Elastodynamic Equations: Characteristics, Wavefronts and Rays

ELASTODYNAMIC EQUATIONS:CHARACTERISTICS, WAVEFRONTS AND RAYS

byLEN BOS†

(Department of Mathematics and Statistics, University of Calgary, Calgary,Alberta T2N 1N4, Canada)

and

MICHAEL A. SLAWINSKI ‡

(Department of Earth Sciences, Memorial University, St. John’s, NewfoundlandA1B 3X5, Canada)

[Received 3 January 2009. Revised 9 August 2009. Accepted 14 September 2009]

Summary

We derive the characteristic equations, and the so-called Christoffel equation, for the vectorelastodynamic equations in terms of both hypersurfaces of nonuniqueness and as wavefrontsbased on a physical definition. We follow Courant and Hilbert in defining a wavefront as asurface for which a solution may be zero on one side but nonzero on the other. We show thatwavefronts defined in this way must be characteristics. Moreover, we give a partial converseshowing that in the case of analytic coefficients, characteristics must (locally) be wavefronts.Furthermore, we show how the equations defining wavefronts, which are characteristics, canbe obtained by letting frequency tend to infinity in a certain trial solution expressed in thefrequency domain. The last approach might suggest that ray theory, which results from theChristoffel equation, is an asymptotic theory. Taking the limit, however, is just a technicalityand is unnecessary to obtain the Christoffel equation, which is contained in the elastodynamicequations and their characteristic equations.

1. Introduction

The purpose of this paper is twofold. First, we show that in an anisotropic inhomogeneous, Hookeansolid wavefronts are tantamount to characteristic hypersurfaces of the elastodynamic equations.In doing so, we extend the definition proposed by Courant and Hilbert for the scalar wave equation(1). Second, we show that the Christoffel equations, which are the equations for wavefronts andequations from which the eikonal equations and the concept of rays emerge, are not intrinsicallylinked to an infinite-frequency formulation. Hence, ray theory is not an asymptotic theory, in thesense of an infinite-frequency approximation, as it is commonly stated (see, for example (2)).

In this paper, we derive the Christoffel equations by three different methods. First, we formulatethem as the equations of the characteristics of the elastodynamic equations, viewed as surfaces ofnonuniqueness. Second, we show that these same equations determine wavefronts defined, follow-ing Courant and Hilbert, as hypersurfaces with the property that, locally, there is a solution that iszero on one side of the surface and nonzero on the other (1). Third, we postulate a trial solution of

†〈[email protected]〉‡Corresponding author〈[email protected]〉

c© The author 2009. Published by Oxford University Press; all rights reserved.For Permissions, please email: [email protected]

doi:10.1093/qjmam/hbp021

2 of 15 L. P. BOS AND M. A. SLAWINSKI

a certain form in the frequency domain and derive the Christoffel equation by letting the frequencytend to infinity.

Our definition of a wavefront (see Definition4.1) is only local since this restriction allows fora partial converse: if the coefficients are analytic, then characteristics are wavefronts. A globaldefinition and a true converse would be more satisfying, but to the best of our knowledge, suchrelationships are not completely understood. We note that many authors define wavefronts to becharacteristic surfaces (2, Section 2.5,3). Such an approach circumvents technical difficulties byabandoning physical intuition. Since the notion of wavefront is physical, it requires a physicaldefinition.

The trial solution in the frequency-domain method is simple and, hence, used by many authors,notably (2, Section 2.5,4, Chapter 5,5, Chapter 15,6, p. 12,7, Chapter 6). An important element inthe trial solution derivation is to let the frequency tend to infinity. This technicality has led the aboveauthors to statements that ray theory is an asymptotic theory. Such a nomenclature is misleadingsince the limit is just a technical device to isolate jumps in the derivatives of the trial solutionacross the hypersurface: high frequency is not essential, the jumps are what matter. Moreover, thisapproach suffers from the fact that it is not known if such a trial solution exists, in general, or iseven asymptotic to a true solution (Babich, personal communication (3, p. 9,8, p. 550)).

To set the stage for our study, we begin this paper by stating the elastodynamic equations.

2. Elastodynamic equations

We begin with the constitutive equations of a Hookean solid, namely,

σi j =3∑

k,l=1

ci jkl (x)εkl , i, j ∈ {1, 2, 3}, (2.1)

whereεkl := (∂uk/∂xl + ∂ul /∂xk)/2, which is the form of the strain tensor that assumes infinites-imal displacements. Herein we use the notation := to emphasize that the left-hand side is beingdefined by the equation on the right-hand side or we use=: for the reverse order. The elasticitytensor,ci jkl , is the proportionality constant relatingσi j andεkl and possesses the intrinsic symme-tries,ci jkl = cjikl = ckli j , which result from the symmetries of the stress and strain tensors and theconcept of strain energy; the elasticity tensor is positive definite, namely,

3∑

i, j,k,l=1

ci jkl viw j vkwl > 0 for nonzero vectorsv andw.

