J. Appl. Math. & Computing Vol. 12(2003), No. 1 - 2, pp. 81 - 105

AN INVERSE PROBLEM OF THE THREE-DIMENSIONAL WAVE EQUATION FOR A GENERAL ANNULAR VIBRATING MEMBRANE WITH PIECEWISE SMOOTH BOUNDARY CONDITIONS E. M. E. ZAYED

Abstract. This paper deals with the very interesting problem about the influence of piecewise smooth boundary conditions on the distribution of the eigenvalues of the negative Laplacian in R3 . The asymptotic expansion of the trace of the wave operator μ  (t) = |t| and i =

√

−1 , where

Laplacian −∇2 = −

 ∞

 3

k=1

μν

∂ ∂xk

2

ν=1

∞ 



exp

−i tμ

υ=1

1/2 υ



for small

are the eigenvalues of the negative

in the (x1 , x2 , x3 )-space, is studied for

an annular vibrating membrane Ω in R3 together with its smooth inner boundary surface S1 and its smooth outer boundary surface S2. In the present paper, a finite number of Dirichlet, Neumann and Robin boundary conditions on the piecewise smooth components Si∗ (i = 1, ..., m) of S1 and on the piecewise smooth components Si∗ (i = m + 1, ..., n) of S2 such that S1 =

m

i=1

Si∗ and S2 =

n

i=m+1

Si∗ are considered. The basic problem is to

extract information on the geometry of the annular vibrating membrane Ω from complete knowledge of its eigenvalues by analyzing the asymptotic μ (t) for small |t| . expansions of the spectral function  AMS Mathematics Subject Classification. Primary: 35Kxx, 35Pxx. Key words and phrases: Inverse problem, wave equation, annular vibrating membrane, eigenvalues, piecewise smooth boundary conditions, spectral function, heat kernel.

1. Introduction Received April 1, 2002. Revised March 4, 2003. c 2003 Korean Society for Computational & Applied Mathematics.  81

82

E. M. E. Zayed

The underlying inverse problem is to deduce some geometric quantities associated with a bounded domain in R3 from complete knowledge of the eigenvalues of the negative Laplacian. Let Ω be a simply connected bounded domain in R3 with a smooth bounding surface S . Consider the Robin problem 

where

∂ ∂n

−∇2 u = μu in Ω , ∂ +γ u = 0 on S , ∂n

(1.1) (1.2)

denotes differentiation along the outward normal to S and γ is a −

positive constant impedance, with u ∈ C 2 (Ω) ∩ C(Ω). Denote its eigenvalues, counted according to multiplicity, by 0 < μ1 ≤ μ2 ≤ ... ≤ μν ≤ ... → ∞

as

ν → ∞.

(1.3)

Zayed et al [12] have discussed the problem (1.1)-(1.2) for small / large impedance γ and have determined some geometrical quantities of Ω using the wave equation approach by analyzing the asymptotic expansion of the spectral function ∞  

1/2 μ  (t) = as |t| → 0, (1.4) exp −i tμυ υ=1

which represents a tempered distribution for −∞ < t < ∞ . Note that, when γ is small, the Robin boundary condition (1.2) looks approximately like the Neumann boundary condition, while when γ is large, the Robin boundary condition (1.2) looks approximately like the Dirichlet boundary ∂ condition provided ∂n remains finite. Zayed [19] and Zayed et al [12] have discussed the problem (1.1)-(1.2) in the following cases: Case1.1. If γ=0 (the Neumann problem)  V |S| sign t μ (t) = δ(− |t|) + 2 sign t + H(z)dz + O(t sign t), as |t| → 0. 4πt 8π |t| 12π 2 (1.5) S Case 1.2. If γ → ∞ (the Dirichlet problem)  V |S| sign t μ (t) = δ(− |t|) − 2 sign t + H(z) dz + O(t sign t), as |t| → 0, 4πt 8π |t| 12π 2 (1.6) S where δ(− |t|) is the Dirac delta function and ⎧ ⎨ 1 0 sign t = ⎩ −1

t > 0, t = 0, t < 0.

(1.7)

An inverse problem of the three-dimensional wave equation

83

In these formulae V , |S| and H(z) are respectively, the volume, the surface area, the mean curvature of Ω, such that H(z) = 12 [ R11(z) + R21(z) ], where R1 (z) and R2 (z) are the principal radii of curvature. Note that the sign ± of the second term of μ  (t) determines whether we have the Neumann or Dirichlet problem. Case 1.3. (the mixed problem) If |S1 | is the surface area of the component S1 of the boundary surface S with the Neumann boundary condition, and if |S2 | is the surface area of the remaining component S2 = S \ S1 of S with the Dirichlet boundary condition, then in [8-11], we obtain small μ (t)

(|S1 | − |S2 |) V δ(− |t|) + sign t 4πt 8π 2 |t| ⎫ ⎧   ⎬ sign t ⎨ + H(z) dz + H(z) dz + O(t sign t) as |t| → 0.(1.8) ⎭ 12π 2 ⎩

=

S1

S2

Note that the order term O(t sign t) in these formulae is yet undetermined. So, in the present paper, we discuss what geometric quantities are contained in this order term in the case Ω is an annular vibrating membrane together with piecewise smooth boundary conditions (1.10) and (1.11) stated below. The object of this paper is to discuss the following more general inverse problem: Let Ω be a general annular vibrating membrane in R3 consisting of a simply connected bounded inner domain Ω1 with a smooth bounding surface S1 and a simply connected bounded outer domain Ω2 ⊃ Ω1 with a smooth bounding surface S2 where Ω1 = Ω1 ∪ S1 . Suppose that the eigenvalues (1.3) are given exactly for the Helmholtz equation −∇2 u = μu

in Ω,

(1.9)

together with the following Dirichlet, Neumann and Robin boundary conditions on the piecewise smooth components Si∗ (i = 1, ..., m) of S1 : ⎫ u = 0, on Si∗ (i = 1, ..., k), ⎪ ⎬ ∂u ∗ = 0, on S (i = k + 1, ..., ), i (1.10)   ∂ni ⎪ ∂ ∗ ⎭ u = 0 on S + γ (i =  + 1, ..., m), i i ∂ni where S1 =

m

i=1

Si∗ as well as the following Dirichlet, Neumann and Robin bound-

ary conditions on the piecewise smooth components Si∗ (i = m + 1, ..., n) of S2 : ⎫ u = 0, on Si∗ (i = m + 1, ..., N ), ⎪ ⎬ ∂u ∗ = 0, on S (i = N + 1, ..., p), i (1.11) ∂n i   ⎪ ∂ + γi u = 0 on S ∗ (i = p + 1, ..., n), ⎭ ∂ni

where S2 =

n

i=m+1

i

Si∗ and γi are piecewise smooth positive constant impedances.

84

E. M. E. Zayed

The basic problem is to determine some geometric quantities (e.g., the volume of Ω, the surface area, the mean curvature, and the Gaussian curvature) associated with the main problem (1.9)-(1.11), using the wave equation approach by analyzing the asymptotic expansions of the spectral function μ  (t) for small |t|. Note that the special cases of the main problem (1.9)-(1.11) have been discussed by Abdel-Halim [1], Zayed et al [8-12] and Zayed [13, 15, 16, 19]. Therefore, this problem can be considered as a more general one which does not seem to have been investigated elsewhere. We close this section with the remark that an alternative ∞ to the spectral function (1.4) is to study the trace of the heat kernel Θ(t) = υ=1 exp(−tμυ ) as t → 0 (see for example ([2, 4, 5, 6, 7, 14, 17, 18, 20, 21, 22]). But, it is well known that the wave equation methods have given very strong result; the definitive one .. is that of Hormander [3] who has studied the distribution μ  (t) = tr[exp(−i tP )] near t = 0 for an elliptic positive semi-definite pseudodifferential operator P in Rn of order m. Therefore, in the present paper, we concentrate our efforts on a study of the asymptotic expansion of μ  (t) as |t| → 0 for the main problem (1.9)-(1.11).

