Obstacle scattering

Let $D \subset \mathbf{R} ^ { 3 }$ be a bounded domain with boundary $S$. A large amount of literature, going back to the mid-1930s, deals with wave scattering by obstacles when $S$ is smooth, for example, a $C^{ 1 , \lambda }$-surface, $0 < \lambda \leq 1$. (See [a3], [a17], [a2].)

The obstacle scattering problem consists of finding the solution to the equation

\begin{equation} \tag{a1} ( \nabla ^ { 2 } + k ^ { 2 } ) u = 0 \text { in } D ^ { \prime } : = \mathbf{R} ^ { 3 } \backslash D , k > 0, \end{equation}

\begin{equation} \tag{a2} \Gamma u = 0 \text { on } S, \end{equation}

\begin{equation} \tag{a3} u = e ^ { i k \alpha x } + v , \operatorname { lim } _ { r \rightarrow \infty } \int _ { | s | = r } \left| \frac { \partial v } { \partial | x | } - i k v \right| ^ { 2 } d s = 0. \end{equation}

Here $\Gamma u = u$ ( the Dirichlet condition), or $\Gamma u = u _ { N }$ (the Neumann condition) or $\Gamma u = u _ { N } + h ( s ) u$ (the Robin condition), where $N$ is the unit normal to $S$ pointing into $D ^ { \prime }$ and $h ( s )$ is a continuous function (cf. also Dirichlet boundary conditions; Neumann boundary conditions). Condition (a3) is the radiation condition, which selects a unique solution to problem (a1)–(a3). In (a3), $\alpha \in S ^ { 2 }$ is a given unit vector, the direction of the incident plane wave $e ^ { i k \alpha x}$, and $k > 0$ is the wave number.

The scattering problem (a1)–(a3) has a solution and the solution is unique. This basic result was proved originally by the integral equations method [a3].

There are many different types of integral equations which allow one to study problem (a1)–(a3) (see [a17], where most of these equations are derived).

The scattering field $v$ in (a3) has the following asymptotics:

\begin{equation} \tag{a4} v ( x , \alpha , k ) = \frac { e ^ { i k r } } { r } A ( \alpha ^ { \prime } , \alpha , k ) + o \left( \frac { 1 } { r } \right), \end{equation}

\begin{equation*} r : = | x | \rightarrow \infty , \alpha ^ { \prime } : = \frac { x } { r }. \end{equation*}

The function $A ( \alpha ^ { \prime } , \alpha , k )$ is called the scattering amplitude. This function has the following properties [a17]:

i) realness: $A ( \alpha ^ { \prime } , \alpha , - k ) = \overline { A ( \alpha ^ { \prime } , \alpha , - k ) }$, $k > 0$, the bar stands for complex conjugation;

ii) reciprocity: $A ( \alpha ^ { \prime } , \alpha , k ) = A ( - \alpha , - \alpha ^ { \prime } , k )$;

iii) unitarity:

\begin{equation*} \frac { A ( \alpha ^ { \prime } , \alpha , k ) - \overline { A ( \alpha , \alpha ^ { \prime } , k ) } } { 2 i } = \end{equation*}

\begin{equation*} = \frac { k } { 4 \pi } \int _ { S ^ { 2 } } f ( \alpha ^ { \prime } , \beta , k ) \overline { f ( \alpha , \beta , k ) } d \beta , \end{equation*}

and its consequence, which is called the optical theorem:

\begin{equation*} \operatorname { Im } A ( \alpha , \alpha , k ) = \frac { k } { 4 \pi } \int _ { S ^ { 2 } } | f ( \alpha , \beta , k ) | ^ { 2 } d \beta : = \frac { k \sigma ( \alpha ) } { 4 \pi }, \end{equation*}

where $\sigma ( \alpha ) : = \int _ { S ^ { 2 } } | f ( \alpha , \beta , k ) | ^ { 2 } d \beta$ is called the cross-section.

The function $A ( \alpha ^ { \prime } , \alpha , k )$ is analytic with respect to $k$ in $\mathbf{C} _ { + } : = \{ k : \operatorname { Im } k \geq 0 \}$ and meromorphic in $\mathbf{C}$; it is analytic with respect to $\alpha ^ { \prime }$ and $\alpha$ on the variety $M : = \left\{ \theta : \theta \in \mathbf{C} ^ { 3 } , \theta \cdot \theta = 1 \right\}$, where $\theta . w : = \sum ^ { 3 _{ j = 1}} \theta _ { j } w _ { j }$, see [a17], [a18] (cf. also Analytic function; Meromorphic function).

