Two-Dimensional Elastic Herglotz Functions and Their Applications in ...

Journal of Elasticity 68: 123–144, 2002. © 2003 Kluwer Academic Publishers. Printed in the Netherlands.

123

Two-Dimensional Elastic Herglotz Functions and their Applications in Inverse Scattering VASSILIOS SEVROGLOU1 and GEORGE PELEKANOS2

1 Department of Mathematics, University of Ioannina, 45110, Ioannina, Greece. E-mail: [email protected] 2 Department of Mathematics and Statistics, Southern Illinois University, Edwardsville, IL 62026, USA. E-mail: [email protected]

Received 20 November 2002; in revised form 28 March 2003 Abstract. In this work solutions of the spectral Navier equation that satisfy the Herglotz boundedness condition in two-dimensional linear elasticity are presented. Navier eigenvectors in polar coordinates are introduced and it is established that they form a linearly independent and complete set in the L2 -sense on every smooth curve. It is also proved that the classical solutions of the spectral Navier equation are expressed via Navier eigenvectors, and this expansion converges uniformly over compact subsets of R2 . Two far-field patterns, the longitudinal and the transverse one corresponding to the displacement field are introduced, and the Herglotz norm is expressed as the sum of the L2 -norms of these patterns over the unit circle. It is also established that the space of elastic Herglotz functions is dense in the space of the classical solutions of the spectral Navier equation. Finally, connection to inverse elasticity scattering is established and reconstructions of rigid bodies are presented. Mathematics Subject Classifications (2000): 45A05, 74B05. Key words: Navier eigenvectors, Herglotz functions, inverse scattering, simple method.

1. Introduction In 1940s, G. Herglotz established several asymptotic properties for solutions of the scalar Helmholtz equation. Later on, Müller [23] published a paper on the properties of entire solutions of the wave equation, where he included all the basic ideas of previous work due to Herglotz as well as many other new properties that came from Herglotz’s results. Herglotz functions proved to be very useful in inverse scattering theory. The main contribution in this area is due to Colton, Kirsch, Kress and Monk [4–6, 8], who used Herglotz functions to develop shape reconstruction algorithms. Comparing to acoustics and electromagnetics not much has been done for the elastic scattering process. The difficulty in passing from electromagnetic waves to elastic ones is due to the fact that in elasticity we have to deal with two kinds of waves which propagate at different phase velocities [3, 20]. A second but more important difference between electromagnetic and elastic waves is associated with

124

V. SEVROGLOU AND G. PELEKANOS

the fact that the two elastic waves are independent of each other, and hence we cannot recover any one of them once the other is known. Results for elastic Herglotz functions in three dimensions can be found in [10–12], whereas results concerning the representation of Herglotz functions in two-dimensional linear elasticity as superposition of plane waves are presented in [16]. The present paper is concerned with elastic Herglotz wave functions in two dimensions which are defined as entire solutions to the Navier equation satisfying an appropriate growth condition at infinity. We constitute an attempt to study basic properties of Herglotz functions while at the same time we are trying to show their importance in developing algorithms for solutions of inverse scattering problems. In particular, we are concerned with the extension to elasticity of an algorithm that is recently developed by Colton and Kirsch [4] which is also known as “simple method”. We organize our paper as follows. In Section 2 we present the basic two-dimensional linear elasticity equations as well as the fundamental Green’s function for R2 . In Section 3 the Navier eigenvectors in polar coordinates are introduced, and linear independence as well as completeness results are established. It is also proved that the classical solutions of the spectral Navier equation are expanded in terms of Navier eigenvectors and this expansion converges uniformly over compact subsets of R2 . In Section 4 we study the Herglotz solutions of the spectral Navier equation. The Herglotz condition is presented and the Herglotz wave functions in two dimensions are introduced. Following Hartman’s ideas [15], two far-field patterns, the longitudinal and the transverse one are presented. A one-to-one correspondence between the Herglotz solutions and their far-field patterns is established. The decomposition of an elastic Herglotz solution into plane waves propagating in every direction with amplitude distributions the longitudinal and the transverse Herglotz kernels, which coincide with the corresponding far field patterns is presented. A property known as the density of elastic Herglotz functions is studied. In particular it is proved that the space of Herglotz functions is dense within the space of the classical solutions of the Navier equation. The same result holds for the space of the tractions on the Herglotz functions, in the space of the tractions on the solutions of the spectral Navier equation. Finally, Section 5 is devoted to showing the connection between theory and applications. In particular, a Colton– Kirsch type algorithm [4] is constructed for Dirichlet boundary conditions, and some preliminary numerical results are presented.

2. Statement of the Problem We assume that R2 is filled by an isotropic and homogeneous elastic medium with Lamé constants λ, µ and mass density ρ. For the Lamé constants λ, µ the strong ellipticity conditions µ > 0, λ + µ > 0 are satisfied in order for the medium

TWO-DIMENSIONAL ELASTIC HERGLOTZ FUNCTIONS WITH APPLICATIONS

125

to sustain longitudinal as well as transverse waves. The Navier equation of the dynamic theory of linearized elasticity is written as [20] µ∇ 2 U(r, t) + (λ + µ)∇∇ · U(r, t) = ρ

∂2 U(r, t). ∂t 2

(1)

Assuming the time-spectral decomposition U(r, t) = u(r)e−iωt ,

(2)

where the circular frequency ω denotes the Fourier dual variable of t, we obtain the spectral Navier equation cs2 ∇ 2 u(r) + (cp2 − cs2 )∇∇ · u(r) + ω2 u(r) = 0,

(3)

where the phase velocities cp , cs of the longitudinal and the transverse wave respectively are given by
 λ + 2µ µ , cs = . (4) cp = ρ ρ Equation (3) implies the characteristic equations ω = kp cp = ks cs ,

(5)

where kp , ks are the wave numbers for the longitudinal and transverse waves, respectively. An alternative form of equation (3) is given by (∗ + ρω2 )u(r) = 0,

(6)

∗ = µ + (λ + µ)∇∇.

(7)

with

The Helmholtz decomposition [22] of the time independent displacement field u is given by u(r) = up (r) + us (r),

(8)

where up (r) is the longitudinal part, while us (r) is the transverse one. It is well known that up (r) and us (r) satisfy the Helmholtz equations ( + kp2 )up (r) = 0,

( + ks2 )us (r) = 0.

