Scattering of acoustic waves on a planar screen of arbitrary shape: Direct and inverse problems

International Journal of Engineering Science 92 (2015) 28–46

Contents lists available at ScienceDirect

International Journal of Engineering Science journal homepage: www.elsevier.com/locate/ijengsci

Scattering of acoustic waves on a planar screen of arbitrary shape: Direct and inverse problems S. Kanaun Technological Institute of Higher Education of Monterrey, State Mexico Campus, 52296 Mexico, Mexico

a r t i c l e

i n f o

Article history: Received 14 February 2015 Received in revised form 25 February 2015 Accepted 18 March 2015 Available online 11 April 2015 Keywords: Acoustic wave scattering problem Integral equations of scattering problems Gaussian approximating functions Inverse scattering problem

a b s t r a c t Scattering of plane monochromatic acoustic waves on a planar screen of arbitrary shape is considered (direct problem). The 2D-integral equation for the pressure jump on the screen is discretized by Gaussian approximating functions. For such functions, the elements of the matrix of the discretized problem take the form of a standard one-dimensional integral that can be tabulated. For regular grids of approximating nodes, the matrix of the discretized problem has the Toeplitz structure, and the corresponding matrix–vector products can be calculated by the Fast Fourier Transform technique. The latter strongly accelerates the process of iterative solution. Examples for an elliptic screen subjected to incident fields with various wave vectors are presented. The problem of reconstruction of the screen shape from the experimentally measured amplitude of the far field scattered on the screen (inverse problem) is discussed. Screens which boundaries are defined by a finite number of scalar parameters are considered. Solution of the inverse problem is reduced to minimization of functions that characterize deviation of experimental and theoretical amplitudes of the far field scattered on a screen. Local and global minima of these functions with respect to the screen shape parameters are analyzed. Optimal frequencies for efficient solution of the inverse problem are identified. Ó 2015 Elsevier Ltd. All rights reserved.

1. Introduction The problem of acoustic wave scattering on screens has important applications in hydro-acoustics. This problem is first reduced to a 2D-integral equation for the pressure jump on the screen surface (see, e.g., Colton & Kress, 1987), and the boundary element method is used for its numerical solution (Shaw, 1979). In this method, the integral equation is discretized by division of the screen surface into a set of small subregions (boundary elements), and inside each element, the solution is approximated by standard functions (e.g., polynomial splines) with unknown coefficients. Using the method of moments or the collocation method the problem is reduced to a finite system of linear algebraic equations for these coefficients (discretized problem). Its matrix is non-sparse, and the matrix terms are integrals over the boundary elements. For high frequency of the incident field, this matrix is large, and only iterative methods are efficient. As a result, timeconsuming operation of the matrix–vector product should be performed at every step of the iteration process. An efficient method of solving the integral equations of the scattering problems was proposed in Kanaun and Levin (2013) and Kanaun (2014). In this method, discretization of the integral equations is carried out using a set of identical approximating functions centered at the nodes of a regular grid. As the result, the matrix of the discretized problem has Toeplitz’s

E-mail address: [email protected] http://dx.doi.org/10.1016/j.ijengsci.2015.03.004 0020-7225/Ó 2015 Elsevier Ltd. All rights reserved.

S. Kanaun / International Journal of Engineering Science 92 (2015) 28–46

29

properties, and the corresponding matrix–vector products can be calculated by the Fast Fourier Transform (FFT) technique. The latter accelerates substantially the process of the iterative solution of the discretized problems. In the work of Kanaun (2014), Gaussian approximating functions were used for discretization of the integral equation of elastic wave scattering on a planar crack. The theory of approximation by Gaussian functions was developed by Maz’ya and Schmidt (2007). In the present work, this method is applied to the problem of acoustic wave scattering on a planar screen of arbitrary shape. This allows substantial reduction in time and memory requirements in comparison with conventional numerical methods. The inverse scattering problem – determination of the screen shape from the data of the far scattered field – can be also successfully solved by the method. In the inverse problem, attention is focused on properties of the functionals that characterize deviation of the predicted and experimental amplitude of the far field scattered on the screen (see, e.g., Colton & Kress, 1998). Screens which boundaries are defined by a finite number of scalar parameters are considered. The corresponding functionals become functions of these parameters, and the problem is reduced to seeking minimum of these functions. It is shown that these functions can have a number of local minima. As a result, numerical methods of seeking the minimum can converge to a local but not global minimum depending on the initial guess. The number and positions of the local minima depend on frequency of the incident field. It is shown that the inverse problem can be solved successfully if the wave number a of the incident field satisfies the condition aL ¼ Oð50Þ, where L is the characteristic size of the screen. In this case, the number of minima is reduced to one global minimum. The structure of the paper is as follows. In Section 2, the integral equation of the scattering problem for screens is discussed. In Section 3, approximation by Gaussian functions is considered, and in Section 4, the integral equation of the scattering problem is discretized by Gaussian approximating functions. In Section 5, the far scattered field, differential and total cross-sections of planar screens are considered. Examples of the numerical solutions of the scattering problems for an elliptic screen are presented in Section 6. The inverse problem is considered in Section 7. 2. Integral equations of the scattering problem for a screen Consider an infinite liquid medium that contains a planar screen with surface X bounded by the contour C (Fig. 1). Let a plane monochromatic incident pressure wave P 0 ðx; tÞ

P0 ðx; tÞ ¼ p0 ðxÞeixt ;

0 p0 ðxÞ ¼ aeiaðn xÞ

ð1Þ

propagate in the medium and be scattered on the screen. Here x is frequency, t is time, a ¼ x=c is the wave number of the incident field, c is the wave velocity; n0 is its wave normal, and n0  x is the scalar product of the vectors n0 and x (x is the vector of a point x in the 3D-medium). The pressure Pðx; tÞ in the medium with the screen has the form

Pðx; tÞ ¼ pðxÞeixt ;

ð2Þ

where the amplitude pðxÞ satisfies Helmgholtz equation (Pierce, 1981)

Dp þ a2 p ¼ 0;

ð3Þ

(D is the 3D-Laplace operator) and the boundary condition on the screen surface X is

  @p @p  ¼ ni ¼ 0;  @n X @xi X

ð4Þ

Fig. 1. Planar screen subjected to an incident wave with the wave normal n0 .

