## Abstract

It is shown that certain anisotropic elastic materials can have one or two sheets of spherical slowness surface. The waves associated with a spherical slowness sheet can be longitudinal, transverse or neither. However, a longitudinal wave can propagate in any direction if and only if the slowness sheet is a sphere . The same cannot be said of transverse waves. A transverse wave can propagate in any direction without having a spherical slowness sheet. If a spherical slowness sheet exists, the waves need not be transverse. The existence of a spherical slowness sheet means that the associated velocity surface and the wave surface also have a sphere. Thus, one sheet of the wave front due to a point source is a sphere, a rather unusual phenomenon for anisotropic elastic materials. Particularly interesting anisotropic elastic materials are the ones in which one longitudinal and two transverse waves can propagate in any direction. They have one spherical slowness sheet for the longitudinal waves. In the special case, they have a second spherical slowness sheet which is disjoint from the spherical slowness sheet . The third slowness sheet is a spheroid.

## 1. Introduction

For a plane wave propagating in the direction ** n** the eigenvalues of the acoustic tensor Q() provide three wave speeds (Fedorov 1968). They depend on

**unless the material is isotropic. A**

*n**velocity surface*can be constructed for which the radial distance from a fixed origin to the surface in the direction

**is the wave speed (Musgrave 1970). There are three sheets for the velocity surface, one for each of the wave speeds ,**

*n**i*=1, 2, 3. If the radial distance is ,

*i*=1, 2, 3, we have a

*slowness surface*. Another surface, called the

*wave surface*, is the polar reciprocal of the slowness surface. It represents the wave front due to a point source in the elastic medium. Among the velocity, slowness and wave surfaces, the slowness surface is mathematically the simplest because it is an algebraic equation of degree six. The velocity surface is an algebraic equation of degree twelve while the wave surface is of degree not less than 150. Synge (1957) first emphasized the importance of the slowness surface in the study of wave motion in anisotropic elastic materials. It has wide applications in surface waves and in transmission and reflection of waves at a plane boundary (see Synge 1957; Barnett & Lothe 1974; Chadwick & Smith 1977). The eigenvector

**(**

*a***) of the acoustic tensor Q(**

*n**) associated with a wave speed is the*

**n***polarization vector*. The displacement of the plane wave is proportional to

**(**

*a***). The wave speeds and the polarization vectors have been extensively studied in (Fedorov 1968; Hayes 1972; Alshits & Lothe 1979**

*n**a*,

*b*,

*c*, 2004).

The slowness surface for an isotropic elastic material consists of three spherical sheets. One sphere is for the longitudinal wave and the other two spheres are for transverse waves and are coincident. The slowness surface for the longitudinal wave has the radius where is the longitudinal wave speed with and *ρ* being an elastic stiffness and mass density, respectively. The slowness surface for the transverse wave has the radius where is the shear wave speed with being another elastic stiffness.

The slowness surface for an anisotropic elastic material usually has a complicated geometry. Chadwick & Norris (1990) proved that there are special orthotropic materials for which the slowness surface is a union of three aligned ellipsoids. Some of the ellipsoids degenerate into spheres. In particular, two of the five cases they studied have one spherical slowness sheet but they are not associated with transverse waves. One of the five cases they studied has two coincident spherical slowness sheets . The waves associated with the double spherical slowness sheets can have a transverse wave component.

Ting (2006) presented anisotropic elastic materials for which the elastic stiffness is a constant, independent of the choice of the coordinate system. The materials can only be orthotropic or hexagonal. For these materials, a longitudinal wave can propagate in any direction. One of the slowness sheets is a sphere . He also studied anisotropic elastic materials for which the elastic stiffness referred to any coordinate system remains the same. Again, the materials can only be orthotropic or hexagonal. For these materials a transverse wave can propagate in any direction. One sheet of the slowness surface is a sphere .

The purpose of this paper is to search for materials that have a spherical slowness sheet other than the ones discovered by Chadwick & Norris (1990) and by Ting (2006). To this end, we will not insist that the slowness sheets are aligned ellipsoids or certain elastic stiffness is independent of the choice of the coordinate system. We recover all spherical slowness sheets obtained by Chadwick & Norris (1990) and by Ting (2006). Of course we obtain some new spherical slowness sheets.