2. Statement of the results Suppose that the outer bounding surface S 2 of the region Ω are given locally by infinitely differentiable functions x β = y β (σ2 ), (β = 1, 2, 3) of the parameters σ2i (i = 1, 2). If these parameters are chosen so that σ2i =constant, are lines of curvature, the first and second fundamental forms of S 2 can be written respectively, in the following forms : II 1 (σ2 , Δσ2 ) =

2

2  gii (σ2 ) Δσ2i ,

(2.1)

2  dii (σ2 ) Δσ2i .

(2.2)

i=1

and II 2 (σ2 , Δσ2 ) =

2

i=1

In terms of the coefficients gii and dii , the principal radii of curvatures for S 2 are given by Rii = gii /dii . Consequently  the mean  curvature H1 and the 1 1 1 Gaussian curvature N1 of S 2 , are H1 = 2 R11 + R22 and N1 = R111R22 . Similarly, Suppose that the inner bounding surface S 1 of the region Ω are given locally by infinitely differentiable functions x β = y β (σ1 ), (β = 1, 2, 3) of the parameters σ1i (i = 1, 2). If these parameters are chosen so that σ1i =constant, are lines of curvature, the first and second fundamental forms of S 1 can be written

An inverse problem of the three-dimensional wave equation

85

respectively, in the following forms : II ∗1 (σ1 , Δσ1 ) =

2

2  ∗ gii (σ1 ) Δσ1i ,

(2.3)

2  d∗ii (σ1 ) Δσ1i .

(2.4)

i=1

and II ∗2

(σ1 , Δσ1 ) =

2

i=1

∗ In terms of the coefficients gii and d∗ii , the principal radii of curvatures for ∗ ∗ ∗ ∗ S 1 are given by Rii = gii /dii . Consequently  the mean curvatures H1 and the Gaussian curvature N1∗ of S 1 , are H1∗ = 12 R1∗ + R1∗ and N1∗ = R∗ 1R∗ . Let 11

22

11

22

|Si∗ | , (i = 1, ..., m) be the surface areas of the components Si∗ of the inner bounding surface S1 and Let |Si∗ | , (i = m + 1, ..., n) be the surface areas of the components Si∗ of the outer bounding surface S2 respectively. Let hi > 0 (i = 1, ..., n) be sufficiently small numbers. Let ni (i = 1, ..., n) be the minimum distances from a point x = (x1 , x2 , x3 ) of the region Ω to the components Si∗ (i = 1, ..., n) respectively. Let ni (σ1 ) (i = 1, ..., m) denote the inward unit normals to the components Si∗ (i = 1, ..., m) of the inner bounding surface S1 and let ni (σ2 ) (i = m + 1, ..., n) denote the inward unit normals to the components Si∗ (i = m + 1, ..., n) of the outer bounding surface S2 respectively. Then, we note that the coordinates in the neighborhood of the components Si∗ (i = 1, ..., m) of S1 are in the same form as in Sec.5.2 of Zayed [18] with the interchanges n1 ↔ ni , h1 ↔ hi , I1 ↔ Ii , D(I1 ) ↔ D(Ii ) and δ1 ↔ δi (i = 1, ..., m). Thus, we have the same formulae (5.2.1)- (5.2.5) of the Sec.5.2 in Zayed [18] with the interchanges n1 ↔ ni , n1 (σ1 ) ↔ ni (σ1 ). Similarly the coordinates in the neighborhood of the components Si∗ (i = m + 1, ..., n) of S2 are similar to that obtained in Sec.5.1 of Zayed [18] with the interchanges n2 ↔ ni , h2 ↔ hi , I2 ↔ Ii . D(I2 ) ↔ D(Ii ) and δ2 ↔ δi (i = m + 1, ..., n). Thus, we have the same formulae (5.1.1)- (5.1.6) of the Sec.5.1 in Zayed [18] with the interchanges n2 ↔ ni , n2 (σ2 ) ↔ ni (σ2 ).

Theorem 2.1. With the assumptions stated above, the asymptotic expansion of the trace of the wave operator μ (t) for small |t| of the main problem (1.9) (1.11) can be written in the form: μ (t) =

a1 a2 δ(− |t|) + sign t + a3 sign t + a4 t sign t + O(t2 sign t), t |t|

as |t| → 0, (2.5)

where 0 < γi << 1 (i =  + 1, ..., b), γi >> 1 (i = b + 1, ..., m), 0 < γi << 1 (i = P + 1, ..., c), and γi >> 1 (i = c + 1, ..., n).

86

E. M. E. Zayed

Here the coefficients aν (ν = 1 − 4) have the forms a1 a2

=

V , 4π

=

1 8π 2





|Si∗ |

+

i=k+1

b

 |Si∗ |

−

⎞⎤

 m

⎜ ∗ ⎟⎥ −1 H1∗ dSi∗ ⎠⎦ + + ⎝|Si | − 2γi i=b+1

=

m 

Si

H1∗ dSi∗ +

c

|Si∗ | +

i=N +1

 c

+

 N

 |Si∗ |

i=P +1

⎧i ⎪ k  ⎨

1 256π 2 ⎪ ⎩ i=1

 n

H1 dSi∗

i=c+1S ∗ i

(H1∗2

−

(H1∗

Si

H1 dSi∗

i=N +1S ∗

N1∗ ) dSi∗

+7

 

⎫ ⎪ ⎬ ⎪ ⎭

i

,

(H1∗2 − N1∗ ) dSi∗

i=k+1S ∗ i

Si∗

 % b

 P

H1 dSi∗ +

(H1 − 3γi ) dSi∗ +

i=P +1S ∗

+7

Si

i=m+1S ∗ i

i=b+1S ∗ i

=

P

⎤⎫ ⎪  N n ⎬

⎥ ⎢ |Si∗ | + (|Si∗ | − 2γi−1 H1 dSi∗ )⎦ , −⎣ ⎪ ⎭ i=m+1 i=c+1 Si∗ ⎧ ⎪ k     b

1 ⎨ ∗ ∗ ∗ ∗ H1 dSi + H1 dSi + (H1∗ − 3γi ) dSi∗ 12π 2 ⎪ ⎩ i=1 ∗ i=k+1 ∗ i=+1 ∗ +

a4



Si∗

⎡

a3

|Si∗ |

i=1

i=+1

⎛

 k

− 3γi ) − 2

(N1∗

i=+1S ∗ i

& 26 47 2 ∗ − γi H1 + γi ) dSi∗ 7 7

 m  N

' ∗2 ( (H1 − (N1∗ − 16γi−1 H1∗ ) dSi∗ + (H12 − N1 ) dSi∗

+

i=b+1S ∗ i

+7

 P

(H12 − N1 ) dSi∗ +

( (H12 − (N1 − 16γi−1 H1 ) dSi∗

i=c+1S ∗ i

i=N +1S ∗ i

+7

n 

'

 % c

&

(H1 − 3γi )2 − (N1 −

i=P +1S ∗ i

i=m+1S ∗ i

⎫ ⎪ ⎬

26 47 γi H1 + γi2 ) dSi∗ ⎪ 7 7 ⎭

An inverse problem of the three-dimensional wave equation

87

With reference to the formulae (1.5)-(1.8) and to the articles [1, 12, 19], the asymptotic expansion (2.5) may be interpreted as follows: (i) Ω is a general annular vibrating membrane in R3 and we have the piecewise smooth boundary conditions (1.10) and (1.11) with small/large impedances γi . (ii) For the first four terms, Ω is a general annular vibrating membrane in R3 k of volume V, the components Si∗ (i = 1, ..., k) of S1 are of surface areas i=1 |Si∗ | , mean curvature H1∗ and Gaussian curvature N1∗ together with Dirichlet boundary conditions, the components Si∗ (i = k + 1, ..., ) of S1 are of surface areas  ∗ ∗ ∗ i=k+1 |Si | , mean curvature H1 and Gaussian curvature N1 together with ∗ Neumann boundary conditions, the components Si (i =  + 1, ..., b) of S1 are of  surface areas bi=+1 |Si∗ | , mean curvatures (H1∗ − 3γi ) and Gaussian curvatures 47 2 ∗ (N1∗ − 26 7 γi H1 + 7 γi ) together with Neumann boundary conditions, and the remaining components Si∗ (i = b + 1, ..., m) of S1 are of surface areas  m