30

S. Kanaun / International Journal of Engineering Science 92 (2015) 28–46

where ni is the normal to the screen. The solution of this problem can be given in the form of the potential of the double layer concentrated on X (see, for example, Colton & Kress, 1987)

pðxÞ ¼ p0 ðxÞ 

Z

@ g ðx  x0 Þni ðx0 Þbðx0 ÞdX0 ; @xi

X

ð5Þ

where gðxÞ is the Green function of the Helmgholtz operator

g ðxÞ ¼

eiajxj : 4pjxj

ð6Þ

For an arbitrary density bðxÞ, the potential (5) satisfies the Helmgholtz equation (3) everywhere outside X and has the jump bðxÞ on X:

h i    pðxÞ ¼ pþ ðxÞ  p ðxÞ  ¼ bðxÞ: X

ð7Þ

X

Here pþ ðxÞ is the limit value of the pressure on X on the side of the positive direction of the normal ni , and p ðxÞ is its value on the opposite side. The normal derivative of the potential (5) is continuous on X

   þ  @pðxÞ @p ðxÞ @p ðxÞ  ¼   ¼ 0: @n X @n @n X

ð8Þ

Substituting Eq. (5) into the boundary condition (4) we obtain the equation for the function bðxÞ:

Z

Tðx  x0 Þbðx0 ÞdX0 ¼

X

@p0 ðxÞ ; @n

x 2 X;

ð9Þ

The kernel TðxÞ ¼ Tðx1 ; x2 ; x3 Þ of the integral operator in this equation is

Tðx1 ; x2 ; x3 Þ ¼ ni

@ 2 gðx1 ; x2 ; x3 Þ nj þ dðx1 ; x2 ; x3 Þ: @xi @xj

ð10Þ

Here dðx1 ; x2 ; x3 Þ is 3D-Dirac delta-function that eliminates a singular component of the derivative @pðxÞ=@n concentrated on

X. This singularity is generated by differentiating the discontinuous function pðxÞ in Eq. (5). Eq. (9) belongs to the class of pseudo-differential equations; existence and uniqueness of their solution is proved in Eskin (1981) and Il’inskiy and Smirnov (1994). Let /ðx1 ; x2 Þ be a smooth function in the plane x1 ; x2 , and its Fourier transform

/ ðk1 ; k2 Þ ¼

Z

1

Z

/ðx1 ; x2 Þ exp½iðk1 x1 þ k2 x2 Þdx1 dx2

ð11Þ

1 2

2

be smooth and tend to zero at infinity faster than jk1 þ k2 j3=2 . Using the convolution property, action of the integral operator with the kernel Tðx1 ; x2 ; 0Þ on such a function can be calculated as follows

Z

1

Z 1

0

Z

1

0

Tðx1  x01 ; x2  x02 ; 0Þ/ðx01 ; x02 Þdx1 dx2 ¼

ð2pÞ2

1

Z

T  ðk1 ; k2 Þ/ ðk1 ; k2 Þ exp½iðk1 x1 þ k1 x1 Þdk1 dk2 :

ð12Þ

1

Here T  ðk1 ; k2 Þ is the Fourier transform of the kernel Tðx1 ; x2 ; 0Þ with respect to variables ðx1 ; x2 Þ. The explicit form of T  ðk1 ; k2 Þ is (see details in the text to follow)

T  ðk1 ; k2 Þ ¼

1 2

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 k1 þ k2  a2 :

ð13Þ

Note that for the considered function /ðx1 ; x2 Þ, the integral (12) converges absolutely. 3. Gaussian approximating functions For numerical solution of Eq. (9), the latter should be firstly discretized using an appropriate class of approximating functions. In this work, Gaussian functions are used for this purpose. According to Maz’ya and Schmidt (2007), a bounded smooth function uðxÞ defined in d-dimensional space can be presented in the form of the following series:

  X uðxÞ  uh ðxÞ ¼ uðrÞ u x  xðrÞ ; r

uðxÞ ¼

1 ðpHÞd=2

exp 

jxj2 2

Hh

! :

ð14Þ

Here xðrÞ ðr ¼ 1; 2; . . .Þ are the nodes of a regular node grid, h is the grid step, uðrÞ ¼ uðxðrÞ Þ is the value of function uðxÞ at the node xðrÞ , and H is a dimensionless parameter of the order 1. In Maz’ya and Schmidt (2007), Eq. (14) is called the ‘‘approximate approximation’’ because its error does not vanish when h ! 0. But the non-vanishing part of the error (the ‘‘saturation error’’) has the order of expðp2 HÞ and may be neglected in calculations.

S. Kanaun / International Journal of Engineering Science 92 (2015) 28–46

31

For the numerical solution of the integral equation (9), the screen region X is placed inside a rectangle X with sizes 2L1  2L2 (Fig. 2)

X : fL1 6 x1 6 L1 ; L2 6 x2 6 L2 g

ð15Þ

and covered by a square node grid with the step h. For such a grid, the coordinates of the nodes in the two-index numeration are

  ðiÞ ðjÞ xði;jÞ ¼ x1 ; x2 ; ðiÞ x1

¼ L1 þ hði  1Þ;

ð16Þ ðjÞ x2

¼ L2 þ hðj  1Þ;

ð17Þ

where

1 6 i 6 N1 ; N1 ¼

1 6 j 6 N2 ;

2L1 þ 1; h

N2 ¼

2L2 þ 1: h

ð18Þ

The one-index numerations of the nodes is introduced by the equation

rði; jÞ ¼ i þ N 1 ðj  1Þ;

1 6 r 6 N; N ¼ N1 N 2 :

ð19Þ

Here N is the total number of the nodes in X. The indices i; j of the two-index numeration are expressed in terms of the index r of the one-index numeration by the equations

  r1 þ 1; jðrÞ ¼ Floor N1 iðrÞ ¼ Mod½r  1; N1 ðjðrÞ  1Þ þ 1;

if r > N1 ;

iðrÞ ¼ r if r 6 N1 :

ð20Þ

where Floor½z is the closest integer to the number z that is smaller than or equal to z, and Mod½a; b is the remainder on divi  ðiðrÞÞ ðjðrÞÞ , where r is the sion of real numbers a by b. Thus, according to Eq. (19), Cartesian coordinates of the node xðrÞ are x1 ; x2 node number in the one-index numeration. Finally, in the region X, an arbitrary function bðxÞ can be approximated by the series

bðxÞ 

N X ðrÞ b uðx  xðrÞ Þ; r¼1

ðrÞ

where b

uðxÞ ¼

! 1 jxj2 exp  2 ; pH Hh

are coefficients of the approximation that coincide with the values of the function bðxÞ at the nodes xðrÞ .

Fig. 2. Rectangular region and the node grid for numerical solution of the scattering problem on a screen.