Inserting constitutive equations (2.1) into Cauchy’s equations of motion,

ρ∂2ui

∂t2 =3∑

j =1

∂σi j

∂xj, i ∈ {1, 2, 3},

we get the elastodynamic equations, namely,

ρ∂2ui

∂t2 =1

2

3∑

j,k,l=1

[∂ci jkl

∂xj

(∂uk

∂xl+∂ul

∂xk

)

+ ci jkl

(∂2uk

∂xj ∂xl+

∂2ul

∂xj ∂xk

)]

, i ∈ {1, 2, 3}, (2.2)

which describe propagation of a disturbance in a Hookean solid. Let us examine these equations.

ELASTODYNAMIC EQUATIONS 3 of 15

3. Characteristic surfaces as sets of nonuniqueness

In this section, we obtain the characteristic equations of elastodynamic equations (2.2) as hyper-surfaces where it is not possible to determine uniquely the second derivatives of the solution givenfunction values and first-order normal derivatives as initial data. (Here, we follow the approach pre-sented by Bona and Slawinski (9, Sections 2.2.2 and 2.4.1).) Since these equations form a system ofthree linear second-order partial differential equations in four independent variables, let us considerthe more general

3∑

k=1

4∑

i, j =1

Ai jkl (y)∂2 fk∂yi ∂yj

+ Bi jkl

(

y, f,∂f∂y

)

= 0, (3.1)

wherel ∈ {1, 2, 3} andy stands fory1, . . . , y4. We want to find a characteristic surface, which wespecify in the form

yi = yi (s1, s2, s3),

along which it is impossible to determine uniquely the second derivatives from the initial conditionsthat are given along this hypersurface. Specifically, these initial conditions are

fk(y1(s1, s2, s3), . . . , y4(s1, s2, s3)) = f 0k (s1, s2, s3),

DV fk(y1(s1, s2, s3), . . . , y4(s1, s2, s3)) = f Dk (s1, s2, s3),

wherek ∈ {1, 2, 3} and whereV denotes the vector normal to the surface and DV stands for thedirectional derivative operator along the surface, namely,V ∙ ∇.

We may view a differential equation as a relation among derivatives of its solution. We can usethis relation to determine all the derivatives, and construct the solution using a Taylor series. Hence,to find the characteristic surface of a second-order differential equation is to find a hypersurfacealong which we cannot solve uniquely for the second-order derivatives.

To proceed with finding the characteristic surface, we choose a new coordinate system,y1, . . . , y4,in such a way that, at least locally, the hypersurface is expressed in these coordinates asy4 = 0 andthe transverse vector along which we know the derivative off is expressed in the new coordinatesas [0, 0, 0, 1]. This way, we replace the four parameters,sj , by three coordinates,y1, y2 and y3, andrewrite the initial conditions as:

fk(y1, y2, y3, 0) = f 0k (y1, y2, y3), (3.2)

∂ fk∂ y4

(y1, y2, y3, 0) = f Dk (y1, y2, y3), (3.3)

wherek ∈ {1, 2, 3}. Differentiating expressions (3.2) with respect toy1, y2 and y3, we obtain

∂ fk∂ yi

(y1, y2, y3, 0) =∂ f 0

k

∂ yi(y1, y2, y3), for i ∈ {1, 2, 3}.

Differentiating again, we get

∂2 fk∂ yj ∂ yi

(y1, y2, y3, 0) =∂2 f 0

k

∂ yj ∂ yi(y1, y2, y3), for i, j ∈ {1, 2, 3}. (3.4)


Also, we differentiate expressions (3.3) to get

∂2 fk∂ yj ∂ y4

(y1, y2, y3, 0) =∂ f D

k

∂ yj(y1, y2, y3), for i, j ∈ {1, 2, 3}. (3.5)

Equations (3.4) and (3.5) form a set of 36 equations for 30 unknowns. However, due to the equal-ity of mixed partial derivatives, there are only 27 independent equations. To complete the system,we invoke the three original differential equations given in expression (3.1) and write the system forthe second derivatives as follows:

1 0 . . . 0 . . . 0 . . . 0

0 1 . . . 0 . . . 0 . . . 0

......

. . ....

......

0 0 . . . 1 . . . 0 . . . 0

......

.... . .

......

A1111 A1211 . . . Ai jk1 . . . A4411 A4421 A4431

A1121...