−1 ∗ (|Si | − 2γi H1∗ dSi∗ ), i=b+1

Si∗

mean curvature H1∗ and Gaussian curvatures (N1∗ − 16γi−1 H1∗ ) together with Dirichlet boundary conditions. Similarly, the components Si∗ (i = m + 1, ..., N ) N of S2 are of surface areas i=m+1 |Si∗ | ,mean curvature H1 and Gaussian curthe components Si∗ (i = vature N1 together with Dirichlet boundary p conditions, ∗ N + 1, ..., p) of S2 are of surface areas i=N +1 |Si | , mean curvature H1 and Gaussian curvature N1 together with Neumann boundary conditions, the compo nents Si∗ (i = p + 1, ..., c) of S2 are of surface areas ci=p+1 |Si∗ | , mean curvatures 47 2 (H1 − 3γi ) and Gaussian curvatures (N1 − 26 7 γi H1 + 7 γi ) together with Neumann boundary conditions, and the remaining components Si∗ (i = c + 1, ..., n) of S2 are of surface areas  n

−1 ∗ (|Si | − 2γi H1 dSi∗ ), i=c+1

Si∗

mean curvature H1 and Gaussian curvatures (N1 − 16γi−1H1 ) together with Dirichlet boundary conditions. (iii) The order term O(t2 sign t) may contain further information about the geometry of the annular region Ω ⊆ R3 and its determination is still an open problem, which has been left for the interested readers.

3. Formulation of the mathematical problem

88

E. M. E. Zayed

With reference to the articles [8-12], it can be easily seen that the spectral function μ (t) associated with the main problem (1.9)-(1.11) is given by  μ (t) =

G (x, x; t)dx,

(3.1)

Ω

where G (x1 , x2 ; t) is the Green’s function for the wave equation  ∇2 −

∂2 ∂t2

G (x1 , x2 ; t) = 0

in Ω × {−∞ < t < ∞} ,

(3.2)

subject to the boundary conditions (1.10)-(1.11) and the initial conditions lim G (x1 , x2 ; t) = 0, lim

t→0

t→0

∂ G (x1 , x2 ; t) = δ(x1 − x2 ). ∂t

(3.3)

Let us write G (x1 , x2 ; t) = G0 (x1 , x2 ; t) + χ(x1 , x2 ; t) ,

(3.4)

where 1 δ(|x1 − x2 | − |t|), (3.5) 4πt is the “fundamental solution” of the wave equation (3.2) while χ(x1 , x2 ; t) is the “regular solution” chosen in such a way that G (x1 , x2 ; t) satisfies the piecewise smooth boundary conditions (1.10)-(1.11). On setting x1 = x2 = x, we find that G0 (x1 , x2 ; t) =

a1 δ(− |t|) + K(t), t

μ (t) = where a1 =

V 4π

(3.6)

and  χ (x, x; t)dx.

K(t) =

(3.7)

Ω

The problem now is to determine the asymptotic expansions of K(t) for small |t| . In what follows, we shall use Fourier transforms with respect to −∞ < t < ∞ and use −∞ < η < ∞ as the Fourier transform parameter. Thus, we define  (x1 , x2 ; η) = G



+∞

−∞

e−2π i η t G (x1 , x2 ; t)dt.

(3.8)

An application of the Fourier transform to the wave equation (3.2) shows that  (x1 , x2 ; η) satisfies the reduced wave equation G

An inverse problem of the three-dimensional wave equation

89

 (x1 , x2 ; η) = −δ(x1 − x2 ) in Ω, (∇2 + 4π 2 η 2 )G

(3.9)

together with the boundary conditions (1.10)-(1.11). The asymptotic expansions of K(t), for small |t| , may then be deduced di rectly from the asymptotic expansions of K(η), for large |η| , where   χ  (x, x; η)dx. (3.10) K(η) = Ω

4. Proof of Theorem 2.1

It is well known (see for example [8-12] ) that the reduced wave equation (3.9) has the fundamental solution

 0 (x1 , x2 ; η) = G

exp(−2πi η rx

1 x2

4π rx

) ,

(4.1)

1 x2

where rx x = |x1 − x2 | is the distance between the points x1 = (x11 , x21 , x31 ) and 1 2 x2 = (x12 , x22 , x32 ) of the annular region Ω ⊆ R3 . The existence of the solution  1 , x2 ; η) satisfying the (4.1) enables us to construct integral equations for G(x boundary conditions (1.10)-(1.11). Therefore, if we consider the problem (1.9)(1.11) with the case 0 < γi << 1 (i = +1, ..., b), γi >> 1 (i = b+1, ..., m), 0 < γi << 1 (i = p + 1, ..., c), and γi >> 1 (i = c + 1, ..., n) then, Green’s theorem gives the following integral equation:  1 , x2 ; η) G(x

 0 (x1 , x2 ; η) = G % & k 

∂   0 (y, x2 ; η)dy −2 G(x1 , y; η) G ∗ ∂n i y S i i=1 % &  

∂   +2 G(x1 , y; η) G (y, x2 ; η) dy ∂ni y 0 i=k+1S ∗ i

+2

 b

i=+1S ∗ i

% & ∂   + γi )G 0 (y, x2 ; η) dy G(x1 , y; η) ( ∂ni y

90

E. M. E. Zayed

& m  %

∂  −1 ∂  0 (y, x2 ; η)dy G −2 G(x1 , y; η) 1 + γi ∂ni y ∂ni y i=b+1S ∗ i

%

& ∂   0 (y, x2 ; η)dy G(x1 , y; η) G ∗ ∂n i y S j i=m+1 % &  p

∂   −2 G(x1 , y; η) G (y, x2 ; η) dy ∂ni y 0 +2

 N

i=N +1S ∗ i

% &  0 (y, x2 ; η) dy  1 , y; η) ( ∂ + γi )G G(x ∗ ∂ni y i=p+1 Sj & n  %

∂ ∂   0 (y, x2 ; η)dy.(4.2) G +2 G(x1 , y; η) 1 + γi−1 ∂n ∂n i y i y i=c+1 −2

 c

Si∗

By applying the iteration methods (see for example [12, 18, 19]) to the integral  1 , x2 ; η) which has a regular equation (4.2), we obtain the Green’s function G(x part in the following form :

χ  (x1 , x2 ; η) = 2

72

Aω ,

(4.3)

ω=1

where

A1

=

−

k  %

i=1 S ∗

& ∂   0 (y, x2 ; η)dy, G ( x1 , y; η)) G ∂ni y 0

i

A2

=

 

i=k+1S ∗ i

A3

=

 b

i=+1S ∗ i

A4

=

−

 0 ( x1 , y; η) G

=

m 

 % N

i=m+1S ∗ i

& ∂  G (y, x2 ; η) dy, ∂ni y 0

% & ∂   + γi )G 0 (y, x2 ; η) dy, G 0 ( x1 , y; η) ( ∂ni y

i=b+1S ∗ i

A5

%

∂  G ( x1 , y; η) ∂ni y 0

% 1+

γi−1

∂ ∂ni y



& ∂   0 (y, x2 ; η)dy, G ( x1 , y; η) G ∂ni y 0

&  G 0 (y, x2 ; η) dy,

An inverse problem of the three-dimensional wave equation

A6

=

−

 P

 0 ( x1 , y; η) G

i=N +1S ∗ i

A7

=

−

%

91

& ∂  G 0 (y, x2 ; η) dy, ∂ni y

% &  0 (y, x2 ; η) dy,  0 ( x1 , y; η) ( ∂ + γi )G G ∂ni y

 c

i=P +1S ∗ i

A8

&% &  % n

∂  −1 ∂  = )G 0 (y, x2 ; η) dy, G ( x1 , y; η) (1 + γi ∂ni y 0 ∂ni y i=c+1 Si∗

A9

=

k   %

&  ∂   0 (y  , x2 ; η)dy dy  , G 0 ( x1 , y; η) M1 (y, y ) G ∂ni y

i=1 S ∗ S ∗ i i

A10

=



  ∂  0 ( x1 , y; η)M2 (y, y )  0 (y , x2 ; η) dy dy  , G G ∂ni y

  