(9)

The free-space fundamental dyadic 
(r, r ) which satisfies the following equation ˜
(r, r ) = −Iδ(r − r ), (∗ + ρω2 )

r, r ∈ R2 ,

(10)

126

V. SEVROGLOU AND G. PELEKANOS

is given by [1] as

 i 1 ˜ (1)  
(r, r ) = IH (ks |r − r |) 4 µ 0    (1) 1 (1)   ∇r ∇r H0 (kp |r − r |) − H0 (ks |r − r |) , − ρω2

(11)

where I˜ is the identity dyadic, δ(r − r ) represents the Dirac measure concentrated at r, H0(1) is the Hankel function of the first kind and zeroth order, and ˜ on the top means dyadic. 3. The Navier Eigenvectors We begin with some results concerning the Navier eigenvectors on a circular disk. In the polar coordinate system the position vector r is expressed as r = (r cos ϕ, r sin ϕ), with r ∈ [0, +∞) and ϕ ∈ [0, 2π ). The circular disk centered at the origin is denoted by C(0, R) = {r: r = |r| < R}. The Navier eigenvectors that correspond to the longitudinal and transverse parts of the displacement field are given respectively by e,i e,i imϕ rˆ + e,i m (r) = ∇(ψm (kp r, ϕ)) = kp Zm (kp r)e

im e,i Z (kp r)eimϕ ϕˆ r m

(12)

and e3 ψme,i (ks r, ϕ)) = me,i (r) = ∇ × (

im e,i Z (ks r)eimϕ rˆ − ks Zme,i (ks r)eimϕ ϕ, ˆ r m (13)

for m = 0, ±1, ±2, . . . , where Zme (ka r) = Hm(1) (ka r),

a = p, s,

(14)

are the Hankel functions of the first kind and m-order, Zmi (ka r) = Jm (ka r),

a = p, s

(15)

are the Bessel functions of the first kind, ψme,i (ka r, ϕ) = Zme,i (ka r)eimφ are the polar solutions of the Helmholtz equation. The prime denotes differentiation with respect to ka r whereas the superscripts e and i denote the exterior and interior eigenvectors, respectively. In addition rˆ , ϕˆ are the unit vectors of the polar coordinate system and e3 is a unit vector perpendicular to the (x1 , x2 )-plane. After some computational effort we conclude that the following orthogonality relations hold among the Navier eigenvectors: e,i (e,i n , m )R =

∗ 2π  2 2 e,i kp R Zn (kp R) Ze,i m (kp R) R  ∗ + n2 Zne,i (kp R) Ze,i m (kp R) δnm ,

(16)

TWO-DIMENSIONAL ELASTIC HERGLOTZ FUNCTIONS WITH APPLICATIONS ∗ 2π  2 2 e,i ks R Zn (ks R) Ze,i m (ks R) R  ∗ + n2 Zne,i (ks R) Ze,i m (ks R) δnm , ∗ 2inπ  (kp R)Zne,i (kp R) Ze,i = − m (ks R) R  ∗ + (ks R)Zne,i (kp R) Ze,i m (ks R) δnm ,

127

(ne,i , me,i )R =

e,i (e,i n , m )R

(17)

(18)

where ( , )R is the L2 -inner product defined on a circle of radius R, δnm is the Kronecker’s delta and ∗ over characters denotes the complex conjugate. We will now establish completeness and linear independence results for the Navier eigenvectors. To this end let ∂D be the boundary of a finite region Di in R2 , which is simply connected and contains the origin with D i = Di ∪ ∂D. We also assume ∂D to be a Lyapunov surface [19] with a connected exterior De . We denote the spectrum of eigenvalues of the homogeneous Dirichlet problem in Di by σ (Di ), and we let L2 (∂D) be the space of square integrable vector functions on ∂D. We are now ready to prove the following two lemmas. / σ (Di ). In addition, LEMMA 1. The set {im , mi } is complete in L2 (∂D) if ω2 ∈ e e 2 the set {m , m } is also complete in L (∂D) for any value of ω2 . Proof. It is sufficient to prove∗ that if the inner product of any vector function ϕ(r) ∈ L2 (∂D) with an element X (r) of the aforementioned sets equals zero then ϕ(r) is zero. To this end, let ∗    (19) X (r ) · ϕ(r ) ds(r ) = 0. ∂D

We now consider the single-layer potential (Sϕ)(r) with density ϕ(r), that is  (20)
(r, r ) · ϕ(r ) ds(r ), r ∈ R2 , (Sϕ)(r) = ∂D

where 
(r, r ) is the fundamental Green’s function for R2 . Using the addition theorem and relation (11), we conclude after some Navier eigenvector manipulations that 
(r, r ) can be expressed as  2  ∞ 

∗ ∗ i k p  e i e i  m (kp r> ) ⊗ m (kp r< ) ,  (ks r> ) ⊗ m (ks r< ) +
(r, r ) = − µ m=−∞ m ks (21) with r> := r, r< := r,

if r > r  or r  , if r < r  , if r < r  or r  , if r > r 

and r = |r|, r  = |r |.

128

V. SEVROGLOU AND G. PELEKANOS

From (19) and the expression of the fundamental solution given by (21) which converges uniformly on compact subsets of r > r  we obtain that (Sϕ)(r) = 0 outside a large circle containing Di . But since the single-layer potential is a solution of the Navier equation, analyticity yields (Sϕ)(r) = 0, r ∈ De . Therefore we can write [Sϕ]+ (r ) = 0,

(22)

for r → ∂D from outside. The single-layer potential Sϕ is an everywhere continuous function including the boundary ∂D; hence [Sϕ]− (r ) = 0,

(23)

/ σ (Di ) for r → ∂D from inside. The last conclusion with the assumption that ω2 ∈ leads to (Sϕ)(r) = 0,

r ∈ Di .