... A4412 A4422 A4432

A1131 A1231 . . . Ai jk3 . . . A4413 A4423 A4433

∂2 f1∂ y2

1

∂2 f1∂ y1∂ y2

...

∂2 fk∂ y j ∂ yi

...

∂2 f1∂ y2

4

∂2 f2∂ y2

4

∂2 f3∂ y2

4

=

∂2 f 01

∂ y21

∂2 f 01

∂ y1∂ y2

...

∂2 f 0k

∂ y j ∂ yi

...

−B1

−B2

−B3

. (3.6)

Herein B` denotes the accumulated function value and first-order derivative terms that appearin (3.1).

This system has nonunique solutions if the determinant of the coefficient matrix is zero. In viewof the upper left unit submatrix, this determinant is tantamount to the determinant of the lowerright 3 × 3 submatrix,A44kl(y1, y2, y3, 0), wherek, l ∈ {1, 2, 3}. To find an expression for thedeterminant of the 3× 3 submatrix in terms of the original coordinates of (3.1), let us assume thatthe hypersurface can be expressed byy4 = ψ(y1, y2, y3). Consequently,

yi = yi , i ∈ {1, 2, 3}, y4 = y4 − ψ(y1, y2, y3),

wherey4 = 0 as required. We write the second derivatives as follows:

∂2

∂yj ∂yi=

∂2

∂ yj ∂ yi−∂ψ

∂yj

∂2

∂ y4∂ yi−∂ψ

∂yi

∂2

∂ yj ∂ y4+∂ψ

∂yj

∂ψ

∂yi

∂2

∂ y24

.

From this expression, we infer that the second derivatives in (3.1) are related to the second deriva-tives in the new coordinates as follows:

3∑

k,l=1

A44kl∂2

∂yk∂yl=

3∑

k,l=1

A44kl

(∂2

∂ yl ∂ yk−∂ψ

∂yl

∂2

∂ y4∂ yk−∂ψ

∂yk

∂2

∂ yl ∂ y4+∂ψ

∂yl

∂ψ

∂yk

∂2

∂ y24

)

.


To find the expression forA44kl , we look for all the terms in the above expression that contain∂2/∂ y2

4. These terms are as follows:

A44kl(y1, y2, y3, 0)

=3∑

i, j =1

Ai jkl (y1, y2, y3, ψ)∂ψ

∂yi

∂ψ

∂yj− 2

3∑

i =1

Ai 4kl(y1, y2, y3, ψ)∂ψ

∂yi+ A44kl(y1, y2, y3, ψ),

wherek, l ∈ {1, 2, 3}. Hence, the determinant of the coefficient matrix of system (3.6) is

det

3∑

i, j =1

Ai jkl∂ψ

∂yi

∂ψ

∂yj− 2

3∑

i =1

Ai 4kl∂ψ

∂yi+ A44kl

, for k, l ∈ {1, 2, 3}. (3.7)

This determinant is a polynomial of degree 3. Setting this polynomial to zero, we obtain threehypersurfaces along which it is impossible to determine uniquely the second derivatives from theconditions that are given along a given hypersurface.

To express our result in terms of the elastodynamic equations, we letyi = xi , fi = ui , Ai jkl =cl jki , for i ∈ {1, 2, 3}, y4 = t and A44kl = −ρδkl for k, l = 1, . . . , 4. We haveAi 4kl = A4ikl = 0since there are no mixed partial derivatives containing time,t . Using (3.7), we write

det

3∑

i, j =1

cl jki (x)∂ψ

∂xi

∂ψ

∂xj− ρ (x) δkl

= 0, for k, l ∈ {1, 2, 3}, (3.8)

wheret = ψ(x) represents a characteristic surface, which is a hypersurface in thex-space where thebehaviour of the solutions is predetermined by (2.2), and hence, one cannot place other constraintsin the form of initial conditions. Equation (3.8) is referred to as the Christoffel equation.

4. Characteristics and wavefronts

In their famous treatise (1, Vol. II, Chapter 6), Courant and Hilbert say that a hypersurfaceS, whichin our case ist = ψ(x), is a wavefrontif there is a solution,u(x, t), of a scalar second-orderhyperbolic partial differential equation with continuous coefficients with the property thatu itself,all its first-order derivatives and any tangential derivative alongSare continuous but that its secondnormal derivative has a jump discontinuity acrossS. If, in particular,u may be chosen to be zeroon one side ofS and nonzero on the other, then it is clear that such a wavefront is the leading edgeof some disturbance and the definition is eminently sensible.

They proceed to show that such a wavefront must be a characteristic surface. Moreover, theyindicate that even if the first discontinuity in the solution appears in a higher order derivative thanthe second one, the surface must still be a characteristic.

