theoretical and computational aspects of scattering from...

Theoretical and computational aspects of scattering

from rough surfaces: one-dimensional perfectly

reflecting surfaces

J DeSanto, G Erdmann, W Hereman and M Misra

Department of Mathematical and Computer Sciences, Colorado School of Mines,Golden, CO 80401-1887, USA, Phone: (303) 273-3036, Fax: (303) 273-3875, email:jdesanto/gerdmann/whereman/[email protected]

Abstract. We discuss the scattering of acoustic or electromagnetic waves from onedimensional rough surfaces. We restrict the discussion in this report to perfectlyreflecting Dirichlet surfaces (TE-polarization). The theoretical development is for bothinfinite surfaces and periodic surfaces, the latter equations derived from the former. Weinclude both derivations for completeness of notation. Several theoretical developmentsare presented. They are characterized by integral equation solutions for the surfacecurrent or normal derivative of the total field. All the equations are discretized to amatrix system and further characterized by the sampling of the rows and columns ofthe matrix which is accomplished in either coordinate space (C) or spectral space (S).The standard equations are referred to here as CC equations of either first kind (CC1)or second kind (CC2). Mixed representation equations or SC type are solved as wellas SS equations fully in spectral space.

Computational results are presented for scattering from various periodic surfaces.The results include examples with grazing incidence, a very rough surface and ahighly oscillatory surface. The examples vary over a parameter set which includes thegeometrical optics regime, physical optics or resonance regime, and a renormalizationregime.

The objective of this study was to determine the best computational method forthese problems. Briefly, the SC method was the fastest but did not converge for largeslopes or very rough surfaces for reasons we explain. The SS method was slower andhad the same convergence difficulties as SC. The CC methods were extremely slow butalways converged. The simplest approach is to try the SC method first. Convergence,when the method works, is very fast. If convergence doesn’t occur then try SS andfinally CC.

2

1. Derivation of CC Equations For an Infinite One-Dimensional Perfectly

Reflecting Rough Surface

We consider the scattering from an infinite one-dimensional rough surface specified by

z = s(x) (see Figure 1). For the examples we consider in this report the surface is

perfectly reflecting and periodic. In this section we first consider the surface to be

infinite and specify it to be periodic later in Section 4. Our computational results

are restricted to periodic surface cases. Notationally we have a spatial 2-vector

x = (x, z) = (x1, x2) and its restriction to the surface xs = (x, s(x)). The gradient

operator is ∂i = ∂∂xi

(i = 1, 2) and the normal derivative ∂n = ni∂i where ni is

the normal to the surface and repeated subscripts are summed (here from 1 to 2).

Fields are represented by ψ and correspond to a velocity potential (acoustics) [6] or the

y−component of the electric (Dirichlet boundary value problem) or magnetic (Neumann

boundary value problem) fields. Since the surface is one-dimensional its generator is

parallel to the y−axis and no polarization change occurs during scattering from such a

surface. The electromagnetic problem thus reduces to a scalar one, which is what we

treat. All fields are time-harmonic so that a factor exp(−iωt) is suppressed throughout

(ω is circular frequency and t is time).

The scattered field satisfies the scalar Helmholtz equation (k1 = 2πλ

is the

wavenumber and λ is wavelength)

(∂i∂i + k21)ψ

sc(x) = 0, x ∈ D+R . (1.1)

The free-space two-dimensional Green’s function G(2) for this problem satisfies the non-

homogeneous Helmholtz equation

(∂i′∂i

′ + k21)G

(2)(x,x′) = −δ(x − x′), (1.2)

where the right hand side is the Dirac delta function in two-dimensions. G(2) is explicitly

given by [6, pg. 54]

G(2)(x,x′) =i

4H

(1)0 (k1|x − x′|), (1.3)

the Hankel function of zeroth-order, first kind. Its Fourier transform relation is

G(2)(x,x′) =1

(2π)2

∫ ∫ei�α . (x−x′)G(α)d�α, (1.4)

where

G(α) =1

α2 − k21+

(1.5)

and we have chosen k1+ = limε→0+(k1 + iε) to indicate that we have an outgoing wave.