(24)

On the other hand, if nˆ is the outward unit normal on the surface ∂D, the action of the surface stress operator T , ˆ + µnˆ × ∇×, T = 2µnˆ · ∇ + λn∇·

(25)

on the single-layer potential is not a continuous function on the boundary ∂D. We thus have the jump relations [21] [T Sϕ]± (r) = ∓ϕ(r) + T (r) (26)
(r, r ) · ϕ(r ) ds(r ), ∂D

where +(−) indicates that r → ∂D from outside (inside), and the superscript r denotes the result of T operating relative to the point r on 
(r, r ). The vanishing 2 of the single layer in all of R yields [T Sϕ]+ (r) = [T Sϕ]− (r),

r ∈ ∂D,

(27)

from which we conclude that ϕ = 0 for r ∈ ∂D. Hence completeness of the set {im , mi } in L2 (∂D) is now proved. We would also like to mention that for the case in which ω2 is an eigenvalue, the set {im , mi } is not complete in L2 (∂D) unless some suitable vector functions are added into it. This last result can be proved using similar arguments as those in [2]. A similar proof for the exterior eigenvectors as for the interior ones can be ✷ established for any value of ω2 . / σ (Di ), LEMMA 2. The set {im , mi } is linearly independent in L2 (∂D) for ω2 ∈ where ∂D is a closed smooth Liapunov type curve in R2 . A similar result for the exterior eigenfunctions holds for any value of ω2 .

129

TWO-DIMENSIONAL ELASTIC HERGLOTZ FUNCTIONS WITH APPLICATIONS

Proof. Assume that there exist scalars {am , bm }, m = 0, 1, . . . , N, such that N



am im (r) + bm mi (r) = 0,

∀r ∈ ∂D.

(28)

m=0

We now consider the expansion u(r) =

N



am im (r) + bm mi (r) ,

r ∈ Di .

(29)

m=0

From (28) follows that u is a solution of (3) that vanishes on ∂D. Since we have / σ (Di ), we have that u(r) = 0 for every r ∈ Di . We then have a assumed that ω2 ∈ vanishing u on every circle C(0, α) inscribed to ∂D. That is, N



am im (r) + bm mi (r) = 0,

r ∈ C(0, α).

(30)

m=0

Forming the inner product of (30) first with im and then with mi and using the orthogonality equations (16)–(18), we obtain   (31) am im 2a + bm im , mi a = 0, m = 0, 1, . . . , N, and   am im , mi a + bm mi 2a = 0,

m = 0, 1, . . . , N.

(32)

But the determinant of the homogenous linear system (31)–(32) is positive due to the Cauchy–Schwarz inequality and to the linear independence of im and mi on the circle of radius α. Thus the system has the trivial solution, that is am = bm = 0, m = 0, 1, . . . , N, and the linear independence of im , mi has been established. Following similar arguments, linear independence for the exterior eigenvectors ✷ for every value of ω2 can be established. The following lemma establishes the uniform convergence of the expansions of Navier eigenvectors over compact subsets. More precisely we have: LEMMA 3. Let u satisfy the spectral Navier equation (3) in the classical sense. If / σ (Di ), then every classical solution u of the spectral Navier equation taking ω2 ∈ the continuous value ϕ on the boundary can be expanded as ∞



am im (r) + bm mi (r) , u(r) =

(33)

m=0

where the series converges uniformly to u in closed subsets of Di . A similar result for the exterior eigenvectors holds for any value of ω2 .

130

V. SEVROGLOU AND G. PELEKANOS

Proof. The proof follows the ideas of Aydin and Hizal [2]. From Lemma 1 and the definition of the completeness of a system of functions we can conclude that any square integrable vector function on a Lyapunov surface can be approximated in the mean square sense arbitrarily closely by a linear combination of {im , mi } if and only if ω2 ∈ / σ (Di ). Let M

 am im (r) + bm mi (r) .

uM (r) =

(34)

m=0

Then



|ϕ(r ) − ϕM (r )|2 ds(r ) = 0

lim

M→∞

(35)

∂D

with ϕM (r) =

M



am im (r) + bm mi (r) ,

r ∈ ∂D.

(36)

m=0

/ σ (Di ), we can express the Since we have an interior Dirichlet problem and ω2 ∈ solution u(r) in Di as a double layer potential via the relation  ϕ(r ) · T (r ) (37)
(r, r ) ds(r ). u(r) = ∂D

Similarly every vector wave function can be expressed in the form (37). We then obtain the following relation:     (38)
(r, r ) ds(r ). ϕ(r ) − ϕM (r ) · T (r ) u(r) − uM (r) = ∂D

Taking into account the Cartesian components of (38), uj , uMj , j = 1, 2, and using the Cauchy–Schwarz inequality we obtain the following relations for each Cartesian component:  1/2   2  |ϕj (r ) − ϕMj (r )| ds(r ) |uj (r) − uMj (r)|
∂D



×

|T

(r ) 



2



1/2


(r, r )| ds(r )

,

r ∈ Di .

(39)

∂D


(r, r )|2 is integrable Using relations (35) and (39) along with the fact that |T (r ) on ∂D we conclude that 

lim |u(r) − uM (r)| = 0.

M→∞

(40)

Thus, uM converges uniformly to u in closed subsets of Di and the proof has been completed.

131

TWO-DIMENSIONAL ELASTIC HERGLOTZ FUNCTIONS WITH APPLICATIONS

Same arguments hold for the exterior Navier eigenvectors em , me , but in this case the expansion converges uniformly for all values of ω2 . ✷

4. Properties of Herglotz Solutions Let

 C(r0 ; R) = r ∈ R2 : r = |r − r0 | < R

(41)

be a circular disk of radius R centered at r0 , and

 ∂C(r0 ; R) = r ∈ R2 : |r − r0 | = R

(42)

be its boundary. A continuous vector field u defined in all of R2 satisfies the Herglotz condition if 1 lim sup |u(r )|2 r  dr  dϑ < +∞. (43) r→∞ r C(0,r) An elastic Herglotz function is defined to be a classical solution of the spectral Navier equation (3), in all of R2 , which satisfies condition (43). In the sequel we will study the basic properties of elastic Herglotz functions and their connection with the far-field patterns they generate. Let u be a Herglotz function. From the completeness of the Navier eigenvectors established in Lemma 1, and the general theory of eigenfunction expansions [26], we have that u(r) =