It would be desirable to know if the converse is true: if a characteristic surface is a wavefront asdefined by Courant and Hilbert. For this purpose, one can invoke Goursat’s nonuniqueness theorem(see (10, Theorems 5.1.3 and 5.2.1)), but it provides only a local solution with a jump in a normalderivative of some order, and it requires the coefficients to be analytic.

Hence, we adjust their definition to allow the invocation of this inverse theorem.

DEFINITION 4.1. Consider the space–time hypersurface S⊂ R3 × R defined byφ(x, t) := t −ψ(x) = 0. We will say that S is awavefrontif for each point(x0, t0) ∈ S, there is a neighbourhood


(x0, t0) ∈ � ⊂ R4 and a solutionu(x, t) of the elastodynamic equations(2.2) such that

u(x, t) =

{0 for t 6 ψ(x),

not identically 0 fort > ψ(x),

and there is a minimal integer m> 2 so thatu(x, t) and all its derivatives up to order m− 1are continuous as are all tangential derivatives along S of any order, but the mth-order normalderivative ofu(x, t) across S has a jump discontinuity at(x0, t0), in one of its components.

Note that this wavefront surface is defined inR4: space–time. The corresponding physical wave-front is obtained by projectingSonto the space coordinates,R3:

St := {x ∈ R3: ψ(x) = t}.

We now give the vector version of Courant and Hilbert’s elegant argument by stating thefollowing theorem.

THEOREM4.2. Suppose that S⊂ R4 is a wavefront and that the coefficients ci jk` ∈ C1(R3). Then,S must be a characteristic surface; in other words, it must satisfy the Christoffel equation(3.8).

Proof. We consider the case of the jumps being in the second normal derivative. The vectors

T jk :=

⟨

0, ∙ ∙ ∙ ,∂φ

∂xj︸︷︷︸kth

, 0 ∙ ∙ ∙ ,−∂φ

∂xk︸︷︷︸j th

, 0 ∙ ∙ ∙ , 0

⟩

, j, k ∈ {1, 2, 3, 4},

where, for convenience, we have setx4 := t are orthogonal to∇φ and hence are tangential to thesurfaceS. Let us denote

φ j :=∂φ

∂xj.

Then, by assumption, the directional derivatives

DT jk

∂ui

∂xn=

∂2ui

∂xn∂xkφ j −

∂2ui

∂xn∂xjφk (4.1)

are continuous. Similarly,

DTn`

∂ui

∂xj=

∂2ui

∂xj ∂x`φn −

∂2ui

∂xj ∂xnφ` (4.2)

are also continuous. Takingφ` × (4.1) − φk × (4.2), it follows that the expressions

∂2ui

∂xn∂xkφ jφ` −

∂2ui

∂xj ∂x`φnφk

are also continuous. If we let [f ] denote the jump in the functionf across the surfaceSat (x0, t0),we must have

[∂2ui

∂xn∂xk

]

φ jφ` −

[∂2ui

∂xj ∂x`

]

φnφk = 0.


Consequently, there must exist scalarsλi such that, fori ∈ {1, 2, 3},[∂2ui

∂xn∂xk

]

= λiφnφk, n, k ∈ {1, 2, 3, 4}. (4.3)

Not all theλi can be zero as then there would be no jumps in any second normal derivatives.Now, consider elastodynamic equations (2.2). Taking jumps acrossSat (x0, t0), we have

ρ

[∂2ui

∂t2

]

=1

2

3∑

j,k,l=1

ci jkl

([∂2uk

∂xj ∂xl

]

+

[∂2ul

∂xj ∂xk

])

, i ∈ {1, 2, 3}

=1

2

3∑

j,k,l=1

ci jkl (λkφ jφ` + λ`φ jφk) by (4.3)

=1

2

3∑

k=1

λk

3∑

j,`=1

ci jk`φ jφ` +1

2

3∑

`=1

λ`

3∑

j,k=1

ci jk`φ jφk

=3∑

k=1

λk

3∑

j,`=1

ci jk` + ci j `k

2φ jφk =

3∑

k=1

3∑

j,`=1

ci jk`φ jφ`

using the intrinsic symmetry properties of theci jk`. Hence,

λi ρφ24 =

3∑

k=1

3∑

j,`=1

ci jk`φ jφ`, i ∈ {1, 2, 3}.

In matrix form, using the fact thatφ4 = 1, we write

(ρ I3 − 0)

λ1

λ2

λ3

=

0

0

0

for the matrix0 ∈ R3×3 with entries

0ik =3∑

j,`=1

ci jk`φ jφk

and the vectorλ 6= 0. This has a solution if and only if

det(ρ I3 − 0) = 0,

which we recognize immediately as the Christoffel equation (3.8), after the appropriate change ofindices.