The region D+R is specified by the characteristic function

θ+(x) = θ(z − s(x))θ(R − r), (1.6)

3

in

sc

sc

sc

R

D

s(x)

H

H

D

R

R

R

R

+

+

-

-

Figure 1. Infinite perfectly reflecting one-dimensional rough surface z = s(x). Incident(in) and scattered (sc) wave are indicated. The region D+

R is specified by z > s(x)and r = |x| < R and is bounded by the rough surface and the semicircle H+

R at radiusR. The complementary region (D−

R) and semi-circle (H−R ) below the surface are also

illustrated.

in the limit as R→ ∞. Here θ is the usual Heaviside function

θ(x) =

{1, x > 0,

0, x < 0,(1.7)

and r = |x|. The function θ+ thus represents the region bounded by the rough surface

s(x) truncated at R and the upper semicircle at radius R denoted by H+R (see Figure

1). Vertical segments joining the surface and the semicircle can also be included [8] but

are omitted in the interests of brevity.

To form equations for the scattered field we use Green’s theorem. Multiply Eq. (1.1)

by G(2) and Eq. (1.2) by ψsc and subtract the resulting equations. We get

∂i′ {∂i

′G(2)(x,x′)ψsc(x′) −G(2)(x,x′)∂i′ψsc(x′)

}= −δ(x − x′)ψsc(x′). (1.8)

Next, multiply Eq. (1.8) by θ+(x′), integrate over all space (in x′) and then integrate

4

by parts using the vector derivative of the characteristic function

∂j′θ+(x′) = nj(x

′)δ(z′ − s(x′))θ(R− r′) − nj(R)δ(r′ − R)θ(z′ − s(x′)), (1.9)

where

nj(x′) = δj2 − δj1s

′(x′), (1.10)

is the non-unit normal to the surface s and

nj(R) = ∂′jr′|r′=R, (1.11)

the radial normal to the semicircle H+R . Two surface integrals result. The integral over

the semicircle is∫ ∫H+

R

[∂r′G

(2)(x,x′)ψsc(x′) −G(2)(x,x′)∂r′ψsc(x′)

]r′=R

Rdθ′, (1.12)

where θ′ is the integration angle in [−π2, π

2]. If ψsc satisfies a Sommerfeld radiation

condition this integral vanishes as R → ∞. More generally, so long as ψsc does not

contain any horizontally propagating plane waves, the integral vanishes as R → ∞[8]. We include the latter restriction since we treat the case of plane wave incidence in

our periodic surface examples later. If there is an incident plane wave we must admit

scattered plane waves in order to balance the total energy on this far away semicircle

[7]. For horizontal plane wave incidence and scattering other equations result than the

ones we quote below [8].

We thus assume that Eq. (1.12) vanishes as R → ∞. The result using Eq. (1.9) is

a single integral over the infinite surface s∞(x). To write this result in convenient form,

define single (S) and double (D) layer acoustic potentials with respective densities u

and v as

(Su)(x) =∫

s∞G(2)(x,x′

s)u(x′s)dx

′, (1.13)

and

(Dv)(x) =∫

s∞∂n

′G(2)(x,x′s)v(x

′s)dx

′, (1.14)

as well as the normal derivative of ψsc

N sc(x) = ∂nψsc(x). (1.15)

The resulting equations can then be written as

(Dψsc)(x) − (SN sc)(x) = θ+(x)ψsc(x), (1.16)

which for x ∈ D+∞ gives the representation of the scattered field, and for x ∈ D−

∞ (the

lower region below the surface) the equation is referred to as an extinction theorem [13].

5

To form surface integral equations use the limiting properties of single and double

layer potentials [3]. Define the limits from above (+) and below (−) as

limx→x±

s

(Su)(x) = (Su)±(xs), (1.17)

and

limx→x±

s

(Dv)(x) = (Dv)±(xs). (1.18)

The single layer is continuous

(Su)+(xs) = (Su)−(xs), (1.19)

and the double layer has a jump discontinuity

(Dv)+(xs) − (Dv)−(xs) = v(xs), (1.20)

with each limit defined as

(Dv)±(xs) = PV(Dv)(xs) ± 1

2v(xs), (1.21)

where PV stands for Cauchy Principal Value. Using these limiting values the surface

integral equation which follows from Eq. (1.16) is

PV(Dψsc)(xs) − (SN sc)(xs) =1

2ψsc(xs). (1.22)

The kernel terms in this integral equation have both arguments in coordinate space (on

the surface) so that a discretized version of them will yield (kernel) matrices whose rows

and columns result from coordinate-space sampling. We thus refer to this equation as

a coordinate-coordinate (CC) equation.