i=k+1S ∗ S ∗ i i

A11

=



  ∂  0 (y , x2 ; η) dy dy  ,  0 ( x1 , y; η) Mγi (y, y ) ( + γi )G G ∂ni y

  b



i=+1S ∗ S ∗ i i

A12

=

& m   %

 ∂  G 0 ( x1 , y; η) Mγ −1 (y, y ) × i ∂ni y

i=b+1S ∗ S ∗ i i

   ∂  )G 0 (y , x2 ; η) dy dy , (1 + ∂ni y % &   N

 ∂   0 (y  , x2 ; η)dy dy  , = G 0 ( x1 , y; η) M3 (y, y ) G ∂n iy i=m+1 

γi−1

A13

Si∗ Si∗

A14

=

  P

i=N +1S ∗ S ∗ i i

A15

=

  c

i=P +1S ∗ S ∗ i i

A16



 0 ( x1 , y; η) M4 (y, y  ) G

   ∂  G 0 (y , x2 ; η) dy dy , ∂ni y

   G 0 ( x1 , y; η)Lγi (y, y ) (

   ∂  + γi )G 0 (y , x2 ; η) dy dy ,  ∂ni y

&   % n

 ∂  = G 0 ( x1 , y; η) Lγ −1 (y, y ) × i ∂n iy i=c+1 Si∗ Si∗

  ∂  0 (y , x2 ; η) dy dy  , )G (1 + γi−1 ∂ni y



92

A17

E. M. E. Zayed

=

⎤ ⎡  k  

⎢  0 (y  , x2 ; η) dy  ,  0 ( x1 , y; η) M5 (y, y  )dy ⎥ − G ⎦G ⎣ i=1 S ∗ i

A18

=

−



i=k+1S ∗ i

⎧  ⎪ k  ⎨

i=k+1S ∗ i

A19

=

i

=

=

−

b

⎧ ⎪ k   ⎨

k 

i

A23

A24

A25

⎫ ⎪ ⎬

[

 ∂  G 0 ( x1 , y; η)] M6 (y, y )dy × ⎪ ∂ni y ⎭

⎪ ⎩ i=1 ∗ Si )   ∂  0 (y , x2 ; η) dy  , + γi )G ( ∂ni y

i=1 S ∗

A22

Si∗

 ∂  G 0 ( x1 , y; η)] M6 (y, y )dy × ⎪ ∂ni y ⎭

i=+1S ∗ i

i=+1S ∗ i

A21

[

  ∂  G 0 (y , x2 ; η) dy , ∂ni y ⎤ ⎡  k  b

⎢  0 (y  , x2 ; η) dy  ,  0 ( x1 , y; η)Mγ∗ (y, y  )dy ⎥ − G ⎦G ⎣ i

i=1 S ∗

A20

⎪ ⎩ i=1

⎫ ⎪ ⎬

⎫ ⎧ ⎪ ⎪ & m  % ⎬ ⎨  ∂   0 (y  , x2 ; η)dy  , G 0 ( x1 , y; η) Mγ −1 (y, y )dy G i ⎪ ⎪ ∂ni y ⎭ ⎩ i=b+1S ∗ i

⎧ m  ⎪ k  ⎨

⎫ ⎪ ⎬

 ∂  [ G 0 ( x1 , y; η)] M6 (y, y )dy × ⎪ ⎪ ∂n iy ⎭ i=b+1S ∗ ⎩ i=1 S ∗ i i  ) ∂  0 (y  , x2 ; η) dy  , (1 + γi−1 )G ∂ni y ⎫ ⎧   ⎪  ⎪   b ⎬ ⎨

  ∂  0 ( x1 , y; η) Mγ∗ (y, y )dy  0 (y , x2 ; η) dy  , = G G i  ⎪ ⎪ ∂n ⎭ iy i=k+1S ∗ ⎩i=+1S ∗ i i ⎫ ⎧ ⎪  ⎪  b  ⎬ ⎨

  = G 0 ( x1 , y; η) M5 (y, y )dy × ⎪ ⎪ ⎭ i=+1S ∗ ⎩i=k+1S ∗ i i  )   ∂  ( + γi )G 0 (y , x2 ; η) dy , ∂ni y ⎫ ⎧ ⎪ ⎪    m ⎬ ⎨

 ∂  = − [ G 0 ( x1 , y; η)] Mγ −1 (y, y )dy × i ⎪ ⎪ ∂ni y ⎭ i=k+1S ∗ ⎩i=b+1S ∗ i i     ∂  G 0 (y , x2 ; η) dy , ∂ni y

=

An inverse problem of the three-dimensional wave equation

A26

=

A27

=

A28

=

A29

=

A30

=

A31

=

A32

=

A33

=

−

⎧  m  ⎪  ⎨

⎪ i=b+1S ∗ ⎩i=k+1S ∗ i i 

 0 ( x1 , y; η) M5 (y, y  )dy G

⎫ ⎪ ⎬ ⎪ ⎭

93

×

)  ∂  0 (y , x2 ; η) dy  , (1 + γi−1 )G ∂ni y ⎫ ⎧ ⎪  ⎪ b m  ⎬ ⎨

 ∂  − [ G 0 ( x1 , y; η)] Mγ −1 (y, y )dy × i ⎪ ⎪ ∂ni y ⎭ i=+1S ∗ ⎩i=b+1S ∗ i i  ) ∂  0 (y  , x2 ; η) dy  , ( + γi )G ∂ni y ⎫ ⎧ ⎪  m  ⎪ b ⎬ ⎨

 ∗  − G 0 ( x1 , y; η) Mγi (y, y )dy × ⎪ ⎪ ⎭ i=b+1S ∗ ⎩i=+1S ∗ i i  ) ∂  0 (y  , x2 ; η) dy  , (1 + γi−1 )G ∂ni y ⎫ ⎧ ⎪  ⎪  N P ⎬ ⎨

 0 (y  , x2 ; η) dy  ,  0 ( x1 , y; η) M7 (y, y  )dy G − G ⎪ ⎪ ⎭ i=m+1S ∗ ⎩i=N +1S ∗ i i ⎫ ⎧ ⎪ ⎪   P N ⎬ ⎨

 ∂  − [ G 0 ( x1 , y; η)] M8 (y, y )dy × ⎪ ⎪ ∂ni y ⎭ i=N +1S ∗ ⎩i=m+1S ∗ i i     ∂  G 0 (y , x2 ; η) dy ,  ∂ni y ⎫ ⎧ ⎪ ⎪   N c ⎬ ⎨

  0 (y  , x2 ; η) dy  ,  − G 0 ( x1 , y; η) Mγi (y, y )dy G ⎪ ⎪ ⎭ i=m+1S ∗ ⎩i=P +1S ∗ i i ⎫ ⎧ ⎪  ⎪  c N ⎬ ⎨

 ∂  − [ G 0 ( x1 , y; η)] M8 (y, y )dy × ⎪ ⎪ ∂ni y ⎭ i=P +1S ∗ ⎩i=m+1S ∗ i i  ) ∂  0 (y  , x2 ; η) dy  , ( + γi )G ∂ni y ⎫ ⎧ ⎪ ⎪   N n ⎬ ⎨