If all the second derivatives are continuous and the first jump occurs in the third normal derivative,consider

DT jk

∂2ui

∂xα∂xβ=

∂3ui

∂xα∂xβ∂xkφ j −

∂3ui

∂xα∂xβ∂xjφk (4.4)


and

DTαγ∂2ui

∂xj ∂xβ=

∂3ui

∂xj ∂xβ∂xγφα −

∂3ui

∂xα∂xβ∂xjφγ , (4.5)

which are both continuous by assumption. We now takeφγ × (4.4) − φk × (4.5) to conclude that

∂3ui

∂xα∂xβ∂xkφ jφγ −

∂3ui

∂xj ∂xβ∂xγφαφk

is also continuous acrossS. It follows that there exist scalarsλi , i ∈ {1, 2, 3}, not all zero, such thatthe jumps

[∂3ui

∂xα∂xk∂xβ

]

= λiφαφk, α, β, k ∈ {1, 2, 3, 4}.

We now take elastodynamic equations (2.2) and differentiate with respect tot to obtain

ρ∂3ui

∂t3 =1

2

3∑

j,k,l=1

[∂ci jkl

∂xj

(∂2uk

∂xl ∂t+∂2ul

∂xk∂t

)

+ ci jkl

(∂3uk

∂xj ∂xl ∂t+

∂3ul

∂xj ∂xk∂t

)]

.

Taking jumps acrossS gives exactly the same equations as previously. Higher order jumps arehandled in an analogous manner. �

The converse is also true in the analytic coefficient case.

THEOREM 4.3. Suppose that S= {(x, t): φ(x, t) = 0} is a characteristic surface; in other words,that it satisfies the Christoffel equation(3.8) and thatφ(x, t) and the coefficients ci jk`(x) are lo-cally analytic in a neighbourhood of a point(x0, t0) ∈ S. Then, S is a wavefront in the sense ofDefinition4.1.

Proof. This is essentially the content of the Goursat nonuniqueness theorem, compare Theo-rems 5.1.3 and 5.2.1 of (10). The fact that there is a jump in a finite derivative follows from thefact that the solution so constructed is piecewise analytic. �

5. Fourier transform and regularity properties

We have seen that characteristic surfaces and wavefronts are intimately connected to jumps inderivatives of solutions of the elastodynamic equations. Such jumps can also be analysed in theFourier domain, which we proceed to do.

As is well known, the high-frequency behaviour of the Fourier transform gives important infor-mation about the smoothness of the function and vice versa. In order to explain this phenomenonmore precisely, we collect here some of the basic facts.

Suppose thatf ∈ L1(R) then we define the Fourier transform off (t) to be

f (ω) :=1

√2π

∫ ∞

−∞e−iωt f (t)dt. (5.1)

If also g ∈ L1(R), then we may define the inverse transform by

g(t) :=1

√2π

∫ ∞

−∞e+iωt g(ω)dω, (5.2)


so that f = ˜f for f , f ∈ L1(R). Already a connection between the regularity of a function and thegrowth of its Fourier transform as|ω| → ∞ is evident in the Riemann–Lebesgue lemma:

LEMMA 5.1. (Riemann–Lebesgue)Suppose that f∈ L1(R) then

lim|ω|→∞

f (ω) = 0.

In other words, if f has the regularity property of being absolutely integrable overR (that is, fis in L1(R)), then necessarily, its Fourier transform must decay to 0 as|ω| → ∞. Conversely, ifa Fourier transform does not decay to 0, then it cannot be the Fourier transform of an absolutelyintegrable function.

Integrability is a weak form of regularity. Much more is true. The Fourier transform replacesdifferentiation in the time domain by multiplication in the frequency domain. Specifically, undersuitable conditions onf , the Fourier transform of

dn

dtnf (t) is (iω)n f (ω) (5.3)

and that of

tn f (t) is i n dn

dωnf (ω). (5.4)

Then, applying the inverse Fourier transform (5.2) to relation (5.3), we obtain

dn

dtnf (t) =

∫ ∞

−∞e+iωt (iω)n f (ω)dω.

Of course, this only makes sense ifωn f (ω) is integrable, which places implicitly a condition on thedecay ofωn f (ω) as|ω| → ∞. Conversely, if indeedωn f (ω) is integrable, then this relation wouldimply that f (t) has annth-derivative.

Here is a more formal statement of this behaviour; a proof may be found, for example, in (11,Corollary 5.2.3).

THEOREM 5.2. Suppose that f(t) ∈ L2(R) and thatωk f (ω) ∈ L1(R) for 0 6 k 6 n. Then, f(t)is n times continuously differentiable for all t∈ R.