A second equation can be formed by taking the normal derivative of Eq. (1.16) for

x ∈ D+∞. The normal derivative of the single layer potential is discontinuous with limits

∂n(Su)±(xs) = PV∂n(Su)(xs) ∓ 1

2u(xs), (1.23)

and the normal derivative of the double layer has the same limit from above and below

but is singular and we take its Hadamard Finite Part (FP) [9]

limx→x±

s

∂n(Dv)(x) = FP∂n(Dv)(xs). (1.24)

[Note: For now the use of PV and FP notation is purely formal.] The result is another

CC integral equation

FP∂n(Dψsc)(xs) − PV∂n(SN sc)(xs) =1

2N sc(xs). (1.25)

The usual boundary value problems we wish to discuss involve total field quantities. We

treat the incident field next in Section 2 and combine the results into integral equations

on the total field.

6

2. Incident and Total Fields

We form integral equations of the total field (ψT) and normal derivative (NT). These

are defined by

ψT(x) = ψi(x) + ψsc(x), (2.1)

and

NT(x) = N i(x) +N sc(x), (2.2)

in terms of the incident (i) field and its normal derivative. ψi(x) satisfies the

homogeneous Helmholtz equation

(∂j∂j + k21)ψ

i(x) = 0. (2.3)

Our examples later are for a single plane wave

ψi(x) = φp(x) = Deik1(α0x−β0z), (2.4)

where α0 = sin(θi), β0 = cos(θi), and θi is the angle of incidence defined from the

z−direction. We choose D = 1 for computations. We also omit horizontally incident

waves, so β0 > 0. More generally, we could have a continuous superposition (or spectral

decomposition) of plane waves

ψi(x) = φc(x) =∫ ∞

−∞I(µ)eik1[µx−m(µ)z]dµ, (2.5)

where

m(µ) =

⎧⎪⎨⎪⎩

(1 − µ2)12 , µ ≤ 1,

i(µ2 − 1)12 , µ > 1.

(2.6)

The density I(µ) is a continuous function. Eq. (2.4) is a special case if we let I(µ) be

the distribution Dδ(µ− α0).

Asymptotically for r = (x2+z2)12 large, Eqs. (2.4) and (2.5) behave quite differently.

Eq. (2.4) has no limit and Eq. (2.5) (for continuous I(µ)) behaves like a cylindrical

wave and satisfies the incoming cylindrical radiation condition in D+∞ and an outgoing

cylindrical wave radiation condition in D−∞, the domain below the surface.

The results of Green’s theorem can be summarized by defining the bracket integral

of Green’s theorem on a surface P for any field φ

[G(2), φ;x, P ] =∫

P

[G(2)(x,x′

p)∂′nφ(x′

p) − ∂′nG(2)(x,x′

p)φ(x′p)

]dx′. (2.7)

The result is [8]

limR→∞

[G(2), φp;x, H+R ] = φp(x), (2.8)

7

and

limR→∞

[G(2), φp;x, H−R ] = 0. (2.9)

For a single plane wave the equation corresponding to Eq. (1.16) is

(Dφp)(x) − (SNp)(x) = θ+(x)φp(x) − φp(x), (2.10)

where the last term follows from Eq. (2.8) and Np is the normal derivative of the plane

wave. Combining Eqs. (2.10) and (1.16) we have a representation for the total field at

x (off the surface)

θ+(x)ψT(x) = φp(x) + (DψT)(x) − (SNT)(x), (2.11)

and in the limit as x approaches the surface s(x) the surface integral equation

1

2ψT(xs) = φp(xs) + PV (DψT)(xs) − (SNT)(xs). (2.12)

A second equation can be derived using the normal derivative of Eq. (2.11) for x ∈ D+∞

and subsequently taking the limit as x approaches the surface. It is

1

2NT(xs) = Np(xs) + FP ∂n(DψT)(xs) − PV ∂n(SNT)(xs). (2.13)

Further, using Green’s theorem in D−∞ on G(2) and φc and combining the result with

the scattered field results in Section 1, an analogous set of equations to Eqs. (2.12)

and (2.13) results (with the modification that the plane wave term is replaced by the

continuous superposition).

