far

considered only the first term in Eq. (17); however, the character of the remaining three terms is the same as that of the first. The resonance conditions for the

remaining three terms are, respectively,

E(•i-•)/C•]•-(k•-k•) = 0

[_(w•+w•)/C•-(k•+k•) = 0

•(w•-w•)/C•?,-(k•-k•) = O.

II. INTERACTION CASES

Three cases of interaction between two intersecting waves must be considered, namely: (1) two transverse waves, (2) two longitudinal waves, and (3) one transverse and one longitudinal wave. We shall consider the case of two transverse waves in detail by way of example. The results for all three cases are given later in Table I.

For the case of two transverse waves we have

Ao-kl=Bo-k:•=O and 0olfCt=k1, 0o2fCt:k2.

We want first to see if the resonance condition Eq. (18) can be satisfied. We must have

[ (c0 •+ c0 2)/C•] 2= k z2+ k•q - 2k 1' k• or

w 1--]-oJ 2• 2 c012 c022 2WlW2 -_+_+• C• / C? Ct • Ct •

COSq•

where e is the angle between kl and k•. The above leads to

cos•=Ct____•+_l(Ct____ •_ C• 2 2\C• • xco2 (.01/

In order that this equation be satisfied we must have the right side less than one and greater than minus one. This leads to the condition that

1--St/St 001 1 +St/St ß

i +Ct/C• 002 1--Ct/C•

For any o•/oa in this range we can choose the angle e between the primary wave vectors k• and k• so that we get a scattered wave (appearing in the direction of k•+k•). If one examines the resonance conditions for the last three terms in Eq. (17) it turns out that none of them can be satisfied for this particular choice of primary waves. For the scattered wave then, we have from Eq. (17) only the longitudinal wave

u(r,t) (l+.f•)f• (•) t- . (19) 4rC•po r

Using the fact that Ao-k•=Bo.k•=0 in this case we have from Eq. (4)

I+= -} (u+•A) { (Ao. Bo)(k:•k•+k•k•) + (Bo. k•) (k•+ 2k•. k:)Ao + (Ao. k•) (k•+ 2k•-k•)Bo} -} }u+ In + e)(Ao. Bo)(k,. k,)

} (•A +B) (Ao-k•) (Bo-k•) (k:+k•).

The amplitude of the scattered wave depends on the polarization of the primary waves. If one primary wave


WAVE INTERACTION IN A SOLID 9

is polarized perpendicular to the k•, k2 plane and the other is polarized in the k•, k•. plane, then I+.•,=0, and there is no scattered wave even though the resonance condition is met. If both primary waves are polarized perpendicular to the k•, k• plane, the scattered wave amplitude is

AoBoVoo•a( 1 • Amp. = • . 16rp0r \Ct4CU

x {-

XEd(3c " l) (a+as +d (d -1)(aa+l)•}, (20)

where a is the frequency ratio o0s/o0• and c is the velocity ratio Ct/C•. If both primary waves are polarized in the kl, ksplane, the scattered wave amplitude is similar in form to Eq. (20). In fact, the scattered wave amplitudes for all three general interaction cases are of similar form. Since these amplitudes may be obtained from straightforward expansion of the terms in Eq. (17), they will not be included in the present paper. The general character of Eq. (20) will be discussed in a later section.

It remains to consider the case where both primary waves are longitudinal, and the case where one is longitudinal and one transverse. Complete results are given in Table I. For the case of two longitudinal waves, the resonance condition can be satisfied for the fourth term of Eq. (17) only. For the remaining case the resonance conditions can be satisfied for the first, second, and fourth terms. In this latter case of interaction between a longitudinal and a transverse primary wave, the amplitude of the longitudinal scattered waves vanish when B0 is polarized perpendicular to the plane of k•, ks. The transverse scattered wave has finite amplitude for any polarization of B0.

The appearance, in Table I, of scattered waves with the difference frequency co•-cos requires some comment. Treatment of the interaction problem has been on a strictly classical basis in this paper. The classical treatment is an approximation to the correct quantum mechanical treatment. A macroscopic plane elastic wave consists of the presence of a very large number of phonons of a particular (long) wavelength in the crystal. With this in mind one can do the following calculations:

(1) Write down the phonon Hamiltonian for the crystal, including the first term giving phonon-phonon interaction.

(2) Assume that at t=0, there are a large number of phonons with wave vectors k• and k• and energies

(3) Assume that first-order time-dependent perturba-

tion theory is valid and compute the state of the system at some time t• later than t=0.

if one does this calculation the following points appear. Due to the structure of the phonon-phonon interaction there is conservation of the phonon wave vector (no Umklapp processes are possible if the initial wavelengths are very long). That is, a phonon of wave vector k• and one of wave vector ks can produce, in interaction, only phonons of wave vector k3=k•q-k,•. The second point is that if t• is not chosen too small, the perturbation theory gives only energy- conserving transitions, that is, we must also have tuo3=•sq-tzo0•. These conditions on the wave vector and energy taken together are equivalent to the resonance condition from Eq. (18). A troublesome point is, however, that this calculation seems to indicate that a scattered wave with the difference of

the primary frequencies cannot arise, in contrast to the classical calculation which permits a scattered wave of either the sum or difference frequencies. It is therefore not clear to us whether or not the difference

frequency waves will actually be produced in an experiment. Since the validity of the assumptions made in both calculations is somewhat of an open question, it is difficult to make a more definite statement at the

present time.