ð21Þ

32

S. Kanaun / International Journal of Engineering Science 92 (2015) 28–46

4. Numerical solution of the integral equation (9) for a planar screen 4.1. Discretization of the integral equation (9) We consider a planar screen that occupies an arbitrary finite region X in the plane (x1 ; x2 ), and the normal n to the screen is directed along the x3 -axis. Because bðxÞ ¼ 0 if x R X, integration in Eq. (9) can be spread over any convenient region X that includes X. Thus, one can take a rectangular X with the screen inside and cover it by a square node grid xðrÞ ; r ¼ 1; 2; . . . ; N (Fig. 2). After approximating the functions bðxÞ by the series (21) and substituting the latter into Eq. (9) we obtain N X ðrÞ Iðx  xðrÞ Þb ¼ t 0 ðxÞ if x 2 X;

ðrÞ

b

¼ 0 if xðrÞ R X;

ð22Þ

r¼1

where IðxÞ ¼ Iðx1 ; x2 Þ; t0 ðxÞ ¼ t0 ðx1 ; x2 Þ, and

Iðx1 ; x2 Þ ¼

Z

Z

1

0

1

0

Tðx1  x01 ; x2  x02 ; 0Þuðx01 ; x02 Þdx1 dx2 ;

t 0 ðx1 ; x2 Þ ¼

 @p0 ðx1 ; x2 ; x3 Þ :  @x3 x3 ¼0

ð23Þ

ðrÞ

The system of linear algebraic equations for the unknowns b in Eqs. (21) follows from Eq. (22) if the latter is satisfied at all the nodes xðrÞ that belong to X (the collocation method). As a result, we obtain the following system of linear algebraic equaðrÞ

tions for b

N X ðrÞ Iðp;rÞ b ¼ t0ðpÞ if xðpÞ 2 X;

ðpÞ

b

¼ 0 if xðpÞ R X; p ¼ 1; 2; . . . ; N;

ð24Þ

r¼1

Iðp;rÞ ¼ IðxðpÞ  xðrÞ Þ;

t 0ðpÞ ¼ t0 ðxðpÞ Þ:

ð25Þ

4.2. Calculation of the integral Iðx1 ; x2 Þ Using the property of convolution the integral in Eq. (23) can be calculated as follows

Iij ðx1 ; x2 Þ ¼ ¼

Z

1

Z Z

1

Z

1 1

  0 0 0 Tðx1  x01 ; x2  x02 ; x3  x03 Þuðx01 ; x02 Þdðx03 Þdx1 dx2 dx3 

Z Z

ð2pÞ3



x3 ¼0



T ðk1 ; k2 ; k3 Þu ðk1 ; k2 Þ exp ½iðk1 x1 þ k2 x2 Þdk1 dk2 dk3 :

ð26Þ

1

Here integration is done over the entire 3D k-space of the Fourier transform parameter,

2

2

2

u ðk1 ; k2 Þ ¼ h exp 4 

Hh

 3 2 2 k1 þ k2 5: 4

ð27Þ

The function T  ðk1 ; k2 ; k3 Þ in Eq. (26) is the 3D-Fourier transform of the function Tðx1 ; x2 ; x3 Þ in Eq. (10) 2

T  ðk1 ; k2 ; k3 Þ ¼ 

2

k3 2 k1

2 k2

þ

þ

2 k3

 a2

þ1¼

2

k1 þ k2  a2 2 k1

þ

2 k2

2

þ k3  a2

:

ð28Þ

Integration over the k3 -variable in Eq. (26) yields

T  ðk1 ; k2 Þ ¼

1 2p

Z

1

T  ðk1 ; k2 ; k3 Þdk3 ¼

1

1 2

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 k1 þ k2  a2 :

ð29Þ

As a result, the integral (26) takes the form

Iðx1 ; x2 Þ ¼

Z

1

1

ð2pÞ2

Z

T  ðk1 ; k2 Þu ðk1 ; k2 Þ exp ½iðk1 x1 þ k2 x2 Þdk1 dk2 :

ð30Þ

1

After introducing polar coordinate system in the plane ðk1 ; k2 Þ and integrating over the polar angle we obtain

Iðx1 ; x2 Þ ¼

1 F ðq; qÞ; h

r h

q ¼ ; q ¼ ah; r ¼

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x21 þ x22 :

ð31Þ

The function F ðq; qÞ in this equation is the following 1D-integral

Fðq; qÞ ¼

1 4p

Z 0

1

  pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi j2 H J 0 ðjqÞjdj; j2  q2 exp  4

ð32Þ

S. Kanaun / International Journal of Engineering Science 92 (2015) 28–46

33

where J 0 ðjqÞ is the Bessel function. This absolutely converging integral can be calculated numerically for various parameters q and q and tabulated. For large values of the distance q, the function Fðq; qÞ has the following asymptotics

  1 þ iqq q2 H b : F ðq; qÞ ¼  exp iq q  4 4pq3

ð33Þ

F ðq; qÞ with sufficient accuracy. For q < 0:5 and q > 10, the function Fðq; qÞ can be replaced by its asymptotics b 4.3. Iterative solution of the discretized system Eq. (24) can be rewritten in the matrix form

AX ¼ F;

ð34Þ

where the vector of unknowns X and of the right hand side F have dimension N:

X ¼ jX 1 ; X 2 ; . . . ; X N jT ; ðrÞ

r

F ¼ jF 1 ; F 2 ; . . . ; F N jT ;

ð35Þ

r

F ¼ ,ðrÞt0ðrÞ ;

X ¼ ,ðrÞb ;

ð36Þ

,ðrÞ ¼ 1 if xðrÞ 2 X; ,ðrÞ ¼ 0 if xðrÞ R X:

ð37Þ





and j . . . jT is the transposition operation. In Eq. (34), A ¼ Iðp;rÞ is a square non-sparse matrix; its dimensions may be large if high accuracy of the solution is required. For linear algebraic systems with such matrices, only iterative methods are efficient. In particular, the Conjugate Gradient Method or its modifications can be used. Detailed description of these methods can be found in Press, Flannery, Teukolsky, and Vetterling (1992). In these methods, the vector of approximate solution is multiplied by the matrix A at each step of the iteration process. For non-sparse matrices of large dimensions, such a product involves substantial computational effort. If, however, a regular grid of approximating nodes is used, the volume of calculations is reduced essentially. Let us consider the sum on the left hand side of Eq. (24)

tðrÞ ¼

N   X ðpÞ I xðrÞ  xðpÞ b ;

ð38Þ

p¼1

where the function IðxÞ is defined in Eq. (31). For two-index numeration of the nodes defined in Eqs. (16) and (17), the sum (38) is the following double sum

trðm;nÞ ¼

N1 X N2   X ðmÞ ðlÞ ðnÞ ðqÞ I x1  x1 ; x2  x2 bðl;qÞ ;

rðl;qÞ

bðl;qÞ ¼ b

;