## 2. Basic equations

In a fixed rectangular coordinate system , the equation of motion for the displacement ** u** is(2.1)where the comma stands for differentiation with respect to , the dot stands for differentiation with time

*t*,

*ρ*is the mass density and is the elastic stiffness. The is positive definite and possesses the full symmetry shown in equation (2.1)

_{2}. The third equality in equation (2.1)

_{2}is redundant because the first two imply the third (Ting 1996, p. 32). For a plane wave propagating in the direction of a unit vector

**with wave speed**

*n**c*, let the displacement

**be given by(2.2)where**

*u**f*is an arbitrary function of its argument and

**is the polarization vector. The equation of motion (2.1)**

*a*_{1}is satisfied if(2.3)where Q(

*) is the acoustic tensor(2.4)For a non-trivial solution of the polarization vector*

**n****we must have(2.5)in which I is the identity tensor.**

*a*We may re-write equation (2.5) as(2.6)in which is the slowness vector and is the Kronecker delta. Equation (2.6) provides the equation for the slowness surface. It is a polynomial in of degree six. Equation (2.5) tells us that is homogeneous in of degree two. This means that(2.7)so that, once is computed, the slowness surface is given by(2.8)

For an isotropic elastic material the three eigenvalues are(2.9)where is the contracted notation of . Writing equation (2.9)_{1} as(2.10)the slowness surface is(2.11)

It is a sphere with radius . Likewise, the slowness surfaces for are coincident sphere(2.12)with radius . The associated polarization vectors , , are(2.13) and are any vectors orthogonal to ** n**. The polarization vector is a longitudinal wave while and are transverse waves.

The slowness surface for an anisotropic elastic material in general has a complicated geometry. In the next section we present conditions under which an orthotropic elastic material has a spherical slowness sheet.

## 3. A spherical slowness sheet for orthotropic materials

There are nine independent elastic constants for an orthotropic material. When the coordinate system are taken along the cryptographic axes the 6×6 stiffness matrix has the structure(3.1)Only the upper triangle of the matrix is shown because ** C** is symmetric. The acoustic tensor Q(

*) in equation (2.4) has the expression(3.2)where(3.3)*

**n**If , , are the eigenvalues of equation (2.3) we have(3.4)When a spherical slowness sheet exists, one of the has the expression where is independent of ** n**. Hence equation (3.4)

_{2}can be written as(3.5)where(3.6)In the above () are independent of

**. Comparison of the coefficients of , , on both sides of equation (3.5) gives us(3.7)**

*n*Comparison of the coefficients of , , , , , leads to(3.8)The two expressions for , , are compatible if(3.9)Comparison of the coefficients of on both sides of equation (3.5) yields a rather complicated expression and is not very useful.

The three equations in (3.9) are *necessary* conditions for the existence of a spherical slowness sheet in orthotropic materials. There are three possibilities.(3.10)(3.11)(3.12)In equation (3.11),(3.13)Other possibilities can be obtained from the above by re-naming the coordinate system. In the following sections, we study the three cases separately.

## 4. Orthotropic materials with

When , the 6×6 stiffness matrix ** C** can be written as(4.1)All equations in (3.7) and (3.8) have as a common factor. Hence, we may choose . From equation (2.5), it can be shown that is an eigenvalue if(4.2a)This can be re-written as(4.2b)Subjected to the condition (4.2

*a*) or (4.2

*b*), equation (4.1) has six independent elastic constants. Anisotropic elastic materials represented by equations (4.1) and (4.2

*a*) or (4.2

*b*) have a spherical slowness sheet . If

**is the**

*D**adjoint*matrix of , the eigenvector

**associated with the eigenvalue is proportional to any column of**

*a***(Hohn 1965). Taking the third column of**

*D***we obtain(4.3)The other two eigenvalues obtained from equation (2.5) can be shown to be(4.4a)where(4.4b)In the remainder of this section we consider special cases of equations (4.1) and (4.2**

*D**a*,

*b*).

*Special case A1.* The polarization vector ** a** shown in equation (4.3) is orthogonal to

**when(4.5)Equations (4.5) and (4.2**

*n**a*,

*b*) can be solved for and as(4.6)where(4.7)Anisotropic elastic materials represented by equation (4.1) subjected to equation (4.6) have a spherical slowness sheet for which a transverse wave can propagate in any direction. These materials have five independent elastic constants.

*Special case A2.* Equation (4.6) is not valid when *g*=0. However, by taking the limit or , one of the two solutions becomes unbounded while the other gives the solution(4.8)By a direct computation, it can be shown that equation (4.8) satisfies equations (4.5) and (4.2*a*,*b*). Thus equation (4.8) is a special case of equation (4.6). With equation (4.8), equation (4.1) reduces to(4.9)It has only four independent elastic constants. Equation (4.9) recovers the one obtained by Ting (2006). When referred to any new coordinate system, the value of does not change its value. The polarization vector of equation (4.3) for the materials given in equation (4.9) simplifies to, after ignoring a common factor (Ting 2006),(4.10)By inspection, it is clear that .

We have thus obtained new materials given by equation (4.1) subjected to equation (4.6) that is more general than equation (4.9) that have a spherical slowness sheet for which a transverse wave can propagate in any direction.