Thus, for an incident field of either form (excluding horizontal plane waves) we can

write Eqs. (2.12) and (2.13) as

1

2ψT(xs) = ψi(xs) + PV (DψT)(xs) − (SNT)(xs), (2.14)

and

1

2NT(xs) = N i(xs) + FP ∂n(DψT)(xs) − PV ∂n(SNT)(xs). (2.15)

These are both coordinate-coordinate (CC) integral equations on the boundary

unknowns ψT and NT. Both are valid for an infinite surface with the restriction that

no horizontal plane waves occur. For completeness and later use we include the field

representation which follows from Eq. (2.11) for an incident field satisfying the above

restrictions. It is

θ+(x)ψT(x) = ψi(x) + (DψT)(x) − (SNT)(x). (2.16)

8

3. Dirichlet Problem

For the Dirichlet (D) boundary value problem

ψT(xs) = 0. (3.1)

Acoustically this describes a soft surface and electromagnetically it is the case of TE-

polarization. With this condition Eq. (2.14) becomes

ψi(xs) = (SNT)(xs), (3.2)

which is referred to as CC1, a first-kind integral equation [15] for the remaining boundary

unknown NT. Eq. (2.15) becomes

1

2NT(xs) = N i(xs) − PV ∂n(SNT)(xs), (3.3)

which is an integral equation of the second kind, and we refer to it as the CC2 equation.

Often the two equations are linearly combined as follows. Choose real constants α and

β, multiply Eq. (3.2) by α and Eq. (3.1) by β, and add the resulting equations. Since all

the functions are evaluated on the surface they are functions of a single variable. Define

the incident field function by

F i(x) = βψi(xs) + αN i(xs). (3.4)

The resulting added equations can be put in the impedance form

F i(x) =∫ ∞

−∞ZD(x, x′)NT(x′)dx′, (3.5)

where the “impedance” kernel is defined symbolically as

ZD(x, x′) =1

2αδ(x− x′) + αPV ∂nG

(2)(xs,x′s) + βG(2)(xs,x

′s). (3.6)

For α = 0 we have CC1. For β = 0, CC2, and for β = 1 and α arbitrary we have

what is referred to as the Combined Field Integral Equation (CFIE) (see [12]). It

becomes particularly important for scattering from bounded bodies where the CC1 and

CC2 solutions contain different resonances but the CFIE solutions remain finite at all

frequencies.

4. Periodic Surface

For a periodic surface with period L, s(x+ L) = s(x). We can then reduce Eq. (3.5) to

an integration over a single period cell, say from −L2

to L2. The single layer potential

term in Eq. (3.2) can be written as

(SNT)(xs) =∫ ∞

−∞G(2)(xs,x

′s)N

T(x′)dx′ (4.1)

9

=∞∑

n=−∞In(x), (4.2)

where

In(x) =∫ (2n+1)L

2

(2n−1)L2

G(2)(xs,x′s)N

T(x′)dx′. (4.3)

In Eq. (4.3) use the Weyl representation for the Green’s function [6, pg. 63]

G(2)(xs,x′s) =

πi

(2π)2

∫ ∞

−∞1

m(µ)eik1[µ(x−x′)+m(µ)|s(x)−s(x′)|]dµ, (4.4)

(with m(µ) defined by Eq. (2.6)), the Floquet (pseudo-)periodicity of the boundary

unknown

NT(x+ nL) = eik1α0nLNT(x), (4.5)

and the change of variables x′′ = x′ − nL.

We can then write Eq. (4.1) on the domain [−L2, L

2] as

(SNT)(xs) =∫ L

2

−L2

Gp1(x, x′)NT(x′)dx′, (4.6)

where Gp1 is the periodic Green’s function with wavenumber k1 given by

Gp1(x, x′) =

πi

(2π)2

∫ ∞

−∞1

m(µ)eik1[µ(x−x′)+m(µ)|s(x)−s(x′)|] .

∞∑n=−∞

eink1L(α0−µ)dµ. (4.7)

For scalar arguments x and x′, Gp1 is confined to the surface.