 ∂  [ G 0 ( x1 , y; η)] Lγ −1 (y, y )dy × i ⎪ ⎪ ∂ni y ⎭ i=m+1 ∗ ⎩i=c+1 ∗ Si

Si

 0 (y , x2 ; η) dy  , G 

94

E. M. E. Zayed

A34

=

A35

=

A36

=

A37

=

A38

=

A39

=

A40

=

⎧  ⎪  n N ⎨

⎫ ⎪ ⎬

 ∂  [ G 0 ( x1 , y; η)] M8 (y, y )dy × ⎪ ⎪ ∂n iy ⎭ i=c+1S ∗ ⎩i=m+1S ∗ i i  ) ∂  0 (y  , x2 ; η) dy  , (1 + γi−1 )G ∂ni y ⎫ ⎧ ⎪  ⎪  P c ⎬ ⎨

  G 0 ( x1 , y; η)Mγi (y, y )dy × ⎪ ⎪ ⎭ i=N +1S ∗ ⎩i=P +1S ∗ i i     ∂  G (y , x2 ; η) dy , ∂ni y 0 ⎫ ⎧ ⎪  ⎪  c P ⎬ ⎨

  G 0 ( x1 , y; η) M7 (y, y )dy × ⎪ ⎪ ⎭ i=P +1S ∗ ⎩i=N +1S ∗ i i  ) ∂  0 (y  , x2 ; η) dy  , ( + γi )G ∂ni y ⎫ ⎧ ⎪  ⎪  P n ⎬ ⎨

 ∂  − [ G 0 ( x1 , y; η)] Lγ −1 (y, y )dy × i ⎪ ⎪ ∂ni y ⎭ i=N +1S ∗ ⎩i=c+1S ∗ i i     ∂  G (y , x2 ; η) dy , ∂niy 0 ⎫ ⎧ ⎪  n  ⎪ P ⎬ ⎨

 0 ( x1 , y; η) M7 (y, y  )dy × − G ⎪ ⎪ ⎭ i=c+1S ∗ ⎩i=N +1S ∗ i i  ) ∂  0 (y  , x2 ; η) dy  , (1 + γi−1 )G ∂ni y ⎫ ⎧ ⎪ ⎪   c n ⎬ ⎨

 ∂  − [ G 0 ( x1 , y; η)] Lγ −1 (y, y )dy × i ⎪ ⎪ ∂ni y ⎭ i=P +1S ∗ ⎩i=c+1S ∗ i i )  ∂  0 (y  , x2 ; η) dy  , + γi )G ( ∂ni y ⎫ ⎧ ⎪ ⎪   n c ⎬ ⎨

 0 ( x1 , y; η) Mγi (y, y  )dy × − G ⎪ ⎪ ⎭ i=c+1S ∗ ⎩i=P +1S ∗ i i  ) ∂  0 (y  , x2 ; η) dy  , (1 + γi−1 )G ∂ni y

An inverse problem of the three-dimensional wave equation

A41

=

A42

=

A43

=

A44

=

A45

=

A46

=

A47

=

A48

=

A49

=

A50

=

⎧  k  ⎪ N ⎨

⎫ ⎪ ⎬

95

 ∂   0 (y  , x2 ; η) dy  , [ G 0 ( x1 , y; η)] M1 (y, y )dy G ⎪ ⎪ ∂n iy ⎭ i=1 S ∗ ⎩i=m+1S ∗ i i ⎫ ⎧ ⎪ ⎪   N k ⎬ ⎨

 ∂   0 (y  , x2 ; η) dy  , − [ G 0 ( x1 , y; η)] M3 (y, y )dy G ⎪ ⎪ ∂ni y ⎭ i=m+1S ∗ ⎩ i=1 S ∗ i i ⎫ ⎧ ⎪  k  ⎪ P ⎬ ⎨

  0 (y  , x2 ; η) dy  ,  G 0 ( x1 , y; η) M7 (y, y )dy G ⎪ ⎪ ⎭ i=1 S ∗ ⎩i=N +1S ∗ i i ⎫ ⎧ ⎪ ⎪   P k ⎬ ⎨

 ∂  [ G 0 ( x1 , y; η)] M8 (y, y )dy × ⎪ ⎪ ∂ni y ⎭ i=N +1S ∗ ⎩ i=1 S ∗ i i     ∂  G 0 (y , x2 ; η) dy , ∂ni y ⎫ ⎧ ⎪ ⎪   k c ⎬ ⎨

  0 (y  , x2 ; η) dy  ,  G 0 ( x1 , y; η) Mγi (y, y )dy G ⎪ ⎪ ⎭ i=1 S ∗ ⎩i=P +1S ∗ i i ⎫ ⎧ ⎪  ⎪ c k  ⎬ ⎨

 ∂  [ G 0 ( x1 , y; η) ] M8 (y, y )dy × ⎪ ⎪ ∂ni y ⎭ i=P +1S ∗ ⎩ i=1 S ∗ i i  ) ∂  0 (y  , x2 ; η) dy  , ( + γi )G ∂ni y ⎫ ⎧ ⎪  k  ⎪ n ⎬ ⎨

 ∂   0 (y  , x2 ; η) dy  , − [ G 0 ( x1 , y; η)] Lγ −1 (y, y )dy G i ⎪ ⎪ ∂n i y ⎭ i=1 S ∗ ⎩i=c+1S ∗ i i ⎫ ⎧ ⎪ n  ⎪ k  ⎬ ⎨

 ∂  − [ G 0 ( x1 , y; η) ] M8 (y, y )dy × ⎪ ⎪ ∂ni y ⎭ i=c+1S ∗ ⎩ i=1 S ∗ i i  )   ∂ −1  (1 + γi )G 0 (y , x2 ; η) dy ,  ∂ni y ⎫ ⎧ ⎪  ⎪   N ⎬ ⎨

 ∂  [ G 0 ( x1 , y; η) ] M6 (y, y )dy × ⎪ ⎪ ∂ni y ⎭ i=k+1S ∗ ⎩i=m+1S ∗ i i     ∂  G 0 (y , x2 ; η) dy ,  ∂ni y ⎫ ⎧ ⎪  ⎪  N  ⎬ ⎨

  0 (y  , x2 ; η) dy  ,  G 0 ( x1 , y; η) M7 (y, y )dy G ⎪ ⎪ ⎭ i=m+1 ∗ ⎩i=k+1 ∗

−

Si

Si

96

E. M. E. Zayed

A51

=

−

⎧  ⎪   P ⎨

i=k+1S ∗ i



A52

=

A53

=

A54

=

A55

=

A56

=

A57

=

⎪ ⎩i=N +1

Si∗

 0 ( x1 , y; η) M7 (y, y  )dy G

⎫ ⎪ ⎬ ⎪ ⎭

×

   ∂  G (y , x2 ; η) dy , ∂ni y 0 ⎫ ⎧ ⎪  ⎪  P  ⎬ ⎨

  − G 0 ( x1 , y; η) M4 (y, y )dy × ⎪ ⎪ ⎭ i=N +1S ∗ ⎩i=k+1S ∗ i i     ∂  G (y , x2 ; η) dy , ∂ni y 0 ⎫ ⎧ ⎪  ⎪   c ⎬ ⎨

  − G 0 ( x1 , y; η) Mγi (y, y )dy × ⎪ ⎪ ⎭ i=k+1S ∗ ⎩i=P +1S ∗ i i     ∂  G (y , x2 ; η) dy , ∂ni y 0 ⎫ ⎧ ⎪  ⎪  c  ⎬ ⎨

 0 ( x1 , y; η) M4 (y, y  )dy × − G ⎪ ⎪ ⎭ i=P +1S ∗ ⎩i=k+1S ∗ i i  ) ∂  0 (y  , x2 ; η) dy  , ( + γi )G ∂ni y ⎫ ⎧ ⎪  ⎪   n ⎬ ⎨

 ∂  [ G 0 ( x1 , y; η)] Mγ −1 (y, y )dy × i ⎪ ⎪ ∂ni y ⎭ i=k+1S ∗ ⎩i=c+1S ∗ i i     ∂  G (y , x2 ; η) dy , ∂ni y 0 ⎫ ⎧ ⎪ ⎪   n  ⎬ ⎨