∞



am im (r) + bm mi (r)



m=0 ∞ 

  im  imϕ imϕ am kp Jm (kp r)e rˆ + Jm (kp r)e ϕˆ = r m=0   im  Jm (ks r)eimϕ rˆ − ks Jm (ks r)eimϕ ϕˆ , + bm r which converges uniformly over compact subsets of R2 [11]. From the asymptotic relations for Bessel functions [22]
  π 2 cos (2m + 1) − ka r + O(r −3/2 ), Jm (ka r) = π ka r 4
  π 2  sin (2m + 1) − ka r + O(r −3/2 ), Jm (ka r) = π ka r 4

(44)

r → ∞,

(45)

r → ∞,

(46)

132

V. SEVROGLOU AND G. PELEKANOS

and equations (12), (13), we obtain 

  π 2kp sin (2m + 1) − kp r eimϕ rˆ + O(r −3/2 ), = πr 4    π 2ks i sin (2m + 1) − ks r eimϕ ϕˆ + O(r −3/2 ) m (r) = − πr 4

im (r)

(47) (48)

as r → ∞. Following [14] the asymptotic forms (47), (48) are used to define in a formal way the longitudinal pattern L(ˆr) ∼ =

∞ 1 −m+1 kp i am eimϕ rˆ , 2 m=0

(49)

while the transverse pattern is given by ∞ 1 −m+1 ∼ T(ˆr) = − ks i bm eimϕ ϕ. ˆ 2 m=0

(50)

The functions L, T are called far-field patterns for u, and they are defined over the unit circle. In particular L coincides with the radial amplitude while T with the tangential one. If the Herglotz condition (43) holds we will show that the formal series (49) and (50) converge in the L2 -sense. Using relations (44)–(50) we construct the asymptotic form (1) (2) u(r) ∼ = L(ˆr)H0 (kp r) + L(−ˆr)H0 (kp r)

+ T(ˆr)H0(1)(ks r) + T(−ˆr)H0(2)(ks r)

(51)

as r → ∞. The terms that involve the radiating Hankel functions of the first kind H0(1) represent outgoing waves, while the terms that involve the Hankel functions of the second kind H0(2) represent incoming waves. The incoming patterns L(−ˆr), T(−ˆr) are determined from the outgoing ones L(ˆr), T(ˆr), respectively, by a simple inversion of the direction of observation. Let u be an elastic Herglotz function that satisfies condition (43). Then there exists a constant c > 0 such that the condition 1 |u(r)|2 r dr dϕ < c, ∀R > 0, (52) R C(0,R) is satisfied. In what follows L2 [0, 2π ] denotes the L2 -functions on the unit circle. Let R > R0 , with R0 being the starting point for the asymptotic behavior of Bessel functions which is given by equations (45), (46). We then write

TWO-DIMENSIONAL ELASTIC HERGLOTZ FUNCTIONS WITH APPLICATIONS

1 R

133

|u(r)|2 r dr dϕ C(0,R)

1 R u(r rˆ )2L2 [0,2π] r dr = R 0 1 R 1 R0 2 u(r rˆ )L2 [0,2π] r dr + u(r rˆ )2L2 [0,2π] r dr. = R 0 R R0

Using the uniform convergence of the expansion (44) over the unit circle, orthogonality arguments, asymptotic relations for Bessel functions and (52), we conclude that 1 |u(r)|2 r dr dϕ R C(0,R) 1 R0 u(r rˆ )2L2 [0,2π] r dr = R 0   R N  π 2

2 2 sin (2m + 1) − kp r dr 2|am | kp + R m=0 4 R0    R π 2 2 sin (2m + 1) − ks r dr + O(ln R) < c. (53) + 2|bm | ks 4 R0 Taking first the limit of (53) as R → ∞, and finally the limit as N → ∞, we obtain ∞



 2kp |am |2 + 2ks |bm |2 < c.

(54)

m=0

By virtue of the Riesz–Fisher representation theorem, (54) implies that the far-field patterns L and T given by (49) and (50) are well-defined in the L2 -sense. Taking into account the Herglotz condition (43) we obtain 1 r→∞ r



lim

|u(r )|2 dr =

C(0,r)

∞



 2kp |am |2 + 2ks |bm |2 < c.

(55)

m=0

From (49) and (50) we obtain L(ˆr)2L2 [0,2π] =

∞ π

|am |2 kp2 , 2 n=0

(56)

T(ˆr)2L2 [0,2π] =

∞ π

|bm |2 ks2 . 2 n=0

(57)

and

134

V. SEVROGLOU AND G. PELEKANOS

It then follows that 4 4 1 |u(r )|2 dr = L(ˆr)2L2 (0,2π) + T(ˆr)2L2 [0,2π] , lim r→∞ r C(0,r) π kp π ks

(58)

where the right-hand side of (58) is the Herglotz norm in two-dimensional elasticity denoted by u2H . The factor 4 in the right-hand side of (58) implies that both incoming and outgoing patterns have the same contribution to the Herglotz norm of u. This is obvious since the incoming patterns depend on the outgoing ones through inversion of the direction of observation. It has also been established [16], that the solution u being Herglotz is equivalent to the corresponding patterns L and T being square integrable. Following Hartman [14], we now define the far-field oscillation functions wa (r) = K(+ˆr)eika r + K(−ˆr)e−ika r ,

(59)

where K(±ˆr) = L(±ˆr) when a = p or T(±ˆr) when a = s. The relations which show a connection between the patterns L, T and the oscillation functions are   1 a e∓ika r a w (r) ± w (r) , (60) K(±ˆr) = 2 ika r where war = ∂wa /∂r and a = p, s. If u is an elastic Herglotz solution of the spectral Navier equation using similar arguments as in [14] it can be shown that the following relations hold: √  1 r  r  ua (r ) − wa (r )2 2 dr  = 0, (61) lim L [0,2π] r→∞ r 0 r √   1  r  ua (r ) − wa (r )2 2 dr  = 0. (62) lim r r L [0,2π] r→∞ r 0 In what follows we will establish a one-to-one correspondence between the Herglotz solutions and their far-field patterns. The importance of the following theorem is that it provides the far-field patterns L and T directly from up and us , respectively, without the need to expand them in eigenvectors and calculate the coefficients. THEOREM 1. If u is a Herglotz solution of the spectral Navier equation (3) and L and T are the corresponding far-field patterns, then   r ∓ika r     2 √  e 1 a  a  r ika u (r rˆ ) ± ur  (r rˆ )¯ dr  = 0, lim K(±ˆr) −  2 r→∞ 2r 0 ika L [0,2π] (63) where K(±ˆr) = L(±ˆr) when a = p or T(±ˆr) when a = s.