Next, use the Poisson sum [17]

∞∑n=−∞

eint = 2π∞∑

j=−∞δ(t+ 2πj), (4.8)

where t = k1L(α0 − µ). The delta function in Eq. (4.8) reduces to

δ(t+ 2πj) =1

k1Lδ(µ− αj), (4.9)

where

αj = α0 + jλ

L. (4.10)

This is the Bragg equation with αj = sin(θj) and θj the angle of the jth outgoing Bragg

wave. Using Eqs. (4.8) to (4.10) to evaluate Eq. (4.7) we get

Gp1(x, x′) =

i

4π

λ

L

∞∑j=−∞

1

βjeik1[αj(x−x′)+βj |s(x)−s(x′)|], (4.11)

10

with

βj =

⎧⎪⎨⎪⎩

(1 − αj)12 , |αj| ≤ 1,

i(αj2 − 1)

12 , |αj| > 1.

(4.12)

Other representations for this periodic Green’s function, useful for its evaluation in

computations, are discussed in Appendix A. Further, it is also convenient to have a

representation for this function off the surface, and it is obviously given by

Gp1(x,x′) =

i

4π

λ

L

∞∑j=−∞

1

βj

eik1[αj(x−x′)+βj |z−z′|]. (4.13)

Similarly, the normal derivative of the single layer potential term in Eq. (3.3) can be

reduced to a single period cell. The result is

PV∫ ∞

−∞∂nG

(2)(xs,x′s)N

T(x′)dx′ = −∫ L

2

−L2

G′p1

(x, x′)NT(x′)dx′, (4.14)

where

G′p1

(x, x′) = [nj∂jGp1(x,x′)]|z=s(x)

|z′=s(x′)(4.15)

follows from Eq. (4.13). The slash on the integral in Eq. (4.14) represents Cauchy

principal value (if appropriate). Using Eqs. (4.6) and (4.14), Eq. (3.4) reduces to

F i(x) =∫ L

2

−L2

ZDp1

(x, x′)NT(x′)dx′, (4.16)

where the impedance kernel is given by

ZDp1

(x, x′) =1

2αδ(x− x′) + αPVG′

p1(x, x′) + βGp1(x, x

′). (4.17)

Again, for α = 0 we have the CC1 equation, for β = 0 the CC2 equation, and for β = 1

and α arbitrary the CFIE equation. These are the equations which are solved for our

examples. The numerical solution is discussed in Appendix B. Discretization of Eq.

(4.16) yields an impedance matrix whose rows and columns result from sampling both

in coordinate space, thus the acronym coordinate-coordinate or CC.

5. Derivation of SC Equations For an Infinite One-Dimensional Perfectly

Reflecting Rough Surface

In the previous four sections we confined our attention to problems where both rows and

columns of the matrix to be inverted were sampled in coordinate space. Here we derive

a mixed representation where the rows are sampled in the conjugate spectral (S) space

and the columns still sampled in the coordinate space. These are the SC equations.

A straightforward derivation without using any of the results in the first four sections

11

can be found in the literature [5]. A different derivation which however yields the same

results proceeds as follows.

Use the representation for the total field given by Eq. (2.16) for the Dirichlet problem

(ψT(xs) = 0). We have

ψsc(x) = −(SNT)(x), x ∈ D+∞, (5.1)

and

ψi(x) = (SNT)(x), x ∈ D−∞. (5.2)

The Weyl representation Eq. (4.4) written off the surface in the x−variable is

G(2)(x,x′s) =

πi

(2π)2

∫ ∞

−∞1

m(µ)eik1[µ(x−x′)+m(µ)|z−s(x′)|]dµ. (5.3)

For z > max[s(x)] we can remove the absolute value in the phase of Eq. (5.3) and for

these values of z write Eq. (5.1) using Eqs. (1.13) and (5.3) as

ψsc(x) =∫ ∞

−∞A(µ)eik1[µx+m(µ)z]dµ, (5.4)

where

A(µ) =−i

4πm(µ)

∫ ∞

−∞e−ik1[µx′+m(µ)s(x′)]NT(x′)dx′. (5.5)

Once we know NT we can thus evaluate A(µ) and the scattered field. Eq. (5.4) is a

spectral representation of the scattered field. For large r = (x2 +z2)12 (where x = r sin θ

and z = r cos θ), a stationary phase evaluation of Eq. (5.4) can be written in terms of

the scattering amplitude T (θ) as

ψsc(x) ∼ T (θ)eik1r

√r, (5.6)

which is an outgoing cylindrical wave where

T (θ) = (−2πi)12A(µsp), (5.7)

and the stationary phase point is µsp = sin θ. The amplitude A(µ) is thus directly related

to the scattering amplitude.