ð39Þ

l¼1 q¼1

where

  ðmÞ ðlÞ ðnÞ ðqÞ I x1  x1 ; x2  x2 ¼ Iðhðm  lÞ; hðn  qÞÞ: ðmÞ

ð40Þ ðpÞ

ðnÞ

ðqÞ

It is seen from this equation that the object Iðx1  x1 ; x2  x2 Þ has the Toeplitz structure: it depends on the differences of the indices: m  l; n  q. As a result, the Fourier transform technique can be used for calculation of the double sum in Eq. (39) and therefore, for the matrix–vector products. Application of the Fast Fourier Transform (FTT) algorithms for the calculation of these sums essentially accelerates the iterative process of solution of Eq. (34). The details of the FFT algorithm for the calculation of the matrix–vector products are described in Golub and Van Loan (1996) and Kanaun (2009). 5. Scattered far field The integral terms in Eq. (5) can be interpreted as the field ps ðxÞ scattered on the screen

ps ðxÞ ¼ 

Z X

@ g ðx  x0 Þni bðx0 ÞdX0 : @xi

ð41Þ

Far from the screen, this field is represented in the form (Pierce, 1981)

ps ðxÞ  AðeÞ

eiajxj ; jxj

ei ¼

xi : jxj

ð42Þ

Here AðeÞ is the amplitude of the far scattered field in the direction e; it can be expressed in terms of integral of the function bðxÞ over the screen area X:

AðeÞ ¼

ia ðei ni Þ 4p

Z X

 bðxÞ exp iaej xj dX:

ð43Þ

34

S. Kanaun / International Journal of Engineering Science 92 (2015) 28–46

For screens in the plane ðx1 ; x2 Þ and bðx1 ; x2 Þ in the form (21), the integral in this equation takes the form

" # N   q 2 h2 H  X  2 ðrÞ ðrÞ ðrÞ bðxÞ exp iaej xj dX ¼ h b exp ia e1 x1 þ e2 x2  e21 þ e22 : 4 X r¼1

Z

ð44Þ

Let S be a spherical surface of a large radius R containing a screen of a characteristic size L in the central region (R  L).

 The averaged over the period of oscillation energy of the scattered field Q s radiated through the surface S is given by the equation (Pierce, 1981)

s 1 Q ¼ xa 2

Z

jAðeÞj2 dS:

ð45Þ

S1

Here S1 is the surface of a unit sphere. For the incident wave propagating in the direction n0

 p0 ðxÞ ¼ a exp ian0  x ;

ð46Þ

let I0 be the energy radiated through a unit surface orthogonal to n0 . The average of this energy over the oscillation period hI0 i is (Pierce, 1981)

D E 1 I0 ¼ xajaj2 : 2

ð47Þ

The total scattering cross-section Q of the screen is the ratio of the average radiation energy of the scattered field hQ s i and the radiation energy hI0 i of the incident wave, i.e.,

s Z 2 Q jAðeÞj Q¼D E¼ dS1 : 2 jaj S1 I0

ð48Þ

where the integrand is called the differential cross-section UðeÞ

UðeÞ ¼

2 jAðeÞj

jaj2

ð49Þ

:

This function is proportional to the density of the scattered energy radiated in the direction e. In the case of a planar screen, the function UðeÞ has mirror symmetry with respect to the screen plane. 6. Elliptic screen We consider a screen that occupies an elliptic region X in the plane x3 ¼ 0:

x21 x22 þ 6 1: a21 a22

ð50Þ

The screen is subjected to an incident wave p0 ðxÞ with wave vector orthogonal to the screen:

p0 ðxÞ ¼ a expðiax3 Þ:

ð51Þ 0

Then, the right hand side of Eq. (9) is constant t ¼ iaa, and this equation takes the form

Z

0

Tðx  x0 Þbðx0 Þdx ¼ iaa;

x 2 X:

ð52Þ

X

For the numerical solution, the elliptic screen is embedded into the rectangular area X ð2a1  2a2 Þ that is covered by a regular node grid with the step h=a1 ¼ 0:001 (2,003,001 nodes). The numerical solution is not sensitive to the value of the parameter H in the approximation (21) if 1 < H < 3. Results of calculations of the function bðxÞ are presented in Figs. 3–5 for various values of the dimensionless wave number aa1 . In these plots, left hand sides are the functions jbð0; x2 Þj=jabs ð0; 0Þj, and the right hand sides are the functions jbðx1 ; 0Þj=jabs ð0; 0Þj. Here bs ðx1 ; x2 Þ is the solution of Eq. (43) for a ¼ 0 and the right hand side equal to the amplitude a of the incident field (the static solution)

2a2 bs ðx1 ; x2 Þ ¼ a EðjÞ

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  2  2 x1 x2  1 ; a1 a2

j¼1

 2 a2 : a1

ð53Þ

where EðjÞ is the elliptic integral of the second kind. The screen differential cross-sections UðeÞ are shown in Figs. 6 and 7 for various values of aa1 . The lines shown in Figs. 6 and 7 are sections of the surface UðeÞ by the planes x1 ¼ 0 (solid lines) and x2 ¼ 0 (dashed lines). It is seen that for long waves (aa1 < 2), the surface UðeÞ is almost symmetric with respect to the axis x3 orthogonal to the screen. For aa1 > 2, the energy

S. Kanaun / International Journal of Engineering Science 92 (2015) 28–46

35

3

1.75

2.5

1.5

2

1.25

1

1

αa1=0.5

0.75

0.5 0.25 0 -0.5

-0.25

0

0.25

0.5

0.75

1

Fig. 3. Dimensionless amplitude of the pressure jump bðx1 ; x2 Þ on an elliptic screen with the semiaxes a1 ; a2 ða2 =a1 ¼ 0:5Þ subjected to the incident field orthogonal to the screen plane; left hand side is the function jbð0; x2 Þj=jabs ð0; 0Þj along the smallest semiaxes of the screen, the right hand side is jbðx1 ; 0Þj=jabs ð0; 0Þj; the frequency parameter aa1 ¼ 0:5; 1; 2; 2:5; 3; bs is the ‘‘static’’ jump – solution of Eq. (9) for a ¼ 0 and right hand side equal to amplitude of the incident field a.

1.5

αa1=4

1.25

1

5 6

0.75

8

0.5

10

0.25

-0.5

-0.25

0

12 0

0.25

0.5

0.75

1

Fig. 4. Same as in Fig. 3 for the frequency parameters aa1 ¼ 4; 5; 6; 8; 10; 12.

16

0.25

14

αa1=14

16

16 20

0.2

20

20

20

0.15

24

24

0.1

24 0.05

14

0 -0.5

-0.25

0

0.25

0.5

0.75

1

Fig. 5. Same as in Fig. 3 for the frequency parameters aa1 ¼ 14; 16; 20; 24.

scattered in the plane x1 ¼ 0 is smaller than such energy scattered in the plane x2 ¼ 0 for the vectors e with the same inclination to the screen plane. Dependence of the normalized total scattering cross-section Q ¼ Q =ðpa1 a2 Þ on the parameter aa1 is in Fig. 8. In Fig. 9, sections of the surface of differential cross-sections UðeÞ by the plane x2 ¼ 0 are presented for the wave vector n0 oblique to the screen surface for various values of the parameters aa1 (n0 ¼ ðsin h; 0;  cos hÞ; h ¼ p=4; h is the angle between the vector n0 and the x3 -axis).