We wish to emphasize that the conditionωk f (ω) ∈ L1(R) forcesωk f (ω) to be in some sense‘small’ for large |ω| which in turn means,informally, that f (ω) decays at least as fast as 1/ωk as|ω| → ∞.

If we consider distributions, then the behaviour at infinity of the Fourier transform gives evenmore specific information. For example, consider

f (t) = Aδ(t)+ g(t),

whereg ∈ L1(R), δ(t) is the Dirac-delta distribution supported att = 0 andA gives the ‘amplitude’of the delta. Then,

f (ω) = Aδ(ω)+ g(ω) = A1

√2π

+ g(ω).

Hence, by the Riemann–Lebesgue lemma,

lim|ω|→∞

f (ω) = A1

√2π

+ 0 = A1

√2π,


which means that we may recover the ‘amplitude’,A, from

A =√

2π lim|ω|→∞

f (ω).

Translatingt to t − a by taking f (t) = Aδ(t − a)+ g(t − a) results in

f (ω) = eiaω{

A1

√2π

+ g(ω)

}

,

so that we may recover|A| by means of

|A| =√

2π lim|ω|→∞

| f (ω)|,

andA itself from

A =√

2π lim|ω|→∞

e−iwa f (ω).

We may also consider a function with a physical jump. For example, take

f (t) = A sgn(t)+ g(t),

where sgnt = 1 for t > 0, sgnt = −1 for t < 0, g is such thatg andg′ are inL1(R) andA givesthe half-amplitude of the jump att = 0. Then, it can be shown that

f (ω) = A

√2

π

1

iω+ g(ω);

but by the Riemann–Lebesgue lemma, lim|ω|→∞ g(ω) = 0 and even, sinceg′ ∈ L1(R), lim|ω|→∞ωg(ω) = 0. Hence,

lim|ω|→∞

ω f (ω) = A

√2

π

1

i+ 0,

so that we may recover the amplitude of the jump att = 0 by

A = i

√π

2lim

|ω|→∞ω f (ω).

If we again translate the jump tot = a by taking f (t) = A sgn(t − a)+ g(t − a) so that

f (ω) = eiaω

{

A

√2

π

1

iω+ g(ω)

}

,

then we may recover|A| from

|A| =

√π

2lim

|ω|→∞|ω f (ω)|

andA itself from

A = i

√π

2lim

|ω|→∞e−iaωω f (ω).


More generally, we may consider a symmetric jump att = 0 by considering

f (t) =∞∑

n=0

antn sgn(t).

In particular, the jump at the origin is given by

limt→0+

f (t)− limt→0−

f (t) = 2a0.

Then, we may calculate

f (ω) =∞∑

n=0

an(−1)n(n − 1)!

√2

π

1

(iω)n+1 =:∞∑

n=0

An+1

(iω)n+1 =∞∑

n=1

An

(iω)n,

where the last equation defines the meaning ofAn+1.We recoverA1, and hence the jump 2a0, from

A1 = i lim|ω|→∞

ω f (ω).

As before, if we translate the jump tot = a by consideringf (t − a), we obtain

f (ω) = eiaω∞∑

n=0

An+1

(iω)n+1 ,

so that we recover

|A1| = lim|ω|→∞

|ω f (ω)|

andA1 itself from

A1 = i lim|ω|→∞

e−iaωω f (ω).

Suppose we have a function depending on space and time,u(x, t), x ∈ R3, t ∈ R. Indeed, morespecifically, consider

u(x, t) = a0(x)δ(t)+∞∑

n=0

an+1(x)tn sgn(t). (5.5)

Then, the Fourier transform ofu with respect to time,t , is

u(x, ω) = a0(x)1

√2π

+∞∑

n=0

an+1(x)(−1)n(n − 1)!

√2

π

1

(iω)n+1 =:∞∑

n=0

An(x)(iω)n

, (5.6)

where again the last equation defines the meaning ofAn(x), n > 0.We now translate the origin,t = 0, to a location depending onx, say, t = ψ(x), to obtain

u(x, t − ψ(x)) and its associated Fourier transform with respect to time,

u(x, ω) = eitψ(x)∞∑

n=0

An(x)(iω)n

. (5.7)


Again, we may recover the amplitude,A0(x), of the Dirac-delta term from

A0(x) = lim|ω|→∞

e−i tψ(x)u(x, ω).

If A0 = 0, then we recoverA1(x) from

A1(x) = i lim|ω|→∞

e−i tψ(x)ωu(x, ω).