To solve for NT use Eq. (5.2) and the representation Eq. (5.3) now for z < min[s(x)].

The result is

ψi(x) =∫ ∞

−∞I(µ)eik1[µx−m(µ)z]dµ, (5.8)

which is just Eq. (2.5) and where

I(µ) =i

4πm(µ)

∫ ∞

−∞e−ik1[µx−m(µ)s(x)]NT(x)dx. (5.9)

12

Given the properties of the incident field, i.e. I(µ), we solve the first kind integral

equation Eq. (5.9) for NT and use this to evaluate the scattered field. The kernel of

the integral equation is now a function of µ (spectral, S) and x (coordinate, C) and the

method is referred to as spectral-coordinate (SC).

We can write Eqs. (5.5) and (5.9) in the symmetric representation

I±(µ) =∫ ∞

−∞e−ik1[µx∓m(µ)s(x)]NT(x)dx, (5.10)

where

I±(µ) = 4πi m(µ)

{ −I(µ)

A(µ) .(5.11)

6. SC Equations For a Periodic Surface

For a periodic surface s(x+ L) = s(x), and we can write Eqs. (5.10) as

∞∑n=−∞

Q±n (µ) = I±(µ), (6.1)

where

Q±n (µ) =

∫ (2n+1)L2

(2n−1)L2

e−ik1[µx∓m(µ)s(x)]NT(x)dx. (6.2)

Again, change variables to x′ = x−nL, and use the Floquet periodicity of NT given by

Eq. (4.5). The result is

Q±n (µ) = eink1L(α0−µ)Q±

0 (µ). (6.3)

Use of the Poisson sum, Eq. (4.8), and Eq. (4.9) yield for Eq. (6.1)

λ

LQ±

0 (µ)∞∑

j=−∞δ(µ− αj) = I±(µ). (6.4)

Integration of both sides of Eq. (6.4) over the µ−domain [αn − ε λL, αn + ε λ

L] where

0 < ε < 1 yields

λ

LQ±

0 (αn) =∫ αn+ε

λL

αn−ελL

I±(µ)dµ. (6.5)

For a single incident plane wave (see Eq. (5.8))

I(µ) = Dδ(µ− α0), (6.6)

and, for a periodic surface, the scattered field spectra are discrete

A(µ) =∞∑

n=−∞Anδ(µ− αn). (6.7)

13

(This can be seen by reducing Eq. (5.5) to the integration over a single periodic cell).

Using these results in Eq. (6.5) and the definitions Eq. (5.11) we get

λ

LQ±

0 (αn) = I±(αn), (6.8)

where

I±(αn) = 4πi βn

{ −Dδn0

An .(6.9)

The integrals Q±0 have dimensions of length times the dimensions of NT. Also, NT has

dimensions of inverse length times the dimensions of the field. It is convenient to scale

out this inverse length by defining the function N(x) as

NT(x) = ik1N(x), (6.10)

so that N(x) has the same dimensions as the field ψT. (The scaling Eq. (6.10) obviously

relates to the fact that differentiation of a wave-like field quantity produces a factor

ik1.) The result can be written as

P±0 (αn) = F±(αn), (6.11)

where

P±0 (αn) =

1

L

∫ −L2

−L2

e−ik1[αnx∓βns(x)]N(x)dx, (6.12)

and

F±(αn) = 2βn

{ −Dδn0

An .(6.13)

The method of solution is to solve the first kind equation for N(x) (the “ + ” equation)

then evaluate the “− ” equation for An. The scattered field from Eqs. (5.4) and (6.7) is

then

ψsc(x, z) =∞∑

n=−∞Ane

ik1(αnx+βnz). (6.14)

The kernels in Eq. (6.12) are functions of µ = αn and x, i.e. a discrete spectral parameter

n and a coordinate variable, thus again the spectral-coordinate (SC) acronym. An

alternative derivation of these results can be found in the literature [4].