TWO-DIMENSIONAL ELASTIC HERGLOTZ FUNCTIONS WITH APPLICATIONS

135

Proof. We define 

f(r ) = f(r  rˆ ) =

 e∓ika r √   r ika ua (r  rˆ ) ± uar (r  rˆ )¯ − 2K(±ˆr). ika

An application of the Cauchy–Schwarz inequality in (64) yields  r ∓ika r  2 1    √  e a  a   r ika u (r rˆ ) ± ur  (r rˆ )¯ dr − 2K(±ˆr) r  2 ika 0 L [0,2π] r 1
f(r  rˆ )2L2 [0,2π] dr  . r 0

(64)

(65)

From (59) and some computational effort we can obtain an estimate for the L2 norm of f(r  rˆ ) as follows:  ∓ika r  2 e   √   2 a  a  ¯  r ika u (r rˆ ) ± ur  (r rˆ ) − 2K(±ˆr) f(r rˆ ) =   2 ika L [0,2π] √  a  2  √  a  2 a 
2 r u (r rˆ ) − w (r rˆ ) + 2  r ur  (r rˆ ) − war (r  rˆ ). (66) ka We can now estimate every term in (66) using the corresponding results for the Helmholtz equation due to Hartman and Wilcox [15]. Taking the limit as r → ∞ of (65), it is easy to see that its right-hand side vanishes due to (66) by virtue of (61), (62). Hence, (63) holds. ✷ In the sequel we will show that the space of the elastic Herglotz functions is dense in the space of classical solutions to the Navier equation in Di with respect to the maximum norm over Di . THEOREM 2. Let u(r) be a classical solution of the spectral Navier equation (3) in a bounded, connected starlike domain Di , with smooth boundary ∂D ∈ C 2,a , where C 2,a denotes the space of twice Hölder continuously differentiable functions of index a with 0 < a
1. We also assume that u(r) ∈ C 2,a (D i ). Then for every ε > 0 there exists an elastic Herglotz function v(r), such that max |u(r) − v(r)|
ε.

(67)

r∈D i

A similar result holds for the acting of the stress operator T on the Herglotz functions. That is, max |T u(r) − T v(r)| < ε,

(68)

r∈D i

where its application on the vector field u ∈ R2 can be written in component form as  ∂ui ∂uj + , i, j = 1, 2. (69) (T u)ij = λδij ∇ · u + µ ∂xj ∂xi

136

V. SEVROGLOU AND G. PELEKANOS

Proof. From the Helmholtz decomposition theorem given by (8), the displacement field u admits the representation u(r) = up (r) + us (r).

(70)

The first part of the right-hand side is the longitudinal wave up (P-wave), while the second one represents the transverse wave us (S-wave). Using the fact that the Herglotz functions are dense within the space of classical solutions of the Helmholtz equation [5], we have that for every εa > 0 there exist Herglotz wave functions vp (r) and vs (r) such that max |uai (r) − via (r)|
εa ,

a = p, s,

r∈D i

(71)

 2 a with ua (r) = 2i=1 uai (r)ˆxi , va (r) = xi , a = p, s and xˆ i , i = 1, 2, i=1 νi (r)ˆ being unit vectors. Starting from the left-hand side of (67) and using (71), we obtain max |u(r) − v(r)| = max |up (r) + us (r) − vp (r) − vs (r)| r∈D i

r∈Di


max |up (r) − vp (r)| + max |us (r) − vs (r)| r∈Di

=

2

i=1

r∈D i

2 

  p  p   max ui (r) − νi (r) + maxusi (r) − νis (r) r∈Di


εp + εs = ε.

i=1

r∈D i

(72)

We will now prove (68) for the set produced by the action of the stress operator T on the elastic Herglotz functions. Indeed, it is known [4, 9] that the following density property holds: max |∇υg (r) − ∇ν(r)|
ε,

(73)

r∈D

where υg is a Herglotz wavefunction with Herglotz kernel g. In other words, the set created from the action of the gradient operator on the Herglotz functions is dense within the set generated from the action of the same operator on the set of the classical solutions of the Helmholtz equation. Taking into account that the stress operator T is given by (69), using relation (73) and some computational effort we reach to (68). Hence the density properties have been established. ✷

5. An Application to Inverse Scattering 5.1. FORMULATION OF THE PROBLEM As it was mentioned earlier, let Di denote a closed, bounded and simply connected subset of R 2 with boundary ∂D, which is assumed to be bounded and Lyapunov. The domain De , lying outside Di , is occupied by a homogeneous and isotropic

TWO-DIMENSIONAL ELASTIC HERGLOTZ FUNCTIONS WITH APPLICATIONS

137

elastic medium with Lamé constants λ, µ and mass density ρ. We suppress the harmonic time dependence e−iωt , where ω denotes the angular frequency. We assume without any loss of generality that ρ = 1. In what follows we formulate our scattering problem using dyadic notation. We choose this alternative way to study this problem due to the dyadic nature of the fundamental Green’s function. The dyadic displacement field  u satisfies the spectral Navier equation u(r) =  0, (∗ + ω2 )

r ∈ R 2 /D,

(74)

where ∗ is given by (7), and as mentioned earlier,on the top means dyadic. Note that (74) is the dyadic form of equation (6). We assume that the scatterer is excited by a complete dyadic field propagating in the direction dˆ which is decomposed into a plane longitudinal wave and a plane transverse one. Such an incident field is given by the form ˆ ikp r·dˆ + (I˜ − dˆ ⊗ d)e ˆ iks r·dˆ ,  uinc (r) = dˆ ⊗ de