Necessary and sufficient conditions for a scatterer to be spherically symmetric is: $A ( \alpha ^ { \prime } , \alpha , k ) = A ( \alpha ^ { \prime } . \alpha , k )$, where $\alpha ^ { \prime } . \alpha$ is the dot product [a18], [a8].

The solution $u ( x , \alpha , k )$ to (a1)–(a3) is called the scattering solution. Any $f ( x ) \in L ^ { 2 } ( D ^ { \prime } )$ can be expanded with respect to scattering solutions:

\begin{equation*} f ( x ) = \frac { 1 } { ( 2 \pi ) ^ { 3 / 2 } } \int _ { {\bf R} ^ { 3 } } \hat { f } ( \xi ) u ( x , \xi ) d \xi , \xi : = k\alpha, \end{equation*}

\begin{equation*} \widehat { f } ( \xi ) = \frac { 1 } { ( 2 \pi ) ^ { 3 / 2 } } \int _ { D ^ { \prime } } f ( x ) \overline { u ( x , \xi ) } d x : = \mathcal{F} f. \end{equation*}

The operator $\mathcal{F} : L ^ { 2 } ( D ^ { \prime } ) \rightarrow L ^ { 2 } ( \mathbf{R} ^ { 3 } )$ is unitary: $\| \mathcal{F} f \| _ { L } 2 (\mathbf{R} ^ { 3 )} = \| f \| _ { L ^ { 2 } ( D ^ { \prime } ) }$, $\mathcal{F} ^ { * } = \mathcal{F} ^ { - 1 }$, see [a17] (cf. also Unitary operator).

The above results hold in ${\bf R} ^ { n }$ with odd $n$. In ${\bf R} ^ { n }$ with even $n$ the scattering amplitude $A ( \alpha ^ { \prime } , \alpha , k )$ as a function of complex $k$ has a logarithmic branch point at $k = 0$.

The scattering problem with minimal assumptions on the smoothness of the boundary $S$ is studied in [a13]. If $\Gamma u = u$, then existence and uniqueness of the scattering solution have been proved without any assumption on the smoothness of the boundary $S$ of a bounded domain $D$. In this case a weak formulation of problem (a1)–(a3) is considered and the limit-absorption principle has been proved.

If $\Gamma u = u _ { N }$, then again a weak formulation of (a1)–(a3) is considered and the only assumption on the smoothness of the boundary $S$ is compactness of the embedding $H ^ { 1 } ( D _ { R } ^ { \prime } )$ into $L ^ { 2 } ( D _ { R } ^ { \prime } )$, where $D _ { R } ^ { \prime } : = D ^ { \prime } \cap B _ { R }$ and $B _ { R } = \{ x : | x | \leq R \}$ is a ball which contains $D$. Existence and uniqueness of the scattering solution have been proved and the limiting-absorption principle has been established. Finally, if $\Gamma u = u _ { N } + h u$, then the same results are obtained under the assumptions of compactness of the embeddings $i _ { 1 } : H ^ { 1 } ( D ^ { \prime R} ) \rightarrow L ^ { 2 } ( D _ { R } ^ { \prime } )$ and $i _ { 2 } : H ^ { 1 } ( D _ { R } ^ { \prime } ) \rightarrow L ^ { 2 } ( S )$, where $S$ is equipped with the $( n - 1 )$-dimensional Hausdorff measure and $H ^ { 1 } ( D _ { R } ^ { \prime } )$ is the Sobolev space.

For example, the embedding $i_1$ is compact for C-domains, that is, domains whose boundary can be covered by finitely many sets open in $\mathbf{R} ^ { 3 }$ and on each of these sets the equation of $S$ in a local coordinate system can be written as $x _ { 3 } = f ( x ^ { \prime } ) , x ^ { \prime } = ( x _ { 1 } , x _ { 2 } )$, where $f ( x ^ { \prime } )$ is a continuous function.

The scattering problem for one obstacle, small in comparison with the wavelength ($k a \ll 1$, where $a$ is the diameter of the small obstacle), for many such bodies (the number $J$ of bodies of order $20$), and in a medium consisting of many such bodies randomly distributed in the space, has been studied in [a7] and later in [a16]. Formulas for the scattering amplitude for acoustic and electromagnetic wave scattering by small bodies of arbitrary shapes are derived in [a16].