6. Christoffel equation via trial solution

It is common to obtain the Christoffel equation (3.8) by postulating a trial solution of the elastody-namic equations (2.2) of the form

u(x, ω) ∼ exp[iωψ(x)]∞∑

n=0

An(x)(iω)n

, (6.1)

where the level sets ofψ describe wavefronts andAα stands for the polarization vector. As men-tioned in section1, it is not known if solutions of this form actually exist or even if such anexpression is asymptotic to a solution asω → ∞.

Referring to expression (5.7), we note thatA0 gives the ‘amplitude’ of the Dirac-delta term,δ(t − ψ(x)), supported on the wavefront, in the solutionu(x, t), and A1 gives the size of thejump discontinuity in the solution across the wavefrontt = ψ(x). Unlike the formulations of theChristoffel equations discussed above, such nondifferentiable solutions need to be interpreted in adistributional sense.

Applying the Fourier transform to (2.2), we get

(iω)2ρ(x)ui (x, ω) =1

2

3∑

j,k,l=1

[∂ci jkl (x)∂xj

(∂uk

∂xl+∂ul

∂xk

)

+ ci jkl (x)

(∂2uk

∂xj ∂xl+

∂2ul

∂xj ∂xk

)]

,

(6.2)wherei ∈ {1, 2, 3} and

ui (x, ω) :=1

√2π

∫ ∞

−∞ui (x, t)e−iωt dt.

Inserting the first two terms of the series (6.1) into (6.2), we write its right-hand side as

1

2exp[iωψ(x)]

3∑

j,k,l=1

[a(x)(iω)2 + b(x)(iω)1 + c(x)(iω)0 + d(x)(iω)−1], (6.3)

where

a(x) = ci jkl (x)(∂ψ

∂xl

∂ψ

∂xjA0k +

∂ψ

∂xk

∂ψ

∂xjA0l

)

,

with i ∈ {1, 2, 3}. Considering the left-hand side of (6.2) that includes the two terms of series (6.1),we write

exp[iωψ(x)][(iω)2ρ(x)A0i (x)+ ωρ(x)A1i (x)], i ∈ {1, 2, 3}. (6.4)


Comparing the right-hand and left-hand sides, which are given by (6.3) and (6.4), respectively, wecanceleiωψ . Then, moving all the terms to one side, we write the resulting coefficient of(iω)2 as

1

2

3∑

j,k,l=1

ci jkl (x)(∂ψ

∂xl

∂ψ

∂xjA0k +

∂ψ

∂xk

∂ψ

∂xjA0l

)

− ρ(x)A0i (x), i ∈ {1, 2, 3}.

Since j , k andl are summation indices andci jkl is symmetric ink andl , we can write this expressionas follows:

3∑

j,k,l=1

ci jkl (x)∂ψ

∂xl

∂ψ

∂xjA0k(x)− ρ(x)A0i (x), i ∈ {1, 2, 3}.

For the equations of motion resulting from inserting the trial solution to be satisfied for alliω, werequire all the coefficients to be zero. Considering the(iω)2 coefficient, we write

3∑

j,k,l=1

ci jkl (x)∂ψ

∂xl

∂ψ

∂xjA0k − ρ(x)A0i = 0, i ∈ {1, 2, 3}. (6.5)

As shown in the section7, these equations result in the eikonal equations, which are associatedwith the trajectories of waves, namely, the rays. The coefficients of(iω)1, (iω)0, (iω)−1, etc., areassociated with the amplitudes of waves and would result in transport equations.

Since (6.5) are expressed only with the zeroth-order term of series (6.1), we rewrite them asfollows:

3∑

j,k,l=1

ci jkl (x)∂ψ

∂xl

∂ψ

∂xjAk − ρ(x)Ai = 0, i ∈ {1, 2, 3}. (6.6)

Factoring the components ofA, we write

3∑

k=1

3∑

j =1

3∑

l=1

ci jkl (x)∂ψ

∂xj

∂ψ

∂xl− ρ(x)δik

Ak(x) = 0, i ∈ {1, 2, 3}. (6.7)

Assuming that the vectorA 6= 0, this system of equations has nontrivial solutions forAk if andonly if

det

3∑

j =1

3∑

l=1

ci jkl (x)∂ψ

∂xj

∂ψ

∂xl− ρ(x)δik

= 0, i, k ∈ {1, 2, 3}, (6.8)

which is the Christoffel equation; it is identical to (3.8).The above derivation relies on the assumption that in solution (6.1), A0 6= 0. However, ifA0 = 0

but A1 6= 0, the highest power ofω in solution (6.1) and (6.2) is ω1. If we collect the termsmultiplyingω1 on both sides of (6.2), we obtain the same equations as in theA0 6= 0 case. Similarly,if A0 = A1 = 0 butA2 6= 0, the highest power ofω isω0; collecting these terms results in the sameequations and so on.