*Special case A3.* If we let(4.11)equation (4.2*b*) is satisfied. With the two eigenvalues obtained from equations (4.4*a*,*b*) we have(4.12)This recovers the case (S1) in Chadwick & Norris (1990). Equation (4.11) tells us that , and must have the same sign. Chadwick & Norris (1990) showed that they could not be negative if the strain energy density is positive. Hence , and must be positive. This means that, from equation (4.11)_{4}, , and are either all positive or only two of them can be negative. It can be shown that one of the first three conditions in equation (4.11) is redundant. Thus, we may delete any one of the first three conditions in equation (4.11). It can also be shown that equation (4.11) is equivalent to(4.13)Equation (4.1) now has only four independent elastic constants.

The polarization vector associated with is (Chadwick & Norris 1990)(4.14)The polarization vectors and associated with the double eigenvalues are any vectors orthogonal to . In particular, we may choose(4.15)so that is a transverse wave. Thus, a transverse wave can propagate in any direction for the material given by equations (4.1) and (4.13).

Equation (4.13) is not valid if anyone of , and vanishes. This means that one of , , vanishes. If , we have . This is studied next.

*Special case A4.* If we let(4.16)while need not vanish, equation (4.2*a*) is automatically satisfied. The elastic stiffness has the structure(4.17)With the two eigenvalues obtained from equations (4.4*a*,*b*) we have(4.18)The polarization vectors associated with , and are(4.19)This recovers the case (S2)_{3} obtained by Chadwick & Norris (1990). It should be noted that associated with is not a transverse wave unless . If , equations (4.17) and (4.19) reduce to(4.20)(4.21)The material represented by equation (4.20) is hexagonal. It has only three elastic constants. The polarization vector is a *quasi*-longitudinal wave, is a vertically polarized wave while is a horizontally polarized transverse wave.

*Special case A5*. When(4.22)equation (4.2*a*) simplifies to(4.23)Hence, there are two possibilities for a spherical slowness sheet with .

*Special case A5(1)*. If(4.24)the elastic stiffness has the structure(4.25)The material is hexagonal. The polarization vector is identical to the one shown in equation (4.21). The hexagonal material shown in equation (4.20) is a special case of equation (4.25) when .

*Special case A5(2)*. If(4.26)the polarization vector is(4.27)It is not a transverse wave unless(4.28)When equations (4.22), (4.26) and (4.28) hold, the elastic stiffness has the structure(4.29)The material is tetragonal. Equation (4.27) simplifies to(4.30)

## 5. Orthotropic materials with

When , equation (3.11) holds. It suggests that , and must have the same sign. All equations in (3.7) and (3.8) have as a common factor. Hence we may choose . It can be shown from equation (2.5) that is an eigenvalue if(5.1)where(5.2)Anisotropic elastic materials subjected to the conditions (3.11) and (5.1) have five independent elastic constants. They have a spherical slowness sheet .

When(5.3a)equation (3.11) gives and equation (5.1) is automatically satisfied. If , equation (3.11) gives , and equation (5.1) yields(5.3b)or(5.3c)We will discuss the three cases separately below.

*Case B1*. When , so that the 6×6 stiffness matrix ** C** has the structure(5.4)The other two eigenvalues obtained from equation (2.5) can be shown to be(5.5a)where(5.5b)We now consider two special cases.

*Special Case B1(1)*. When(5.6)the eigenvalues obtained from equation (5.5*a*,*b*) are(5.7)The polarization vectors associated with , and are(5.8)This is identical to (P4) in Chadwick & Norris (1990), while equation (5.7) can be deduced from (S4) in Chadwick & Norris (1990) by setting . If, in addition to equation (5.6),(5.9)equation (5.8) simplifies to(5.10)The polarization vector is a *quasi*-longitudinal wave, is a horizontally polarized transverse wave while is a vertically polarized wave. It should be noted that is a transverse wave but is not a spherical slowness sheet. is a spheroid. It should also be noted that equations (5.9) and (5.6) imply that . Hence, the material represented by equation (5.4) with and equation (5.9) is hexagonal.

*Special Case B1(2)*. When , equation (5.4) reduces to(5.11)It has only four independent elastic constants. Equation (5.5*b*) simplifies to(5.12)Hence the eigenvalues are(5.13)Equation (5.13) is a special case of (S5) in Chadwick & Norris (1990) who considered all three slowness sheets to be aligned ellipsoids. In (S5) the slowness sheet associated with is an ellipsoid while the in equation (5.13) is a sphere. The polarization vectors associated with equation (5.13) are identical to (P5) obtained by Chadwick & Norris (1990), i.e.(5.14)It is remarkable that, for the materials represented by equation (5.11), not just one of the slowness sheets is a sphere; all waves are polarized along the coordinate axes.