36

S. Kanaun / International Journal of Engineering Science 92 (2015) 28–46 0.004

0.3

αa1=1

αa1=2.5

0.003

0.2

0.002

2

0.1

0.001

1.5

0.5 -0.002

0

-0.001

0

0.001

0.002

-0.2

-0.1

0

0

0.1

0.2

Fig. 6. Differential cross-section of the elliptic screen with the semiaxes a1 ; a2 ða2 =a1 ¼ 0:5Þ subjected to the incident field orthogonal to the screen plane for dimensionless wave numbers aa1 ¼ 0:5; 1 (left figure) and aa1 ¼ 0:5; 1; 2; 2:5 (right figure); solid line is section of the function UðeÞ by the plane x2 ¼ 0, dashed line is the section of UðeÞ by the plane x1 ¼ 0.

2.5

αa1=8

αa1=18

2

10

6 1.5

1

14 4 5

10

0.5

0 -1

-0.5

0

0.5

1

0 -2

-1

0

1

2

Fig. 7. Same as in Fig. 6 for aa1 ¼ 4; 6; 8 (left figure) and aa1 ¼ 10; 14; 18 (right figure).

2.5

2 1.5 1

0.5 0

αa1 0

5

10

15

20

25

Fig. 8. Normalized total scattering cross-section Q of an elliptic screen with semiaxes a1 ; a2 ða2 =a1 ¼ 0:5Þ subjected to the incident field orthogonal to the screen plane as the function of dimensionless wave number aa1 .

37

S. Kanaun / International Journal of Engineering Science 92 (2015) 28–46

αa1=3

αa1=1 0.2

0.002

2.5 0.1

0.001

2

0.5

1.5

0 -0.001

-0.0005

0

0.0005

0.001

0.0015

0 -0.05

0

0.05

2

0.1

0.15

αa1=18

αa1=8 6 1.5

6

4

14

1

10 2

0.5

4

0

0 0

0.5

1

1.5

2

0

2

4

6

8

Fig. 9. Differential cross-section of an elliptic screen with semiaxes a1 ; a2 ða2 =a1 ¼ 0:5Þ subjected to an incident field oblique to the screen plane by the angle h ¼ p=4 for dimensionless wave numbers aa1 ¼ 0:5; 1; 1:5; 2; 2:5; 3; 4; 6; 8; 10; 14; 18; the lines are sections of the functions UðeÞ by the plane x2 ¼ 0.

7. The inverse problem The inverse problem consists in reconstructing of the screen shape from the measured values of the amplitude of the far field scattered on the screen. We focus on a specific situation when the incident wave source (transducer) and a system of discrete receivers are in the plane parallel to the screen (Fig. 10). This plane is at a distance D from the screen, and D  L; L is a characteristic screen size. The transducer is in the axis x3 orthogonal to the screen plane, and the receivers cover a finite region S of the observation plane. In hydro-acoustics, transducers usually generate a short-time (milliseconds) package of waves of basic frequency f(Hz), and receivers measure the pressure field scattered on the screen. Spectral analysis allows calculating the amplitudes of the harmonic component of the pressure waves of the basic frequency produced by the transducer and arrived at the receivers. If the distance D is known, one can estimate the amplitude a of the monochromatic components of frequency f of the incident wave in the screen area. This incident wave can be considered as a plane wave for screens with sizes much smaller than the distance D. It follows from Eq. (42) that the amplitude of the scattered field in the plane of the receivers ðx3 ¼ DÞ is

38

S. Kanaun / International Journal of Engineering Science 92 (2015) 28–46

X2

X1

Fig. 10. Scheme of the acoustic experiment for the solution of the inverse problem for a screen; the transducer of acoustic waves is on the ship, triangles are receivers spread over certain region around the transducer.

Aðx1 ; x2 Þ ¼ ps ðx1 ; x2 Þjxjeiajxj ;

jxj ¼

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x21 þ x21 þ D2 ;

a ¼ 2pf =c:

ð54Þ

Thus the amplitude Aðx1 ; x2 Þ can be calculated from the experimentally measured scattered pressure ps ðx1 ; x2 Þ at the receivers. Elliptic screens. For elliptic screens, the unknowns of the inverse problem are the orientations and sizes of the screen semiaxes. The experimentally measured amplitude of the scattered field in the plane of the receivers is denoted A ðx1 ; x2 Þ. The predicted amplitude in Eq. (54) can be constructed for the screens with various lengths of semiaxes according to the algorithm presented in previous sections. Hence, for solution of the inverse problem, we consider two functions J 0 ða1 ; a2 Þ and Jða1 ; a2 Þ (Colton & Kress, 1998)

J 0 ða1 ; a2 Þ ¼ jAða1 ; a2 ; 0; 0Þ  A ð0; 0Þj; Z jAða1 ; a2 ; x1 ; x2 Þ  A ðx1 ; x2 Þjdx1 dx2 : Jða1 ; a2 Þ ¼

ð55Þ ð56Þ

S

The function J 0 ða1 ; a2 Þ corresponds to the situation when the transducer and receiver are at the same point (echo-signal). For construction of the function Jða1 ; a2 Þ, one has to interpolate the experimental values of A ðx1 ; x2 Þ obtained at the points of the

x2(m)

|A*(x1,x2)|

f=100Hz

40

20

0

0.006

0.0064

0.00658

0.0065

0.0062

20

40

x1(m) 40

20

0

20

40

Fig. 11. Contour plots of the far scattered amplitude of an elliptic screen with semiaxes a1 ¼ 0:8 m; a2 ¼ 0:4 m for the frequency of the incident field f ¼ 100 Hz.

39

S. Kanaun / International Journal of Engineering Science 92 (2015) 28–46

1.0

a2

J0(a1,a2)

1.0

f=100Hz

0.001 0.0006 0.0004

0.8

0.006

0.004

f=100Hz

5

0.003

1 40

5

0.006

0.0004 0.4

J(a1,a2)

0.8

0.0006 0.0004

0.6

a2

40

0.6

0.004 0.003 0.0006 0.0004 0.002

30

3

30

20

3 10

0.4

13

0.001

0.002

0.2

0.2

a1 0.2

0.4

0.6

0.8

a1

1.0

0.2

0.4

0.6

0.8

1.0

Fig. 12. Contour plots of the functions J0 ða1 ; a2 Þ and Jða1 ; a2 Þ in Eqs. (57) and (58) for the frequency of the incident field f ¼ 100 Hz.