(75)

where dˆ = (cos θ, sin θ) is the direction of propagation, I˜ is the identity dyadic, and kp , ks are the wavenumbers of the P (longitudinal) and S (transverse) waves, respectively. In order to obtain the vector incident field from the dyadic one, dotting by the vector dˆ we get the longitudinal wave propagating in the dˆ direction and ˆ we dotting by the vector θˆd (the polarization vector), which is perpendicular to d, obtain a transverse wave propagating in the dˆ direction and polarized in the θˆd direction. The dyadic field (75) can be represented as a dyadic superposition of two vector fields which appear as the first vectors of the dyads, while the second vectors of the ˆ θˆd }, i.e., dyads are provided by the incident orthogonal base {d, ˆ d) ˆ ⊗ dˆ + uinc,s (r; d, ˆ θˆd ) ⊗ θˆd ,  uinc (r) = uinc,p (r; d,

(76)

where ˆ d) ˆ = de ˆ ikp r·dˆ , uinc,p (r; d,

ˆ θˆd ) = θˆd eiks r·dˆ . uinc,s (r; d,

(77)

In (76), as well as in what follows, the first argument represents the vector at the observation point, the second is the propagation vector and the third is the polarization vector. The dyadic character of the incident field  uinc is inherited in the scattered field sct  u it generates by preserving the order of the dyads as in the following form ˆ = usct,p (r; d, ˆ d) ˆ ⊗ dˆ + usct,s (r; d, ˆ θˆd ) ⊗ θˆd ,  usct (r; d)

(78)

where the first vectors of the dyads correspond to the vector scattered fields generated by the first vectors of the dyads of  uinc , respectively. Furthermore, each

138

V. SEVROGLOU AND G. PELEKANOS

displacement field appearing as a first vector of each dyad in (78) has to satisfy the Kupradze radiation conditions at infinity [21]:  √ ∂usct,a sct,a sct,a = 0, a = p, s, (79) lim u − ika u = 0, lim r r→∞ r→∞ ∂r uniformly for all directions r = |r|. From the Helmholtz decomposition theorem the scattered field  usct can be written as usct,p (r) +  usct,s (r),  usct (r) = 

r ∈ Be .

(80)

In what follows we will consider the rigid body problem uinc (r) +  usct (r) =  0,  utot(r) = 

r ∈ ∂D.

(81)

In other words we will assume that on the surface of the scatterer the total displacement field vanishes. We now present the relations that hold for the far-field patterns of the scattered field. It has been shown [17] that at the radiation zone eikp r eiks r us∞ (ˆr) √ + O(r −3/2 ), r → ∞, up∞ (ˆr) √ +  (82)  usct (r) =  r r √ a where the coefficients of the terms eik r / r with a = p, s, are the far-field patterns for the P and S wave, respectively. In particular, the relations which give the longitudinal (P-wave) and transverse (S-wave) far-field patterns for the Dirichlet problem are +1   ˆ = i e−ikp rˆ ·r T (r ) utot(r ) ds(r ), (83) rˆ ⊗ rˆ ·  up∞ (ˆr; d) 4µ π kp ∂D +1 ˜   ˆ = i√ (I − rˆ ⊗ rˆ ) · e−iks rˆ ·r T (r ) utot(r ) ds(r ). (84)  us∞ (ˆr; d) 4µ π ks ∂D One of the main results in the theory of Herglotz functions that finds applications in inverse scattering is given by the following representation theorem. In particular this theorem shows that any elastic Herglotz function can be represented as a superposition of plane waves over the unit circle propagating in every direction. In this plane wave decomposition the amplitude distributions are described by the corresponding outgoing far-field patterns and they are referred to as the longitudinal and transverse Herglotz kernels. Its proof can be found in [16] and we omit it here for brevity. THEOREM 3. If u is an elastic Herglotz solution of the spectral Navier equation (74) there exist functions  L,  T which belong to L2 [0, 2π ] such that 2π 2π ˆ ˆ ikp d·r ˆ ˆ  ˆ  ˆ iks d·r ds(d) + ds(d), (85) L(d)e T(d)e  u(r) = 0

0

TWO-DIMENSIONAL ELASTIC HERGLOTZ FUNCTIONS WITH APPLICATIONS

139

where dˆ = (cos ϑ, sin ϑ) is a unit vector and  L and  T are the dyadic counterparts (see also [24]) of equations (49) and (50), respectively. Conversely, if u is given by (85) with  L,  T ∈ L2 [0, 2π ], then it is an elastic Herglotz function. The following theorem shows that the superposition of incident plane waves for every direction of propagation generates a scattered field which is the superposition of the scattered fields that correspond to a specific direction plane wave incidence. Its proof can be found in [17]. THEOREM 4. For given dyadic densities  L and  T which belong to L2 [0, 2π ], the solution to the exterior Dirichlet scattering problem for the incident wave, 2π 2π inc ikp r·dˆ ˆ ˆ  ˆ  ˆ iks r·dˆ ds(d), ds(d) + (86) L(d)e T(d)e  v (r) = 0

0

is given by the relation 2π   ˆ ·  ˆ + ˆ ds(d), ˆ  usct (r; d) L(d) T(d)  vsct (r) =

(87)

0

where  usct is the scattered field generated by an incident wave of the form (76), and has the far-field patterns 2π   ˆ ·  ˆ + ˆ ds(d), ˆ  up∞ (ˆr; d) L(d) T(d) (88)  vp∞ (ˆr) = 0 2π   s ˆ ·  ˆ + ˆ ds(d), ˆ  us∞ (ˆr; d) L(d) T(d) (89)  v∞ (ˆr) = 0

p us∞ , where  u∞ , 

are the corresponding to the scattered field, longitudinal and transverse far-field patterns.