 0 ( x1 , y; η) M7 (y, y  )dy × G ⎪ ⎪ ⎭ i=c+1S ∗ ⎩i=k+1S ∗ i i )  ∂  0 (y  , x2 ; η) dy  , )G (1 + γi−1 ∂ni y ⎫ ⎧ ⎪ ⎪   b N ⎬ ⎨

 ∂  [ G 0 ( x1 , y; η) ] M6 (y, y )dy × ⎪ ⎪ ∂ni y ⎭ i=+1S ∗ ⎩i=m+1S ∗ i i  ) ∂  0 (y  , x2 ; η) dy  , ( + γi )G ∂ni y

An inverse problem of the three-dimensional wave equation

A58

=

A59

=

A60

=

A61

=

A62

=

A63

=

A64

=

A65

=

⎧  ⎪  N b ⎨

⎫ ⎪ ⎬

97

 0 (y  , x2 ; η) dy  ,  0 ( x1 , y; η) M ∗∗ (y, y  )dy G G γi ⎪ ⎪ ⎭ i=m+1S ∗ ⎩i=+1S ∗ i i ⎫ ⎧ ⎪ ⎪   b P ⎬ ⎨

  − G 0 ( x1 , y; η) M7 (y, y )dy × ⎪ ⎪ ⎭ i=+1S ∗ ⎩i=N +1S ∗ i i  )   ∂  ( + γi )G 0 (y , x2 ; η) dy , ∂ni y ⎫ ⎧ ⎪ ⎪   P b ⎬ ⎨

  − G 0 ( x1 , y; η) Mγi (y, y )dy × ⎪ ⎪ ⎭ i=N +1S ∗ ⎩i=+1S ∗ i i     ∂  G 0 (y , x2 ; η) dy , ∂ni y ⎫ ⎧ ⎪ ⎪   b c ⎬ ⎨

  − G 0 ( x1 , y; η) Mγi (y, y )dy × ⎪ ⎪ ⎭ i=+1S ∗ ⎩i=P +1S ∗ i i )    ∂  + γi )G 0 (y , x2 ; η) dy , ( ∂ni y ⎫ ⎧ ⎪ ⎪   c b ⎬ ⎨

  − G 0 ( x1 , y; η) Mγi (y, y )dy × ⎪ ⎪ ⎭ i=P +1S ∗ ⎩i=+1S ∗ i i  )   ∂  ( + γi )G 0 (y , x2 ; η) dy ,  ∂ni y ⎫ ⎧ ⎪ ⎪   b n ⎬ ⎨

 ∂  [ G 0 ( x1 , y; η) ] Mγ −1 (y, y )dy × i ⎪ ⎪ ∂ni y ⎭ i=+1S ∗ ⎩i=c+1S ∗ i i  )   ∂  ( + γi )G 0 (y , x2 ; η) dy ,  ∂ni y ⎫ ⎧ ⎪  ⎪  n b ⎬ ⎨

 ∗∗  G 0 ( x1 , y; η) Mγi (y, y )dy × ⎪ ⎪ ⎭ i=c+1S ∗ ⎩i=+1S ∗ i i )    ∂ −1  )G 0 (y , x2 ; η) dy , (1 + γi  ∂ni y ⎫ ⎧ ⎪  m  ⎪ N ⎬ ⎨

 ∂  − [ G 0 ( x1 , y; η)] M1 (y, y )dy × ⎪ ⎪ ∂ni y ⎭ i=b+1S ∗ ⎩i=m+1S ∗ i i  )   ∂ −1  (1 + γi )G 0 (y , x2 ; η) dy , ∂ni y

98

E. M. E. Zayed

A66

⎫ ⎧ ⎪ &  ⎪ N m  % ⎬ ⎨

 ∂  = − G 0 ( x1 , y; η) Lγ −1 (y, y )dy × i ⎪ ⎪ ∂ni y ⎭ ⎩ i=m+1 i=b+1S ∗ i

Si∗

A67

A68

 (y  , x2 ; η)dy  , G ⎫ ⎧ 0 ⎪ ⎪   m P ⎬ ⎨

 0 ( x1 , y; η) M5 (y, y  )dy × = G ⎪ ⎪ ⎭ i=b+1S ∗ ⎩i=N +1S ∗ i i  ) ∂  0 (y  , x2 ; η) dy  , (1 + γi−1 )G ∂ni y ⎫ ⎧ ⎪ ⎪   P m ⎬ ⎨

 ∂  = [ G 0 ( x1 , y; η) ] Mγ∗−1 (y, y )dy × ⎪ ⎪ i ∂ni y ⎭ ⎩ i=N +1S ∗ i

A69

=

m



i=b+1S ∗ i

   ∂  G 0 (y , x2 ; η) dy , ∂ni y

⎧  ⎪  c ⎨

⎪ ⎩i=P +1 ∗ Si 

i=b+1S ∗ i

(1 +

A70

=

c

 0 ( x1 , y; η) Mγ∗ (y, y  )dy G i

[

i

i=b+1S ∗ i



dy , ⎫ ⎪ ⎬

 ∂  G 0 ( x1 , y; η)] Mγ∗−1 (y, y )dy × ⎪ i ∂ni y ⎭

Si∗



(1 +

A72

×

⎪ ⎩i=b+1 ∗ Si  )   ∂  ( + γi )G 0 (y , x2 ; η) dy , ∂ni y ⎫ ⎧ ⎪ ⎪   m n ⎬ ⎨

 ∂  = − [ G 0 ( x1 , y; η) ] Mγ −1 (y, y )dy × i ⎪ ⎪ ∂ni y ⎭ ⎩i=c+1 i=P +1S ∗

A71

⎪ ⎭

)

∂  0 (y  , x2 ; η) )G ∂ni y

γi−1

⎧  ⎪ m  ⎨

⎫ ⎪ ⎬

= −

n

γi−1

⎧  ⎪ m  ⎨ ⎪ ⎩i=b+1 ∗ Si 

i=c+1S ∗ i

(1 +

γi−1

)

∂  0 (y  , x2 ; η) )G ∂ni y [



dy , ⎫ ⎪ ⎬

 ∂  G 0 ( x1 , y; η) ] Lγ −1 (y, y )dy × i ⎪ ∂ni y ⎭

)

∂  0 (y  , x2 ; η) )G  ∂ni y



dy ,

An inverse problem of the three-dimensional wave equation

where, we deduce also that 

M1 (y, y ) =

∞

υ

(υ)



(−1) K1 (y , y),

υ=0 (0)  K1 (y ,

y) = 

M2 (y, y ) =

2

 ∂  G (y, y ; η), ∂ni y 0

∞



(υ)

K2 (y , y),

υ=0 (0)



K2 (y , y) = 

Mγi (y, y ) =

 ∂  G 0 (y, y ; η), ∂ni y ∞

 Kγ(υ) (y , y), i

2

υ=0

 ∂  0 (y, y  ; η), 2 + γi G  ∂ni y 

 (y , Kγ(0) i

y) = 

Mγ −1 (y, y ) = i

∞

υ=0

υ

i



 (0) Kγ −1 (y , i

y) =

∂2 ∂ 2 + γi−1 ∂ni y ∂ni y ∂ni y



∞



υ=0 ∞



υ=0 ∞



υ=0 ∞

M3 (y, y ) = M4 (y, y ) = Lγi (y, y ) = Lγ −1 (y, y ) = i



M5 (y, y ) =



(υ)

(−1) Kγ −1 (y , y),

υ=0 ∞



(υ)

K1 (y , y), υ

(υ)



(−1) K2 (y , y), υ



(−1) Kγ(υ) (y , y), i 

(υ)

Kγ −1 (y , y), i

υ

(υ)



(−1) K3 (y , y),

υ=0 (0)



K3 (y , y) = 

M6 (y, y ) =

2

∂2  0 (y, y  ; η), G  ∂ni y ∂ni y

∞



(υ)