0.0012

12

J0(a1,0.32/a1)

f=100Hz

0.001

10

0.0008

8

0.0006

6

0.0004

4

0.0002

2

0

a1 0.3

0.4

0.5

0.6

0.7

0.8

0.9

0

J(a1,0.32/a1)

f=100Hz

a1 0.3

0.4

0.5

0.6

0.7

0.8

0.9

Fig. 13. Graphs of the functions J0 ða1 ; 0:32=a1 Þ and Jða1 ; 0:32=a1Þ for the frequency of the incident field f ¼ 100 Hz.

receivers on the entire region S , and then, to perform integration over this region. In these equations, Aða1 ; a2 ; x1 ; x2 Þ is the predicted amplitude of the scattered field for the screen with semiaxes a1 ; a2 . The amplitude Aða1 ; a2 ; x1 ; x2 Þ is calculated from Eqs. (43) and (44) after constructing the function bðxÞ for the given frequency of the incident field. If the orientations of actual and ‘‘predicted’’ screen are the same, the semiaxes ða1 ; a2 Þ of the actual screen provide minimum to the functions J 0 ða1 ; a2 Þ and Jða1 ; a2 Þ. For a numerical example, let the actual screen be at depth D ¼ 100 m and have semiaxis a1 ¼ 0:8 m; a2 ¼ 0:4 m. The liquid is water with wave velocity c ¼ 1500 m=sec, and the transducer generates wave packages with three basic frequencies: f ¼ 100 Hz;1 kHz, and 10 kHz. The lengths of the corresponding incident waves are 15 m; 1:5 m, and 0:15 m. The receivers are at nodes of a square grid with the step 10 m that covers the square S ð100 m  100 mÞ with the transducer in its center. In Fig. 11, the scattered amplitude of the actual screen for frequency f ¼ 100 Hz is presented. The lines of the constant values of jA ðx1 ; x2 Þj show that the scattered amplitude is symmetric with respect to the x3 -axis. The plots of the functions J 0 ða1 ; a2 Þ and Jða1 ; a2 Þ are shown in Fig. 12 for 0:1 6 a1 ; a2 6 1. It is seen that minima of these functions are on the line defined by the equation a1 a2 ¼ 0:32. This line corresponds to screens of the constant area equal to the area of the actual screen. The graphs of these functions along this line (J 0 ða1 ; 0:32=a1Þ and Jða1 ; 0:32=a1Þ) are shown in Fig. 13. Both functions have two minima at the values of the semiaxes a1 ¼ 0:4; a2 ¼ 0:8 and a1 ¼ 0:8; a2 ¼ 0:4. Thus for frequency 100 Hz, neither the function J 0 ða1 ; a2 Þ nor Jða1 ; a2 Þ can be used to determine the orientation of the screen in plane ðx1 ; x2 Þ. Only the screen area pa1 a2 can be definitely found. The case f ¼ 1 kHz is shown in Figs. 14–16. It is seen from Fig. 14 that the contour plots of the module of the scattered amplitude jA ðx1 ; x2 Þj of the actual screen are not symmetric with respect to the x3 -axis, and the screen orientation can be

40

S. Kanaun / International Journal of Engineering Science 92 (2015) 28–46

x2

|A*(x1,x2)|

40

20

0.9 0.75

0

0.7

0.8

0.85

20

40

x1 40

20

0

20

40

Fig. 14. Same as in Fig. 11 for f ¼ 1 kHz.

1.0

a2

J0(a1,a2) 0.03 0.3

0.3

1000 0.6

0.4

6000

0.7 0.2

4000

300

2000 500 300

0.05 0.03

0.5

f=1kHz

500

0.03 0.1 0.03

0.4

J(a1,a2)

0.8

0.5

0.1 0.05 0.6

a2

0.7

0.05 0.8

1.0

f=1kHz

4000

500

2000

300

1000

0.2

0.9

a1 0.2

0.4

0.6

0.8

1.0

a1 0.2

0.4

0.6

0.8

1.0

Fig. 15. Same as in Fig. 12 for f ¼ 1 kHz.

easily found from these figure: the ellipse semiaxes are directed along the axes of symmetry of the surface jA ðx1 ; x2 Þj. For f ¼ 1 kHz, the function J 0 ða1 ; a2 Þ has two zero-minima, as in the case f ¼ 100 Hz, but the function Jða1 ; a2 Þ has only one global minimum at the correct values of the semiaxes a1 ¼ 0:8; a2 ¼ 0:4. It is also seen from Figs. 15 and 16 that the functional Jða1 ; a2 Þ has three local non-zero minima. Their presence creates difficulties in numerical determination of the global minimum of this function by the Newton, Steepest Descent, or Coordinate Descent methods. These methods may converge not to the global but local minimum depending on the initial guess. The case of frequency f ¼ 10 kHz is shown in Figs. 17–19. From the contour plots of the amplitude of the scattered field of the actual screen in Fig. 17, the screen orientation can be clearly identified. The graphs in Figs. 18 and 19 show that the function J 0 ða1 ; a2 Þ has two zero-minima at the same points as for the frequencies f ¼ 100 Hz and f ¼ 1 kHz, but the function Jða1 ; a2 Þ has only one well pronounced zero-minimum for the values of the semiaxes: a1 ¼ 0:8; a2 ¼ 0:4. Absence of other local minima simplifies application of numerical methods for seeking the global minimum of the function Jða1 ; a2 Þ.

41

S. Kanaun / International Journal of Engineering Science 92 (2015) 28–46 0.18

J0(a1,0.32/a1)

1500

f=1kHz

0.15

f=1kHz

J(a1,0.32/a1)

1200

0.12

900

0.09 600

0.06 300

0.03

a1

0 0.3

0.4

0.5

0.6

0.7

0.8

0.9

a1

0 0.3

1

0.4

0.5

0.6

0.7

0.8

0.9

1

Fig. 16. Same as in Fig. 13 for f ¼ 1 kHz.

|A*(x1,x2)|

0.3

40

0.4

10kHz

0.3 0.4

0.65

0.1

0.1

1 3 5

20

0.1

0.1

6.5

0

0.1

0.1 3 1

20

0.65

0.4 40

0.1

0.4 0.3

0.3 40

20

0

20

40

Fig. 17. Same as in Fig. 11 for f ¼ 10 kHz. 1.0

a2

10kHz

J0(a1,a2) 0.3 0.1 0.5

0.8

0.8

4000

0.6

1 0.5

3

10kHz

J(a1,a2)

3000

3 1

0.4

a2

5

0.5

0.6

1.0

2500

1500

2200 0.1 0.3

0.4

2000

300 40 1000

5 0.2

0.2

a1

a1 0.2

0.4

0.6

0.8

1.0

0.2

Fig. 18. Same as in Fig. 12 for f ¼ 10 kHz.