5.2. THE INVERSION SCHEME In what follows we present an inversion algorithm for the rigid body problem in two-dimensional linear elasticity, based on the ideas of Colton and Kirsch [4]. Before we proceed with the method we need to derive far-field equations for the elastic case. Using some asymptotic analysis we can show that i+1  rˆ ⊗ rˆ e−ikp rˆ ·y0 , 4µ π kp i+1 ˜ s  √ (r, y0 ) = (I − rˆ ⊗ rˆ )e−iks rˆ ·y0
∞ 4µ π ks p  (r, y0 ) =
∞

(90) (91)

140

V. SEVROGLOU AND G. PELEKANOS

are the far-field patterns of the longitudinal 
p (r, y0 ) and transverse 
s (r, y0 ) parts, respectively, of the fundamental dyadic 
(r, y0 ) given by equation (11) and placed at the point y0 ∈ Di . We now proceed as in [4] by considering special right-hand sides, given by (90), (91), in equations (88), (89) of Theorem 4. Hence, the far-field equations now take the form 2π   p ˆ ·  ˆ y0 ) +  ˆ y0 ) ds(d) ˆ   up∞ (r; d) L(d; T(d; (92)
∞ (r, y0 ) = 0

and s  (r, y0 ) =
∞

0

2π

  ˆ ·  ˆ y0 ) +  ˆ y0 ) ds(d). ˆ  us∞ (r; d) L(d; T(d;

(93)

We are now ready to formulate an inversion algorithm for the determination of the p u∞ ,  us∞ . The proof of support of the scattering object Di , from the knowledge of  the following theorem, which is the main theorem of the section, is also presented in [24]; however for the convenience of the reader we include it here as well. / σ (Di ) and that on ∂D THEOREM 5. Assume that Di is simply connected, ω2 ∈ the solution  utot of (74) satisfies the boundary condition (81). Then for every ε > 0 L(·, y0 ) and  T(·, y0 ) ∈ L2 [0, 2π ] such and y0 ∈ Di there exists a pair of dyadics  that the approximate far-field equations hold, i.e.,   2π     p p   ˆ  ˆ ˆ  ˆ   u (r; d) · L( d; y ) + T( d; y ) ds( d) −
(r, y ) < ε, (94) 0 0 0 ∞ ∞      

0

2π 0

    s s ˆ ˆ ˆ ˆ     u∞ (r; d) · L(d; y0 ) + T(d; y0 ) ds(d) −
∞ (r, y0 ) 

L2 [0,2π]

< ε, (95) L2 [0,2π]

and  L(·, y0 )L2 [0,2π] → +∞,  T(·, y0 )L2 [0,2π] → +∞

(96) (97)

as y0 → ∂D. / σ (Di ) there exists a unique solution for the boundary value Proof. If ω2 ∈ problem   ∗ w(r) =  0, r ∈ D, (98)  + ω2  0, r ∈ ∂D. (99)  w(r) +  8 (r, y0 ) =  In Theorem 2, it has been shown that the tensor Herglotz functions are dense in the space of classical solutions of the Navier equation. Hence any classical solution of Navier’s equation,  w, can be approximated with respect to the maximum norm by a Herglotz tensor function.

TWO-DIMENSIONAL ELASTIC HERGLOTZ FUNCTIONS WITH APPLICATIONS

141

It now follows that for the solution  w of Navier’s equation in Di we have that for every 9 > 0 there exists a pair of dyadics  L and  T ∈ L2 [0, 2π ] which are kernels for the Herglotz dyadic  vL,T (r) such that   w(r) < ε. (100) vL,T (r) −  max r∈D i

From (100) and an application of the Dirichlet boundary condition it follows immediately that  
(r, y0 ) < ε. (101) vL,T (r) +  max r∈∂D

p us∞ depend Taking into account the asymptotic behavior of 
and since  u∞ and  continuously on the boundary data, we can conclude from (101) to (94), (95). We now let y0 → ∂D and since 
is unbounded on ∂D, from (101), we get that L and  T must be unbounded on ∂D, that is  vL,T (r) is unbounded on ∂D. Hence  relations (96) and (97) are satisfied. ✷

Summarizing, the method of the shape reconstruction consists on finding a pair of densities  L and  T which satisfy the approximate far-field equations (88) and (89). From the knowledge of  L and  T, the boundary of the scatterer can be 2 L and  T has extremely large found at the points where the L -norm of the dyadics  value. 5.3. NUMERICAL RESULTS We will illustrate the applicability of the method for the special case of scattering of an SH wave by prismatic cylinders. Because the polarization of the SH waves is assumed to be parallel to the length of a cylindrical insert, the scattered waves have the same polarization and wave velocity. That results in a scalar problem where only one wave is present in the complete analysis, the SH wave. This is not the case, however, if a harmonic P or SV wave is incident to the obstacle, since then the waves reflected from the boundary involve both P and SV waves. In what follows the incident SH wave uinc = u0 eiks r cos θ propagates in the positive x-direction with wave number ks , and has a constant amplitude u0 . It is easy to see that the far-field equation (93) reduces for the SH wave case to its scalar form 2π iπ/4 ˆ d) ˆ dϑ = √e uSH r; d)g( e−iks rˆ ·y0 , (102) ∞ (ˆ 8π ks 0 where ks is the wavenumber, y0 a point in the interior of the scattering object, ˆ y0 ) + L(d; g = g(.; y0 ) ∈ L2 [0, 2π ] is the scalar Herglotz kernel that replaces [

142

V. SEVROGLOU AND G. PELEKANOS

ˆ denotes the far-field data that are obtained from Nyström’s  ˆ y0 )], and uSH r; d) T(d; ∞ (ˆ method [9]. We now note that since the far field operator is compact, the solution of the farfield equation (102) presents an ill-posed problem in the sense of Hadamard. To resolve this issue a regularization method due to Tikhonov is used. A regularized solution of (102) is then found by minimizing the functional 2  2π iπ/4   e SH −ik r ˆ ·y  ˆ d) ˆ dϑ − √ u∞ (ˆr; d)g( e s 0 + αg(·, y0 )2L2 [0,2π] .  2  8π k 0 s L [0,2π] (103) More details about this approach can be found in Groetsch [13]. As it is pointed out in [7] the regularization parameter α should depend on y0 . This is due to the fact that according to Theorem 5 we are looking for a solution g = g(·, y0 ) that blows up as y0 approaches the boundary of the scatterer. The optimum value of α is determined using Morozov’s discrepancy principle [7, 25]. In Theorem 5 we established that the norm of the Herglotz kernel gL2 [0,2π] becomes unbounded as y0 approaches the boundary of the scatterer. Hence repeating the above minimization process for all y0 on a grid containing the object, we obtain an image of the object by plotting for each y0 the norm of the optimum regularized solution. As in [7], we used the singular system of the far-field operator in order to simplify the computation of the algorithm. For a different kind of regularization see [24]. In our first numerical example we consider a circle of radius a = 0.21 m with Dirichlet boundary conditions. The scatterer is located at the center of the test square which was divided into 21 × 21 subsquares. The wavenumber for the SH-wave is ks = 9.5. We excite our object by a plane SH-wave. We compute the ˆ at 29 equidistantly distributed observation points and 29 r; d) far-field pattern uSH ∞ (ˆ uniformly distributed directions. Figure 1 (left) shows the contour for the circle of radius 0.21 m and center (0, 0). We now repeat the above experiment for two square

Figure 1. Reconstruction of a circle of radius 0.21 m (left); reconstruction of two squares with side 0.41 m and 5% Gaussian noise (right).