These formulas for acoustic wave scattering on a single small body, containing the origin, are

\begin{equation*} A ( \alpha ^ { \prime } , \alpha , k ) \equiv - \frac { C } { 4 \pi } , \text { if } \Gamma u = u , k a \ll 1, \end{equation*}

where $C$ is the electrical capacitance of the body $D$;

\begin{equation*} A ( \alpha ^ { \prime } , \alpha , k ) \approx - \frac { k ^ { 2 } V } { 4 \pi } ( 1 + \beta _ { p q } \alpha _ { q } \alpha _ { p } ^ { \prime } ) \text { if } \Gamma u = u _ { N } , k a \ll 1, \end{equation*}

where over the repeated indices one sums up from $1$ to $3$, $v$ is the volume of $D$ and $\beta _ { p q } = \beta _ { q p }$ is the magnetic polarizability tensor;

\begin{equation*} A ( \alpha ^ { \prime } , \alpha , k ) \approx - \frac { h | S | } { 4 \pi ( 1 + h | S | C ^ { - 1 } ) } \end{equation*}

\begin{equation*} \text{if} \ \Gamma u = u _ { N } + h u , k a \ll 1 , h =\text{const}, \end{equation*}

where $|S|$ is the area of the boundary $S$.

The $S$-matrix for electromagnetic wave scattering by a small homogeneous body of arbitrary shape is:

\begin{equation*} S = \frac { k ^ { 2 } V } { 4 \pi } \cdot \left( \begin{array} { c } { A B } \\ { C D } \end{array} \right), \end{equation*}

\begin{equation*} A = \mu _ { 0 } \beta _ { 11 } + \alpha _ { 22 } \operatorname { cos } \theta - \alpha _ { 32 } \operatorname { sin } \theta , B = \alpha _ { 21 } \operatorname { cos } \theta - \alpha _ { 31 } \operatorname { sin } \theta - \mu _ { 0 } \beta _ { 12 }, \end{equation*}

\begin{equation*} C = \alpha _ { 12 } - \mu _ { 0 } \beta _ { 21 } \operatorname { cos } \theta + \mu _ { 0 } \beta _ { 31 } \operatorname { sin } \theta , D = \alpha _ { 11 } + \mu _ { 0 } \beta _ { 22 } \operatorname { cos } \theta - \mu _ { 0 } \beta _ { 32 } \operatorname { sin } \theta, \end{equation*}

where $\beta _ { i j }$ and $\alpha _ { i j }$ are tensors of magnetic and electric polarizability and $\theta$ is the angle between the direction $e_3$ of the incident field and the direction of the scattered field; $\mu_0$ is the magnetic permeability of the medium in which the small body is embedded (cf. also Scattering matrix).

Formulas for the tensors $\alpha _ { i j }$ and $\beta _ { i j }$ are derived in [a7] and [a16].

One can derive an equation for the average field in the medium which consists of many small obstacles (particles) randomly distributed in a region. This done in [a16] and [a6].

Scattering by random surfaces is studied in [a1].

Inverse obstacle scattering.

Three inverse obstacle scattering problems are of interest:

ISP1) Given $A ( \alpha ^ { \prime } , \alpha_0 , k )$ for all $\alpha ^ { \prime } \in S ^ { 2 }$ and all $k > 0$, $\alpha _ { 0 } \in S ^ { 2 }$ being fixed, find $S$ and the boundary condition on $S$.

ISP2) Given $A ( \alpha ^ { \prime } , \alpha , k _ { 0 } )$ for all $\alpha ^ { \prime } , \alpha \in S ^ { 2 } , k _ { 0 } > 0$ being fixed, find $S$ and the boundary condition on $S$.

ISP3) Given $A ( \alpha ^ { \prime } , \alpha _ { 0 } , k _ { 0 } )$ for all $\alpha ^ { \prime } \in S ^ { 2 } , \alpha _ { 0 } \in S ^ { 2 }$ and $k _ { 0 } > 0$ being fixed, find $S$.

Uniqueness of the solution to ISP1) (for $\Gamma u = 0$) was first proved by M. Schiffer (1964), whose argument is given in [a17].

Uniqueness of the solution to ISP2) was first proved by A.G. Ramm (1985) and his proof is given in [a17].