0.4

0.6

0.8

1.0

42

S. Kanaun / International Journal of Engineering Science 92 (2015) 28–46 0.8 0.7

5000

f=10kHz

J0(a1,0.32/a1)

J(a1,0.32/a1)

f=10kHz

4000

0.6 0.5

3000

0.4

2000

0.3 0.2

1000

0.1 0

a1 0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0

a1 0.3

0.4

0.5

0.6

0.7

0.8

0.9

Fig. 19. Same as in Fig. 13 for f ¼ 10 kHz.

Screens with boundary tortuosity. Let us consider a screen with boundary (contours C) defined by the equations

RðuÞ ¼ r 0 þ r 1 cosðmuÞ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffii pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffii 1h 1h r0 ¼ 1 þ 2ð3A0  1Þ ; r 1 ¼ 2  2ð3A0  1Þ : 3 3

ð57Þ ð58Þ

Here R and uare the radius and polar angle of a point in the screen boundary; the parameter A0 relates to the screen area Ss : Ss ¼ pA0 ; 38 6 A0 6 1. If A0 ¼ 38 ; r0 ¼ r1 ¼ 0:5, if A0 ¼ 1; r0 ¼ 1; r1 ¼ 0. The parameter m is continuous, and 2 6 m 610. These equations define a two-parametric family of screens of the same characteristic size L ¼ 2. Example of the screen with A0 ¼ 0:7 and m ¼ 6 is presented in Fig. 20. Here and thereafter lengths are given in meters (m), areas in m2 , and the incident field has a unit amplitude. The inverse problem consists in determination of the parameters A0 and m on the basis of the far scattered field of the actual screen. For the solution, we consider the functions J 0 and J similar to (55) and (56).

J 0 ðA0 ; mÞ ¼ jAðA0 ; m; 0; 0Þ  A ð0; 0Þj; Z jAðA0 ; m; x1 ; x2 Þ  A ðx1 ; x2 Þjdx1 dx2 : JðA0 ; mÞ ¼ S

ð59Þ ð60Þ

Here AðA0 ; m; x1 ; x2 Þ is the predicted amplitude for the given parameters A0 and m; A ðx1 ; x2 Þ is the actual amplitude that corresponds to the screen with the parameters A0 ¼ 0:7; m ¼ 6. For f ¼ 100 Hz, the contour plots of the functions J 0 ðA0 ; mÞ and JðA0 ; mÞ are shown in Fig. 21. The minima of these functions are on the line A0 ¼ 0:7, and dependencies of J 0 ð0:7; mÞ and Jð0:7; mÞ on the parameter m are shown in Fig. 22. Both functions have two zero-minima at m ¼ 6 and m ¼ 9, and one local minimum at m between 3 and 4. Thus, for this frequency, the actual values of the parameter m cannot be uniquely determined.

Fig. 20. Screen with boundary tortuosity defined by the parameters A0 ¼ 0:7; m ¼ 6.

43

S. Kanaun / International Journal of Engineering Science 92 (2015) 28–46

10

m

10

f=100Hz

m

J0(A0,m)

0

0 5 10

0.001 8

8

0.015

100

0.01

60

0.003 0.003

0

6

0.005

0.01

0.002

0.015

f=100Hz J(A0,m)

100

10 60

0

6

30

0.005

30

0.002 4

10

4

0.001 0.001 A0

2 0.4

0.5

0.6

0.7

0.8

0.9

A0

2 0.4

1.0

0.5

0.6

0.7

0.8

0.9

1.0

Fig. 21. Contour plots of the functions J 0 ðA0 ; mÞ and JðA0 ; mÞ in Eqs. (57) and (58) for the frequency of the incident field f ¼ 100 Hz.

0.003

J0(0.7,m)

25

f=100Hz

0.0025

J(0.7,m)

f=100Hz

20

0.002

15

0.0015 10

0.001 5

0.0005

m

0 2

3

4

5

6

7

8

9

10

m

0 2

3

4

5

6

7

8

9

10

Fig. 22. Functions J0 ð0:7; mÞ and Jð0:7; mÞ for the frequency of the incident field f ¼ 100 Hz.

The case f ¼ 1 kHz is shown in Figs. 23 and 24. Again, the minima of J 0 ðA0 ; mÞ and JðA0 ; mÞ are on the line A0 ¼ 0:7; function J 0 ðA0 ; mÞ has two zero minima at m ¼ 6; m ¼ 9, and a local minimum for m is between 3 and 4. The function Jð0:7; mÞ has one zero-minimum at m ¼ 6, and two local minima at m ¼ 9 and 3 < m < 4. So, the global minimum of JðA0 ; mÞ provides correct values of the parameters A0 and m. But presence of the local minima creates difficulties in seeking the global minimum by numerical methods. In the case f ¼ 10 kHz shown in Figs. 25 and 26, the function JðA0 ; mÞ has only one well pronounced zero-minimum at A0 ¼ 0:7; m ¼ 6, and local minima of this function do not exist. Function J 0 ðA0 ; mÞ has one zero-minimum and two local minima as before. So, for this frequency, the function JðA0 ; mÞ allows one to reconstruct the correct tortuosity of the screen boundary. Combined screen model. The two-parameter models of the screen boundary considered above can be combined in a fourparametric model if the screen boundary is defined by the equations

x1 ðuÞ ¼ a1 RðuÞ cos u; x2 ðuÞ ¼ a2 RðuÞ cos u; 0 6 u 6 2p:

ð61Þ

Here ðx1 ðuÞ; x2 ðuÞÞ are Cartesian coordinates of the boundary point, the function RðuÞ has form (57) and (58). In this case, u does not coincide with the polar angle of the corresponding boundary point. This model describes a wide class of the screen boundaries, and the parameters a1 ; a2 define global sizes of the screen, meanwhile parameters A0 ; m describe tortuosity of the boundary. To solve the inverse problem, one has to find minima of the function Jða1 ; a2 ; A0 ; mÞ similar to (56) and (60). The solid line in Fig. 27 is the screen with the following values of the parameters a1 ¼ 0:8; a2 ¼ 0:4; A0 ¼ 0:7; m ¼ 6. This screen is taken as an actual screen for solution of the inverse problem. The testing frequency was f ¼ 10 kHz. The process of solution starts with the value of A0 ¼ 1 (elliptic screen). Since in this case r1 ¼ 0, the parameter m can be arbitrary.