TWO-DIMENSIONAL ELASTIC HERGLOTZ FUNCTIONS WITH APPLICATIONS

143

scatterers with side 0.41 m; their centers are located at locations (−1.2, 0) and (1.2, 0), respectively, on the test square. The test domain is subdivided again into 21 × 21 subsquares and 5% Gaussian noise is added to the data. The reconstruction results are given in Figure 1 (right) where we clearly observe the presence of two square objects. Note however that the right square is better reconstructed compared to the left one. As it is easily observed, this method has some very attractive features, i.e., no low or high-frequency approximations are made and its implementation is very simple since it only involves matrix manipulations that lead to solutions of linear integral equations. A mathematical difficulty with the procedure just described is / Di . This deficiency led that it is not clear what happens in the case when y0 ∈ Kirsch to introduce a modified version of the linear sampling method [18]. The authors are now working towards this direction and in the near future they expect to present reconstructions from P and SV wave incidence, i.e., for cases where the problem is not scalar anymore. In conclusion, the work at hand is concerned with extending characterizations and denseness properties from acoustic Herglotz wave functions to two-dimensional elastic ones. The elastic Herglotz functions are defined as entire solutions to the Navier equation satisfying an appropriate growth condition at infinity. This is a very interesting and important topic, since as it became apparent in our last section, it provided us with the necessary theoretical background for extending the ideas of Colton and Kirsch [4] from the acoustic case to the more complicated elastic one.

References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11.

T. Arens, The scattering of plane elastic waves by a one-dimensional periodic surface. Math. Methods Appl. Sci. 22 (1999) 55–72. K. Aydin and A. Hizal, On the completeness of the spherical vector wave functions. J. Math. Anal. Appl. 117 (1986) 428–440. A. Ben-Menahem and S.J. Singh, Seismic Waves and Sources. Springer, New York (1981). D. Colton and A. Kirsch, A simple method for solving the inverse scattering problems in the resonance region. Inverse Problems 12 (1996) 383–393. D. Colton and P. Monk, A novel method for solving the inverse scattering problem for timeharmonic acoustic waves in the resonance region. SIAM J. Appl. Math. 45 (1985) 1039–1053. D. Colton and P. Monk, The inverse scattering problem for time-harmonic acoustic waves in an inhomogeneous medium. Quart. J. Mech. Appl. Math. 40 (1987) 189–212. D. Colton, M. Piana and R. Potthast, A simple method using Morozov’s discrepancy principle for solving inverse scattering problems. Inverse Problems 13 (1997) 1477–1493. D. Colton and R. Kress, Dense sets and far field patterns in electromagnetic wave propagation. SIAM J. Math. Anal. 16 (1985) 1049–1060. D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory. Springer, New York (1992). G. Dassios and Z. Rigou, On the density of traction traces in scattering by elastic waves. SIAM J. Appl. Math. 53 (1993) 141–153. G. Dassios and Z. Rigou, Elastic Herglotz functions. SIAM J. Appl. Math. 55 (1995) 1345– 1361.

144 12. 13. 14. 15. 16.

17. 18. 19. 20. 21. 22. 23. 24. 25. 26.

V. SEVROGLOU AND G. PELEKANOS

G. Dassios and Z. Rigou, On the reconstruction of a rigid body in the theory of elasticity. Z. Angew. Math. Mech. 77(12) (1997) 911–923. C.W. Groetsch, The Theory of Tikhonov Regularization for Fredholm Equations of the First Kind. Pitman, Boston (1984). P. Hartman, On the solutions of V +V = 0 in an exterior region. Math. Z. 71 (1959) 251–257. P. Hartman and C. Wilcox, On solutions of the Helmholtz equation in exterior domains. Math. Z. 75 (1961) 228–255. K. Kiriaki and V. Sevroglou, On Herglotz functions in two-dimensional linear elasticity. In: Scattering Theory and Biomedical Engineering Modelling and Applications, Proc. of the 4th Internat. Workshop (1999) pp. 151–158. K. Kiriaki and V. Sevroglou, Integral equations methods in obstacle elastic scattering. Bull. Greek Math. Soc. 45 (2001) 57–69. A. Kirsch, Characterization of the shape of a scattering obstacle using the spectral data of the far field operator. Inverse Problems 14 (1998) 1489–1512. V.D. Kupradze, Dynamical problems in elasticity. In: Progress in Solid Mechanics. NorthHolland, Amsterdam (1963). V.D. Kupradze, Three-Dimensional Problems of the Mathematical Theory of Elasticity and Thermoelasticity. North-Holland, Amsterdam (1979). P.A. Martin, On the scattering of elastic waves by an elastic inclusion in two dimensions. Quart. J. Mech. Appl. Math. 43 (1990) 275–292. P.M. Morse and H. Feshbach, Methods of Theoretical Physics, Vols. I, II. McGraw-Hill, New York (1953). C. Müller, Über die ganzen Lösungen der Wellengleichung. Math. Ann. 124 (1952) 235–264. V. Sevroglou and G. Pelekanos, An inversion algorithm in two-dimensional elasticity. J. Math. Anal. Appl. 263 (2001) 277–293. A.N. Tikhonov, A.V. Goncharsky, V.V. Stepanov and A.G. Yagola, Numerical Methods for the Solution of Ill-Posed Problems. Kluwer, Dordrecht (1995). E.C. Titchmarsh, Eigenfunction Expansions Associated with Second-Order Differential Equations, Vols. I, II. Oxford Univ. Press, Oxford, UK (1946, 1958).