A uniqueness theorem for ISP3) has not yet (2000) been proved: it is an open problem to prove (or disprove the existence of) such a theorem.

One can consider inverse obstacle scattering for penetrable obstacles [a14].

Schiffer's proof of the uniqueness theorem is based on a result saying that the spectrum of the Dirichlet Laplacian in any bounded domain is discrete. This result follows from the compactness of the embedding $i : \overline { H } ^ { 1 } ( D ) \rightarrow L ^ { 2 } ( D )$ for any bounded domain (without any assumptions on the smoothness of its boundary $S$), $\overline { H } ^ { 1 } ( D )$ is the Sobolev space which is the closure in $H ^ { 1 } ( D )$ of $C ^ { \infty_0 }(D)$. It is known [a5] that $i : H ^ { 1 } ( D ) \rightarrow L ^ { 2 } ( D )$ is not compact for rough domains (it is compact for Lipschitz domains, for domains satisfying the cone condition, for C-domains, and for E-domains, i.e. domains for which a bounded extension operator from $H ^ { 1 } ( D )$ into $H ^ { 1 } ( \mathbf{R} ^ { 3 } )$ exists, see [a5]).

Therefore the spectrum of a Neumann Laplacian in such a rough domain for which the imbedding $i : H ^ { 1 } ( D ) \rightarrow L ^ { 2 } ( D )$ is not compact is not discrete. One way out is given in [a12] and another one in [a11].

Suppose that $A _ { 1 } ( \alpha ^ { \prime } , \alpha , k _ { 0 } )$ and $A _ { 2 } ( \alpha ^ { \prime } , \alpha , k _ { 0 } )$ are scattering amplitudes at a fixed $k = k _ { 0 } > 0$ for two obstacles and let $\operatorname { sup } _ { \alpha ^ { \prime } , \alpha \in S ^ { 2 } } | A _ { 1 } - A _ { 2 } | < \delta$.

Assume that the boundaries of the two obstacles are $C^{ 2 , \lambda }$, $0 < \lambda \leq 1$, that is, in local coordinates these boundaries $S _ { m }$, $m = 1,2$, have equations $x _ { 3 } = f _ { m } ( x _ { 1 } , x _ { 2 } )$, where $f \in C ^ { 2 , \lambda }$, $m = 1,2$, $\|\, f _ { m } \| _ { C ^{ 2 , \lambda}} \leq c _ { 0 } = \text{const } > 0$.

Let $\rho$ denote the Hausdorff distance between $S _ { 1 }$ and $S _ { 2 }$: $\rho = \operatorname { sup } _ { x \in S _ { 1 } } \text { inf }_{ y \in S _ { 2 } } | x - y |$.

The basic stability result [a10] is:

\begin{equation*} \rho \leq c _ { 1 } \left( \frac { \operatorname { ln } | \operatorname { ln } \delta | } { | \operatorname { ln } \delta | } \right) ^ { c _ { 2 } }, \end{equation*}

where $c_1$ and $c_2$ are positive constants.

In [a10] a yet open problem (as of 2000) is formulated: Derive an inversion formula for finding $S$, given the data $A ( \alpha ^ { \prime } , \alpha ) : = A ( \alpha ^ { \prime } , \alpha , k _ { 0 } )$, $\forall \alpha ^ { \prime } , \alpha \in S ^ { 2 }$.

The existence of such a formula is proved in [a10]: if $\chi ( x ) : = \chi _ { D } ( x )$ is the characteristic function of $D$ and $\tilde { \chi } ( \xi )$ is its Fourier transform, then there exists a function $v _ { \varepsilon } ( \alpha , \theta ) \in L ^ { 2 } ( S ^ { 2 } )$ such that

\begin{equation*} \tilde { \chi } ( \xi ) = \frac { 8 \pi } { \xi ^ { 2 } } \operatorname { lim } _ { \varepsilon \downarrow 0 } \int _ { S ^ { 2 } } A ( \theta ^ { \prime } , \alpha ) v _ { \varepsilon } ( \alpha , \theta ) d \alpha, \end{equation*}

where $\theta , \theta ^ { \prime } \in M$, $\theta ^ { \prime } - \theta = \xi$, $\xi \in \mathbf{R} ^ { 3 }$ is an arbitrary vector. A formula for calculating $A ( \theta ^ { \prime } , \alpha )$, $\theta ^ { \prime } \in M$, given $A ( \alpha ^ { \prime } , \alpha )$, $\forall \alpha ^ { \prime } , \alpha \in S ^ { 2 }$, is derived in [a10]. The problem is to construct $v _ { \varepsilon } ( \alpha , \theta )$ from the data $A ( \alpha ^ { \prime } , \alpha )$ algorithmically. For inverse potential scattering this is done in [a15].