44

S. Kanaun / International Journal of Engineering Science 92 (2015) 28–46 10

m

0.5

8

0.05 0.03

0.2

300

10000

0.2

6

5000

6

0.1

A0 0.6

2000 1000

4

2 0.5

0

500

0.05 0.03

0.4

2000

1000

0.1

4

J(A0,m)

78

0.03

0.9

f=1kHz

500

J0(A0,m)

0.003 8

m

10

f=1kHz

0.7

0.8

0.9

300 200

A0

2 0.4

1.0

0.5

0.6

0.7

0.8

0.9

1.0

Fig. 23. Same as in Fig. 21 for f ¼ 1 kHz. 0.12

f=1kHz

J0(0.7,m)

0.1

1000

J(0.7,m)

f=1kHz

800

0.08

600

0.06 400

0.04 200

0.02

m

0 2

3

4

5

6

7

8

9

0

10

m 2

3

4

5

6

7

8

9

10

Fig. 24. Same as in Fig. 22 for f ¼ 1 kHz. 10

m

10

f=10kHz

m

f=10kHz

J(A0,m)

J0(A0,m) 0.5 8

4

3

2 1

0.2

1

2

3

8

4

5000

4

A0

2 0.4

0.5

0.6

0.7

200 0 1000 500

6

0.5 0.2

4

3000 2000

0.5 0.2 0

6

4000

0.8

0.9

1.0

A0

2 0.4

0.5

Fig. 25. Same as in Fig. 21 for f ¼ 10 kHz.

0.6

0.7

0.8

0.9

1.0

45

S. Kanaun / International Journal of Engineering Science 92 (2015) 28–46 6000

f=10kHz

J0(0.7,m)

1.6

J(0.7,m)

f=10kHz

5000

1.4 1.2

4000

1 3000

0.8 0.6

2000

0.4

1000

0.2

m

0 2

4

6

8

10

m

0 2

4

6

8

10

Fig. 26. Same as in Fig. 22 for f ¼ 10 kHz.

Fig. 27. Solution of the inverse problem for the screen with the parameters a1 ¼ 0:8; a2 ¼ 0:4; A0 ¼ 0:7; m ¼ 6. Solid line is the actual screen, the curves 1–5 are the screen boundary after the corresponding steps of the Coordinate Descent method, the sixth iteration coincides with the actual screen.

The function Jða1 ; a2 ; 1; mÞ was constructed for 0:1 6 a1 ; a2 6 1. The only minimum of J was found at a1 ¼ 0:744; a2 ¼ 0:3 (the corresponding line is indicated as 1 in Fig. 27) . Note that the area of the ellipse with these semiaxes is very close to the area of the actual screen. Then, for the obtained values of a1 ; a2 , minimum of the function Jð0:744; 0:3; A0 ; mÞ is to be found with respect to the parameters A0 ; m. The single minimum of this function is at A0 ¼ 0:9026; m ¼ 6. For these values of A0 ; m, the minimum of the function J with respect to a1 ; a2 is again to be found. In the literature, this process is called the Coordinate Descent method. It was necessary to perform six iterations in order to obtain the correct solution of the inverse problem. The contours that correspond to each iteration are indicated by the corresponding numbers in Fig. 27. The sixth iteration coincides with the actual screen boundary. At each step of the iteration process the only minimum of the function Jða1 ; a2 ; A0 ; mÞ was identified. 8. Conclusions The developed method can be applied to scattering problems for screens. It allows calculating the jump of the acoustic pressure on the screens, amplitudes of the far scattered field, total and differential cross-sections of planar screens of arbitrary shapes by action of monochromatic incident fields of various frequencies and directions. The method is computationally economical. If the screen boundary is described by a model that depends on a finite number of scalar parameters, the inverse problem is reduced to seeking minima of a function of these parameters. This function may have a number of local minima, and for identification of the global minimum, detailed study of all the minima should be performed. The results presented in this paper show that the inverse problem for the screens can be successfully solved if the frequency of the incident wave f satisfies the condition 2pfL=c ¼ Oð50Þ. For incident waves of lower frequencies, only the screen area can be definitely identified. For incident waves of higher frequencies, the scattered field is concentrated in a small vicinity of the origin, and for the considered system of the receivers, it is impossible to construct the far scattered amplitude A ðx1 ; x2 Þ with sufficient accuracy. Note that in the short wave limit ( f ! 1), the function A ðx1 ; x2 Þ tends to the d-function concentrated at the origin with the coefficient proportional to the screen area. So, details of the screen shape disappear in this limit. In addition, attenuation of waves increases with frequency, and at large distance D, the scattered field can be lost. If the screen is not parallel to the plane of receivers, its orientation can be found by seeking coordinates of the maximum of the module of the scattering amplitude jA ðx1 ; x2 Þj in the observation plane. The normal to the screen surface is directed from the screen center to the point of this maximum. For successful application of the method, the angle between this normal to the screen and the x3 -axis should not be overly large, so that the point of the maximum of jAðx1 ; x2 Þj does not fall outside the observation region S (in which case the screen orientation cannot be found).

46

S. Kanaun / International Journal of Engineering Science 92 (2015) 28–46

References Colton, D., & Kress, R. (1987). Integral equation method in scattering theory. NY: Wiley. Colton, D., & Kress, R. (1998). Inverse acoustic and electromagnetic scattering theory. Berlin, Heidelberg, NY: Springer. Eskin, G. (1981). Boundary-value problems for elliptic pseudo-differential equations. NY, USA: Math Soc. Golub, G., & Van Loan, C. (1996). Matrix computations (3d ed.). The Johns Hopkins University Press. Il’inskiy, A., & Smirnov, G. (1994). Integral equation for problems of wave diffraction at screens. Journal of Communication Technology and Electronics (39/4), 129–136. Kanaun, S. (2009). Fast calculation of elastic fields in a homogeneous medium with isolated heterogeneous inclusions. International Journal of Multiscale Computational Engineering, 7/4, 263–276. Kanaun, S. (2014). Scattering of monochromatic elastic waves on a planar crack of arbitrary shape. Wave Motion, 51, 360–381. Kanaun, S., & Levin, V. (2013). Scattering of elastic waves on a heterogeneous inclusions of arbitrary shape: An efficient numerical method for 3D-problems. Wave Motion, 50, 687–707. Maz’ya, V., & Schmidt, G. (2007). Approximate approximation. Mathematical surveys and monographs (Vol. 141). Providence: American Mathematical Society. Pierce, A. (1981). Acoustics: An introduction to its physical principles and applications. NY: McGraw-Hill. Press, W., Flannery, B., Teukolsky, S., & Vetterling, W. (1992). Numerical recipes in FORTRAN: The art of scientific computing (2nd ed.). Cambridge University Press. Shaw, R. (1979). Boundary integral equation method applied to wave problems. In P. K. Benerdjee & R. Butterfield (Eds.). Development in boundary element method (Vol. 1). London: Applied Science Publisher.