K4 (y , y),

υ=0 (0)



K4 (y , y) =

 0 (y, y ; η), 2G 

  0 (y, y  ; η), G

99

100

E. M. E. Zayed 

Mγ∗i (y, y )

=

∞

υ ∗ (υ)



(−1) K γi (y , y),

υ=0

∗ (0)

K γi

 ∂ ∂2  0 (y, y  ; η), G (y , y) = 2 + γi ∂ni y ∂ni y ∂ni y 





M7 (y, y ) 

M8 (y, y ) 

Mγi (y, y ) 

Mγ −1 (y, y ) i

∗ (0)

K

γi−1

= = = =

∞

υ=0 ∞

υ=0 ∞

υ=0 ∞

υ=0

υ





Mγ∗−1 (y, y ) i

= =

∞

υ=0 ∞



υ ∗ (υ)



(−1) K γi (y , y), ∗ (υ)



K γi−1 (y , y),

(y , y) = 2 1 +

Mγ∗∗ (y, y ) i

(υ)

(−1) K4 (y , y),

%





(υ)

K3 (y , y),

γi−1

∗ (υ)

& ∂  0 (y, y  ; η), G ∂ni y 

K γi (y , y), υ ∗ (υ)



(−1) K γ −1 (y , y). i

υ=0

In these formulae, we note for example that K1 (y , y  ) is the iterates of On the basis of (4.3), the function χ (x1 , x 2 ; η) will be estimated for |η| → ∞ together with small / large impedances γi . The case when x1 and x 2 lie in the neighborhood of the components Si∗ (i = 1, ..., m) of the inner bounding surface S1 or in the neighborhood of the components Si∗ (i = m + 1, ..., n) of the outer bounding surface S2 is particularly interesting. In what follows, we shall use methods similar to that obtained in Pleijel [6], Abdel-Halim [1], Zayed et al [8-12] and Zayed [13, 15, 16, 19] to examine this case. It now follows that the local expansions of the functions : (ν)

(0) K1 (y , y  ).

 0 ( x, y; η), G

∂  G 0 ( x, y; η), ∂niy

(4.4)

when the distance between x and y is small, are very similar to that obtained in [11, 12, 19]. Consequently, the local expansions of the kernels: K1 (y , y  ) , K2 (y , y  ), K3 (y , y  ) , K4 (y , y  ), (0)

(0)

(0)

(0)

(4.5)

when the distance between y and y  is small, follow directly from knowledge of the local expansions of (4.4). Similarly, when the distance between y and y  is

An inverse problem of the three-dimensional wave equation

101

small and for small / large γi , the local expansions of the kernels: ∗ (0)



∗ (0)



Kγ(0) (y , y  ) , K γi (y , y), Kγ −1 (y , y  ) , K γ −1 (y , y), i (0) i

i

(4.6)

follow directly from knowledge of the local expansions of (4.4). 3 4.1. If ξ1 and ξ2 are points in the upper half-plane ξ > 0 of the Definition 1 2 3 ξ , ξ , ξ -space, then we define * ρ12 = (ξ11 − ξ21 )2 + (ξ12 − ξ22 )2 + (ξ13 + ξ23 )2 .

An e λ (ξ1 , ξ2 ; η)-function is defined for points ξ1 and ξ2 belonging to sufficiently small domains D(Ii ), (i = 1, ..., n) except when ξ1 = ξ2 ∈ Ii , and λ is called the degree of this function. For every positive integer ∧ , it has the following local expansions (see [ 11, 12, 19]) :  l1  l2  l3

∂ ∂ ∂ ∗ e λ (ξ1 , ξ2 ; η) = f (ξ11 , ξ12 )(ξ13 )p1 (ξ23 )p2 × ∂ξ11 ∂ξ12 ∂ξ13  0 ( x1 , x2 ; η)]r + R∧ (ξ1 , ξ2 ; η), (4.7) [G x1 x2 =ρ12 ∗ where denotes a sum of a finite number of terms in which f (ξ11 , ξ12 ) is an infinitely differentiable function. In this expansion p1 , p2 , l1 , l2 , l3 are integers, where p1 ≥ 0 , p2 ≥ 0 , l1 ≥ 0 , l2 ≥ 0 , λ = min (p1 + p2 − q) , q = l1 +l 2 +l3 and ∗ the minimum is taken over all terms which occur in the summation . The ∧ remainder R (ξ1 , ξ2 ; η) has continuous derivatives of all orders d ≤ ∧ satisfying Dd R∧ (ξ1 , ξ2 ; η) = O{η −∧ exp(−Aη i ρ12 )} as |η| → ∞ ,

(4.8)

where A is a positive constant . Thus, using methods similar to that obtained in [11, 12, 19], we can show that the functions (4.4) are e λ -functions with degrees λ = −1, −2 respectively. Consequently, the functions (4.5) are e λ -functions with degrees λ = 0, 0, −1, 1 respectively , while for small / large impedances γi the functions (4.6) are e λ functions with degrees λ = 0, −1, 0, 1, respectively. Definition 4.2. If x1 and x 2 are points in large domains Ω + Si∗ , then we define ri = min(rx1 y + rx2 y ) y

if y ∈ Si∗ (i = 1, ..., n).

(4.9)

An E λ (x1 , x 2 ; η)-function is defined and infinitely differentiable with respect to x1 and x2 when these points belong to large domains Ω + Si∗ except when x1 = x2 ∈ Si∗ (i = 1, ..., n). Thus, the E λ -function has a similar local expansion of the e λ - function (see [11, 12, 19]).

102

E. M. E. Zayed

With the help of the articles [11, 12, 19], it is easily seen that (4.3) is an E −2 (x1 , x 2 ; η)- function and consequently, we obtain  ( x1 , x2 ; η) = G

n

  O ri−2 exp (−Ai η i ri )

(4.10)

i=1

which is valid for |η| → ∞ and for small / large impedances γi , where Ai (i =  ( x1 , x2 ; η) is 1, ..., n) are positive constants. The estimate (4.10) shows that G exponentially small for |η| → ∞ . This proves that the integral (3.8) converges for |η| → ∞ . Following the articles [11, 12, 19] , if the e λ -expansions of the functions (4.4) -(4.6) are introduced into (4.3) and if we use formulae similar to (7.4) and (7.10) of Sec.7 in Zayed et al [12], we obtain the following local behavior of χ  (x1 , x 2 ; η) when ri are small , which is valid for |η| → ∞ and for small / large impedances γi : χ  (x1 , x 2 ; η) =

n

χ  i (x1 , x 2 ; η),

(4.11)

i=1

where (a): if x1 and x2 belong to sufficiently small domains D(Ii ) (i = 1, ..., k), then χ i (x1 , x2 ; η) =

, 1 + G 0 ( x1 , x2 ; η) 2 rx1 x2 =ρ12 +O{ρ−1 exp(−Ai η i ρ12 )}, as |η| → ∞. 12

(4.12)

(b): if x1 and x2 belong to sufficiently small domains D(Ii ) (i = k + 1, ..., ), then χ i (x1 , x2 ; η) =

−

, 1 + G 0 ( x1 , x2 ; η) 2 rx1 x2 =ρ12

+O{ρ−1 exp(−Ai η i ρ12 )}, as |η| → ∞. 12

(4.13)

(c): if x1 and x2 belong to sufficiently small domains D(Ii ) (i =  + 1, ..., b), then

χ i (x1 , x2 ; η)

=

1 − 2



 1 − γi

∂ ∂ξ13

−1 ) + ,  0 ( x1 , x2 ; η) G

+O{ρ−1 exp(−Ai η i ρ12 )}, as |η| → ∞. 12

rx1 x2 =ρ12

(4.14)

(d): if x1 and x2 belong to sufficiently small domains D(Ii ) (i = b + 1, ..., m), then

An inverse problem of the three-dimensional wave equation

χ i (x1 , x2 ; η) =

1 2

1−

γi−1

+O{ρ−1 12



∂ ∂ξ13

. +

,  0 ( x1 , x2 ; η) G

103

rx1 x2 =ρ12

exp(−Ai η i ρ12 )}, as |η| → ∞.