References

[a1]	F. Bass, I. Fuks, "Wave scattering from statistically rough surfaces" , Pergamon (1979)
[a2]	D. Colton, R. Kress, "Integral equations methods in scattering theory" , Wiley (1983)
[a3]	V. Kupradze, "Randwertaufgaben der Schwingungstheorie und Integralgleichungen" , DVW (1956)
[a4]	R. Leis, "Initial boundary value problems in mathematical physics" , New York (1986)
[a5]	V. Maz'ja, "Sobolev spaces" , Springer (1985)
[a6]	V. Marchenko, E. Khruslov, "Boundary value problems in domains with granulated boundary" , Nauk. Dumka, Kiev (1974) (In Russian)
[a7]	A.G. Ramm, "Theory and applications of some new classes of integral equations" , Springer (1980)
[a8]	A.G. Ramm, "Necessary and sufficient condition for a scattering amplitude to correspond to a spherically symmetric scatterer" Appl. Math. Lett. , 2 (1989) pp. 263–265
[a9]	A.G. Ramm, "Stability estimates for obstacle scattering" J. Math. Anal. Appl. , 188 : 3 (1994) pp. 743–751
[a10]	A.G. Ramm, "Stability of the solution to inverse obstacle scattering problem" J. Inverse Ill-Posed Probl. , 2 : 3 (1994) pp. 269–275
[a11]	A.G. Ramm, "New method for proving uniqueness theorems for obstacle inverse scattering problems" Appl. Math. Lett. , 6 : 6 (1993) pp. 19–22
[a12]	A.G. Ramm, "Uniqueness theorems for inverse obstacle scattering problems in Lipschitz domains" Applic. Anal. , 59 (1995) pp. 377–383
[a13]	A.G. Ramm, M. Sammartino, "Existence and uniqueness of the scattering solutions in the exterior of rough domains" A.G. Ramm (ed.) P.N. Shivakumar (ed.) A.V. Strauss (ed.) , Operator Theory and Its Applications , Fields Inst. Commun. , 25 , Amer. Math. Soc. (2000) pp. 457–472
[a14]	A.G. Ramm, P. Pang, G. Yan, "A uniqueness result for the inverse transmission problem" Internat. J. Appl. Math. , 2 : 5 (2000) pp. 625–634
[a15]	A.G. Ramm, "Stability estimates in inverse scattering" Acta Applic. Math. , 28 : 1 (1992) pp. 1–42
[a16]	A.G. Ramm, "Iterative methods for calculating the static fields and wave scattering by small bodies" , Springer (1982)
[a17]	A.G. Ramm, "Scattering by obstacles" , Reidel (1986)
[a18]	A.G. Ramm, "Multidimensional inverse scattering problems" , Longman/Wiley (1992)
[a19]	A.G. Ramm, "Spectral properties of the Schroedinger operator in some domains with infinite boundaries" Soviet Math. Dokl. , 152 (1963) pp. 282–285
[a20]	A.G. Ramm, "Reconstruction of the domain shape from the scattering amplitude" Radiotech. i Electron. , 11 (1965) pp. 2068–2070
[a21]	A.G. Ramm, "Approximate formulas for tensor polarizability and capacitance of bodies of arbitrary shape and its applications" Soviet Phys. Dokl. , 195 (1970) pp. 1303–1306
[a22]	A.G. Ramm, "Electromagnetic wave scattering by small bodies of an arbitrary shape" V. Varadan (ed.) , Acoustic, Electromagnetic and Elastic Scattering: Focus on T-Matrix Approach , Pergamon (1980) pp. 537–546
[a23]	A.G. Ramm, "On inverse diffraction problem" J. Math. Anal. Appl. , 103 (1984) pp. 139–147
[a24]	F. Ursell, "On the exterior problems of acoustics" Proc. Cambridge Philos. Soc. , 74 (1973) pp. 117–125 (See also: 84 (1978), 545-548)
[a25]	I. Vekua, "Metaharmonic functions" Trudy Tbil. Math. Inst. , 12 (1943) pp. 105–174 (In Russian)