III. BEAMWIDTH OF SCATTERED WAVE

The scattered wave given by Eq. (17) appears in the form of a conical beam with vertex at the interaction

zone and maximum intensity along the direction of the vector i8. In experiments aimed at detecting the scattered wave it would be advantageous to minimize the spread of this beam. The parameters which deter- mine the angular width of the beam may be identified as follows.

Consider the first term in the right-hand side of Eq. (17). The amplitude of the integral is equal to the volume of interaction V when •=•. We seek here the vector t0 for which the amplitude first becomes zero (or a minimum) as t moves away from t•. This occurs when one full cycle of the wave is fitted into V, or in other words, when the coefficient of r' is approximately equal 2•r/1, 1 being a length characterizing the volume of interaction. It follows that the angular width of the beam is proportional to Xs/1 where X, is the wavelength of the scattered wave. Thus the scattered beam will

be narrow when X,¾¾/. For the case we are considering X, is inversely proportional to co•-+-o0s so the width of the scattered beam may be made small by using large primary frequencies and/or a large interaction volume.

When the other terms in Eq. (17) are considered, the same proportionality to X•/1 is obtained. However, for the second and fourth terms, X• is inversely proportional to o0•-o0s which suggests that in general a scattered wave with the difference frequency will be more spread out.


10 G. I•. JONES AND D. R. KOBETT

IV. NUMERICAL EXAMPLE

We use a numerical example to illustrate the previous results. Choosing polystyrene for the elastic medium we have 3

X= 2.89)<10 •ø dyn/cm s,

/•= 1.38)<10 •ø dyn/cm 2,

K= 3.81)<10 •ø dyn/cm 2,

A = -- 1.00)< 10 u dyn/cm 2,

B= -- 8.3 )< 10 •ø dyn/cm 2,

C= -- 1.06)< 10 n dyn/cm 2,

p0 = 1.056 g/cm a.

Taking again the case of two transverse primary waves we find that resonance can be obtained for 0.338

< a < 2.955. Since the two primary waves are the same type, we may without loss of generality choose co•>o• which restricts a to the range 0.338 < a_< 1. The smallest angle qo between the primary waves for which a scattered (resonant) wave is obtained is 120.8 ø corresponding to a= 1. As a decreases, • approaches 180 ø.

If both primary waves are polarized perpendicular to the kl, k2 plane, the scattered wave amplitude Eq. (20) is largest for a= 1. We then obtain

Amp (max)= 10.32)< 10-•SAoBoVw18/r cm.

(when A0, B0, and r are in cm, V in cm 3andwl in rad/sec). As a decreases, the amplitude passes through a minimum at a=0.583 and we find that

Amp (rain) = 7.45)< 10-•SAoBoVw13/r cm.

Clearly, the amplitude is more sensitive to the primary wave frequencies characterized by co•, than to the frequency ratio.

Numerical results have also been obtained for two transverse waves in copper, using third-order elastic coefficients. 4 The frequency ratio a and angle • for resonance are about the same as those for polystyrene.

However, the scattered wave amplitude is much more sensitive to the frequency ratio and is zero for a=0.5.

Let us further evaluate the interaction in polystyrene of two 10-Mc waves with a volume of intersection equal to 1 cm 3. If the displacement amplitude of the inter- acting waves is approximately 10 -•ø cm and we let r-10 cm, then the displacement amplitude of the scattered wave is calculated to be approximately 10 -•5 cm. By maintaining the same volume, frequency, and observation distance, we see that the amplitude of the scattered wave varies as the product of the primary wave amplitudes. Thus, the difference between the amplitudes of the primary waves and the scattered wave decreases as A 0 and B0 get larger. If we use the unreasonably large value of 10 -ø cm for A0 and B0 (this corresponds to a strain amplitude of about 10 -3 in the above example), then we find that the amplitudes of the intersecting and scattered waves are of the same order of magnitude.

V. CONCLUSIONS

Two intersecting plane elastic waves produce a scattered wave when the resonance condition [-Eq. (18)• is satisfied. The resonance condition is a function of the

ratio of the primary frequencies, but is independent of the absolute frequencies. The scattered wave appears in the approximate form of a conical beam emanating from the volume of interaction and has maximum

intensity along the direction t8 defined by the resonance condition. The width of the beam is proportional to Xs/l where X8 is the wavelength of the scattered wave and 1 is a length characterizing the volume of interaction V. The maximum intensity of the scattered wave is greatest for high primary frequencies, large primary-wave amplitudes, and large interaction volume.

In an experiment aimed at detecting the scattered wave, the distance L from the interaction zone to the point of observation should be large compared with 1. Therefore, optimum experimental conditions are

a D. S. Hughes and J. L. Kelly, Phys. Rev. 92, 1145 (1953). The correlation between the elastic constants l, m, and n given in

, ß 1 the reference and A B, and C used here is A--n, B=m-•n, C=l-m+•n.

4 A. Seeger and O. Buck, Z. Naturforsh. 15a, 1056 (1960).

ACKNOWLEDGMENTS

The authors wish to express their thanks to F. R. Rollins and A.D. St. John for many helpful discussions. This work was supported by the U.S. Air Force.


interaction of elastic waves in an isotropic solid

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