(4.15)

(e): if x1 and x2 belong to sufficiently small domains D(Ii ) (i = m + 1, ..., N ), then χ i (x1 , x2 ; η) =

, 1 + G 0 ( x1 , x2 ; η) 2 rx1 x2 =ρ12

−

+O{ρ−1 exp(−Ai η i ρ12 )}, as |η| → ∞. 12

(4.16)

(f): if x1 and x2 belong to sufficiently small domains D(Ii ) (i = N + 1, ..., p), then , 1 + G 0 ( x1 , x2 ; η) 2 rx1 x2 =ρ12

χ i (x1 , x2 ; η) =

+O{ρ−1 exp(−Ai η i ρ12 )}, as |η| → ∞. 12

(4.17)

(g): if x1 and x2 belong to sufficiently small domains D(Ii ) (i = p + 1, ..., c), then

χ i (x1 , x2 ; η) =

1 2



 1 − γi

∂ ∂ξ13

−1 ) +

,  0 ( x1 , x2 ; η) G

rx1 x2 =ρ12

+O{ρ−1 exp(−Ai η i ρ12 )}, as |η| → ∞. 12

(4.18)

(h): if x1 and x2 belong to sufficiently small domains D(Ii ) (i = c + 1, ..., n), then χ i (x1 , x2 ; η) =

1 − 2

1−

γi−1



∂ ∂ξ13

+O{ρ−1 exp(−Ai 12

. + ,  0 ( x1 , x2 ; η) G

rx1 x2 =ρ12

η i ρ12 )}, as |η| → ∞.

(4.19)

When ri ≥ δi > 0 (i = 1, ..., n), the functions χ i (x1 , x2 ; η) are of order O {exp (−B i η)} as |η| → ∞ where B is a positive constant. Thus, since limri →0 ρr12i = 1 (see [11, 12, 19]) , then we have the asymptotic formulae (4.12)(4.19) with ρ12 in the small domains D (Ii ) (i = 1, ..., n) being replaced by ri in the large domains Ω + Si∗ (i, ..., n) respectively. With reference to the articles [1, 10, 12, 19], it can be seen for ξ 3 ≥ hi > 0, (i = 1, ..., n) that the functions χ i (x, x; η) are of order O{exp(−4η i Ai hi )} and the integral of the function χ  (x, x; η) over the annular region Ω ⊆ R3 can be

104

E. M. E. Zayed

approximated in the following way (see(3.10)):   hi n

   K(η) = χ i (x, x; η) 1 − 2ξ 3 H1 + (ξ 3 )2 N1 dξ 3 dSi∗ i=m+1 S ∗

ξ 3 =0

i

−

m  

i=1 S ∗ i

+

n

hi ξ 3 =0

  χ i (x, x; η) 1 + 2ξ 3 H1∗ + (ξ 3 )2 N1∗ dξ 3 dSi∗

O{exp(−4η i Ai hi )} as

|η| → ∞ .

(4.20)

i=1

If the eλ -expansions of χ i (x, x; η) (see [12], [19]) are introduced into (4.20) and with the help of the formula (7.2) of Sec.7 in [19] (see also [12]), we obtain i a4 i a3  K(η) = − a2 sign η − − 2 +O 2 πη πη



1 η3

as |η| → ∞. (4.21)

On inverting Fourier transforms, we obtain K(t) =

 a2 sign t + a3 sign t + a4 t sign t + O t2 sign t , |t|

as |t| → 0, (4.22)

where the cofficiention a2 , a3 and a4 are given explicity in Section 2. From (3.6) and (4.22) we arrive at the complete proof of Theorem 2.1.

Acknowledgment The author wishes to thank the referee for his comments and suggestions.

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An inverse problem of the three-dimensional wave equation

105

7. R.T. Waechter, On hearing the shape of a drum : An extension to higher dimensions, Proc. Camb. Philos. Soc., 72 (1972) 439 - 447. 8. E.M.E. Zayed and I.H. Abdel-Halim,Short-time asymptotics of the trace of the wave operator for a general annular drum in R2 with Robin boundary conditions, Indian J. Pure Appl. Math., 32 (2001) 493-500. 9. E.M.E. Zayed and I.H. Abdel-Halim,The wave equation approach to an inverse eigenvalue problem for an arbitrary multiply connected drum in R2 with Robin boundary conditions, Int. J. Math. Math. Sci. 25 (2001) 717–726. 10. E.M.E. Zayed and I.H. Abdel-Halim, An inverse problem of the wave equation for a general annular drum in R3 with Robin boundary conditions, Chaos, & Soliton and Fractals,12 (2001) 2259-2266. 11. E.M.E. Zayed and I.H. Abdel-Halim, The 3D inverse problem of the waves with fractals for a general annular bounded domain with Robin boundary, Chaos, & Soliton and Fractals, 12 (2001) 2307-2321. 12. E.M.E. Zayed, M.A. Kishta and A.A.M. Hassan,The wave equation approach to an inverse problem for a general convex domain: an extension to higher dimensions, Bull. Calcutta Math. Soc. 82 (1990) 457–474. 13. E.M.E. Zayed,Short-time asymptotics of the spectral distribution of the wave equation in R3 for a multiply connected domain with Robin boundary conditions, Bull. Greek Math. Soc., 41 (1999) 139-153. 14. E.M.E. Zayed,An inverse eigenvalue problem for a general convex domain: an extension to higher dimensions, J. Math. Anal. Appl., 112 (1985) 455–470. 15. E.M.E. Zayed, The wave equation approach to inverse problems: an extension to higher dimensions, Bull. Calcutta Math. Soc., 78 (1986) 281–291. 16. E.M.E. Zayed,The wave equation approach to Robin inverse problems for a doublyconnected region: an extension to higher dimensions, J. Comput. Math., 7 (1989) 301–312. 17. E.M.E. Zayed,Hearing the shape of a general doubly-connected domain in R3 with impedance boundary conditions, J. Math. Phys., 31 (1990) 2361 - 2365. 18. E.M.E. Zayed,Hearing the shape of a general doubly-connected domain in R3 with mixed boundary conditions, J. Appl. Math. Phys (ZAMP), 42 (1991) 547-564 . 19. E.M.E. Zayed, The wave equation approach to an inverse problem for a general convex domain in R3 with a finite number of piecewise impedance boundary conditions, Bull. Calcutta Math. Soc., 85 (1993) 237–248. 20. E.M.E. Zayed,An inverse problem for a general doubly- connected bounded domain in R3 with a finite number of piecewise impedance boundary conditions, Appl. Anal., 64 (1997) 69-98. 21. E.M.E. Zayed, An inverse problem for a general multiply connected bounded domain: an extension to higher dimensions, Appl. Anal., 72 (1999) 27–41. 22. E.M.E. Zayed, An inverse problem for a general bounded domain in R3 with piecewise smooth mixed boundary conditions, Int. J. Theore. Phys., 39 (2000) 189-205. E.M.E. Zayed received his BSC from Tanta University (Egypt), in 1973, and received his first MSC from Al-Azhar University (Egypt) in 1977, and his second MSC from Dundee University (U.K) in 1978. He received his Ph.D at Dundee University (U.K) in 1981, under the direction of Professor B.D. Sleeman. From 1977-1981 he was a postgraduate student at the University of Dundee.Since 1989 he has been a full Professor of Mathematics at Zagazig University (Egypt). His Research interests focus on inverse problems in differential equations. He published more than 94 papers in this field in high level standard mathematical Journals around the world. Since 1993 he has been a reviewer for the Mathematical Reviews (USA) and has reviewed more than 40 papers. Also, he does mathematical consulting Department of Mathematics, Faculty of Science, Zagazig University, Zagazig, Egypt. e-mail: [email protected]