Since the Christoffel equation (3.8) is obtained by lettingω → ∞, ray theory is often referredto as anasymptotictheory. However, as is especially clear from section5, it is not high frequenciesthat matter but rather the existence of jumps in some derivatives of a solution.


7. Rays

Having obtained the Christoffel equation, we can proceed to get rays. We letp := ∇ψ and writethe Christoffel equation as

det

3∑

j =1

3∑

l=1

ci jkl (x) pj pl − ρ(x)δik

= 0, i, k ∈ {1, 2, 3}.

Assuming thatp2 := p ∙ p 6= 0, we can write this determinant as follows:

(p2)3 det

3∑

j =1

3∑

l=1

ci jkl (x)pj pl

p2 −ρ(x)p2 δik

= 0, i, k ∈ {1, 2, 3}; (7.1)

sinceψ has units of time,∂ψ/∂xj has units of slowness and the level sets ofψ(x) can be viewedas wavefronts at a given time. Consequently,∇ψ(x) is a vector whose direction corresponds to thewavefront normal and whose magnitude corresponds to the wavefront slowness. Lettingp2 = 0would mean that the slowness of the propagation of the wavefront is zero, which would imply thevelocity to be infinite, a nonphysical situation.

Expression (7.1) is a polynomial of degree 3 inp2, where the coefficients depend on the directionof the phase-slowness vector,p. Such polynomial can be factored out as

[

p2 −1

v21(x, p)

][

p2 −1

v22(x, p)

][

p2 −1

v23(x, p)

]

= 0, (7.2)

where 1/v2i are the roots of the determinant in expression (7.1). The existence of three roots corre-

sponds to the three types of waves that propagate in a Hookean solid.Let us consider a root of (7.2), namely,

p2 = 1/v2i (x, p), i ∈ {1, 2, 3}. (7.3)

The characteristic equations of each eikonal equation (7.3) are Hamilton’s equations

dxi

dt=∂H

∂pi,

dpi

dt= −

∂H

∂xi, i ∈ {1, 2, 3},

whereH := 12 p2v2(x, p). Solutions of Hamilton’s equation,x (t), are rays.

8. Conclusions

As shown in section4, characteristics and wavefronts are intimately connected. It is indeed per-fectly reasonable to identify wavefronts and characteristics; however, doing soa priori obscuresthe physical meaning of a wavefront. To avoid such ana priori formulation, we have extended thedefinition of wavefronts proposed by Courant and Hilbert to anisotropic inhomogeneous media andshown that these wavefronts are tantamount to characteristics, which are the only hypersurfacesacross which solutions may have a jump in some derivative.

In section6, we discussed the popular trial-solution approach of obtaining wavefronts and rays.We showed that it is just a technique for the encoding of jumps of a specific type. Letting frequencytend to infinity is a means of recovering a jump from the form of the solution. Hence, particularlyin view of section5, it does not appear justifiable to consider ray theory as an asymptotic theory.


Acknowledgements

The authors are grateful for fruitful discussions with A. Bona and M. G. Rochester during prepara-tion of the manuscript and for comments of N. Bleistein and P. Martin during the revision process.MS wishes to acknowledge M. Epstein whose understanding of rays as intrinsic entities of the elas-todynamic equations, not high-frequency approximations, motivated aspects of this research. Theresearch of MS was done at the Berkeley Seismological Laboratory with the support of an NationalScience and Engineering Research Council of Canada grant during his sabbatical year.

References

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(Cambridge University Press, Cambridge 2008).3. Y. A. Kravtsov and Y. I. Orlov,Geometrical Optics of Inhomogeneous Media(Springer, Berlin

1990).4. C. H. Chapman,Fundamentals of Seismic Wave Propagation(Cambridge University Press,

Cambridge 2004).5. F. A. Dahlen and J. Tromp,Theoretical Global Seismology(Princeton University Press,

Princeton, NJ 1998).6. Y. A. Kravtsov,Geometrical Optics in Engineering Physics(Alpha Science, Harrow, UK 2005).7. A. Udıas,Principles of Seismology(Cambridge University Press, Cambridge 1999).8. V. Cerveny, Seismic Ray Theory(Cambridge University Press, Cambridge 2001).9. A. Bona and M. A. Slawinski,Ray Theory: Characteristics and Asymptotics. Samizdat Press,

Golden, CO 2008. http://samizdat.mines.edu/characteristics.10. L. Hormander,Linear Partial Differential Operators(Springer, Berlin 1963).11. R. J. Zimmer,Essential Results of Functional Analysis(University of Chicago Press, Chicago,

IL 1990).

Elastodynamic Equations: Characteristics, Wavefronts and Rays

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