*Case B2*. When(5.15)the 6×6 stiffness matrix ** C** has the structure(5.16)It has five independent elastic constants. The polarization vector associated with can be shown to be(5.17)In the following, we consider two special cases of equation (5.16).

*Special Case B2(1)*. When , equation (5.16) simplifies to(5.18a)(5.18b)It has four independent elastic constants. This is equivalent to the materials specified in (C3) in Chadwick & Norris (1990). The material is hexagonal. The eigenvalues and the polarization vectors can be computed. They are(5.19)(5.20)Equations (5.19) and (5.20) recover (S3) and (P3) in Chadwick & Norris (1990). Again, is a transverse wave but is not a spherical slowness sheet. is a spheroid. is a spherical slowness sheet but is not a transverse wave unless . This is studied next.

*Special Case B2(2)*. If , equations (5.18*a*,*b*) simplifies to(5.21)It has three independent elastic constants. The material is hexagonal. Equations (5.19) and (5.20) reduce to(5.22)(5.23)Equation (5.22) shows us that the slowness sheets and are spheres. Equation (5.23) tells us that the polarization vector is a longitudinal wave while and are transverse waves. It is remarkable that, for the materials represented by equation (5.21), one longitudinal and two transverse waves can propagate in any direction just like isotropic elastic materials. Chadwick & Norris (1990) and Ting (2006) could have specialized their equation (C3) and equation (3.13), respectively, to equation (5.21). Since they did not do it, they missed the significance of equation (5.23). Equation (5.21) is a special case of equation (3.13) in Ting (2006) which will be derived in equation (6.4) later.

*Case B3*. When(5.24)the polarization vector associated with can be shown to be(5.25)It is a transverse wave if(5.26)The elastic stiffness of the materials subjected to equations (5.24) and (5.26) have the structure(5.27)It has only three elastic constants. The material is tetragonal. A transverse wave can propagate in the material in any direction with(5.28)

## 6. Orthotropic materials with

When , equation (3.12) holds. There are no common factors for equations in (3.7) and (3.8) unless(6.1)We can then choose . Equation (3.12) reduces to(6.2)One solution is(6.3)The 6×6 matrix for the elastic stiffness ** C** has the structure(6.4)It is shown in Ting (2006) that anisotropic elastic materials represented by equation (6.4) have a spherical slowness sheet for which the wave is longitudinal. Moreover, the value of does not change when referred to any new coordinate system. The waves associated with and are necessarily transverse. Hence, one longitudinal and two transverse waves can propagate in any direction for the materials represented by equation (6.4). As pointed out earlier, equation (5.21) is a special case of equation (6.4).

Other solutions to equation (6.2) violate the positive definiteness of strain energy density.

## 7. Concluding remarks

The analyses presented above are valid if the strain energy density of the materials shown is positive and non-zero. This is assured when the 6×6 matrix of the elastic stiffness ** C** is positive definite. The matrix

**is positive definite if and only if () where is the determinant of the leading**

*C**α*×

*α*principal minor of

**(Hohn 1965). It can be shown that all**

*C***presented here can be made positive definite by choosing the elements of**

*C***properly.**

*C*We have shown that, for the materials represented by equation (5.21) and (6.4), one longitudinal and two transverse waves can propagate in any direction. This is like isotropic elastic materials except that isotropic elastic materials have three spherical slowness sheets, while the materials represented by equations (6.4) and (5.27) have one and two, respectively, spherical slowness sheets. We also showed that a spherical slowness sheet does not necessarily provide transverse waves and a slowness sheet, for which a transverse wave can propagate in any direction, need not be a sphere.

For all cases for which a spherical slowness sheet gives a transverse wave in any direction, the elastic stiffness ** C** is not

*structurally invariant*except for the Special case A2 (Ting 2006; Ting & He in press). The elastic stiffness

**for other cases is not structurally invariant. We have confined our attention to orthotropic materials in this paper. It is shown in Ting (in press) that the anisotropic elastic materials obtained by Ting (2006) and represented by equation (6.4) are the only ones that have a spherical slowness sheet for which a longitudinal wave can propagate in any direction. Since the materials represented by equation (6.4) can only be orthotropic, hexagonal or isotropic, no spherical slowness sheet for longitudinal waves exists in other materials. The question is open if there are trigonal, monoclinic or triclinic materials for which a spherical slowness sheet for transverse waves exists.**

*C*## Footnotes

↵† Professor Emeritus of University of Illinois at Chicago and Consulting Professor of Stanford University.

- Received January 3, 2006.
- Accepted March 3, 2006.

- © 2006 The Royal Society