7. SS Equations For a Periodic Surface

We derive the spectral-spectral (SS) equations from the SC equations in Section 6. The

method is to expand the boundary unknown in the topological or surface wave basis

[10] in Eq. (6.12)

N(x) =∞∑

j′=−∞Nj′e

ik1(αj′x−βj′s(x)). (7.1)

14

Note that this “basis” is not a complete set. The justification for its choice rests on the

fact that in many cases it produces an extremely fast and highly accurate result. The

result is first a system of linear equations for the vector of expansion coefficients in the

discrete spectral domain N = {Nj}KN = F+, (7.2)

with the components

F+j = −2β0Dδj0, (7.3)

and the matrix whose entries are

Kjj′ =1

2π

∫ π

−πe−i(j−j′)yeik1(βj−βj′)s(

L2π

y)dy, (7.4)

where the integrals have been scaled to [π, π], and second, the set of equations to evaluate

for the Aj coefficients is

∑j′Mjj′Nj′ = 2βjAj , (7.5)

where the matrix elements are

Mjj′ =1

2π

∫ π

−πe−i(j−j′)ye−ik1(βj+βj′)s(

L2π

)ydy. (7.6)

The scattered field is then given by Eq. (6.14). Rows and columns of both K and

M are indexed in the (discrete) spectral integer j and the method is referred to as

spectral-spectral (SS).

8. Energy

We know that the sum of the scattered energy must equal the energy in the incident

wave

∑j

|Aj |2Re(βj) = β0D2. (8.1)

Only real (Re) orders carry energy away from the surface. We set D = 1 for our trials

and quote the condition as

∑j

|Aj |2 Re(βj)

β0= 1. (8.2)

The left hand side of Eq. (8.2) is referred to as the normalized energy. The Aj are

computed and then we determine how well the energy check Eq. (8.2) is satisfied. It is

a necessary but not sufficient condition of accuracy.

15

9. Computational Results

This section presents timing and reliability results for several formalisms on several

surfaces. λ/∆x is a measure of pulse width, with higher numbers corresponding to

faster sampling. λ/∆x = 10 is often quoted, but it can be either oversampling or

undersampling. The surface graphs use equal scales on the horizontal and vertical axes,

so apparent tangency is true tangency. The best CC method is used for the surface

current and scattered amplitude plots, since spectral methods often have incorrect

currents. An additional result is referred to as CG, a CC Galerkin approach with

Fourier basis functions for a first kind equation. This is very similar to CC1, but the

self-cell integral is performed exactly, not using the first few terms in an expansion.

A discussion of the numerical methods for CC, SC, and CG methods can be found in

Appendices B through D respectively. The SS numerical technique is treated in Section

7. For our calculations with CG, we set α = 0 and β = 1 (see Appendix D).

In Sections 9.1 to 9.5 we treat a variety of rough surface examples. In Sections 9.1

to 9.3 respectively we consider the cases when λ/L � 1 (geometrical optics regime),

λ/L ≈ 1 (resonance or physical optics regime), and λ/L 1 (sometimes referred to as

a renormalization regime) all for a cosine surface. In Section 9.4 we treat a very rough

surface with a maximum slope of about 25. In Section 9.5 we present results of a case

with a highly oscillatory surface with the oscillations increasing as the end points of the

period are approached.

The results of these computations can be summarized as follows. The CC methods

always worked well in the sense that the error was small for a sufficiently large matrix.

This is illustrated in Figures 2-9 and the accompanying tables contained in Sections 9.1

to 9.5 (Examples 1-5). For the CC methods, fill time refers to the time necessary to

compute the matrix elements, here the time to compute the periodic Green’s function

and its normal derivative. This took a great deal of time (see the discussion in Appendix

B), and the result was that the CC methods were extremely slow.

The fill time for the SC method was several orders of magnitude faster than CC

(this is because we were only evaluating a function, as shown in Appendix C). SC was

clearly the solution method of preference when it worked. It failed to work for very rough

(Example 4) and highly oscillatory (Example 5) surfaces. The fill time for the SS method

was between that for CC and SC and consisted of the evaluation of matrix elements

of the form given in Eq. 7.4. The SS method is based on the same type of topological

basis expansion as the SC method and it had the same convergence difficulties as the

SC method.

16

9.1. Example 1, λ/L� 1

Case A, No Grazing

S(x) −(d/2) cos(2πx/L)

d/L 0.075

λ/L 0.01563553622559

θi 20◦

Error = log10 |1 − Normalized Energy|

Matrix Linear Solution

Formalism Size λ/∆x Fill Time Time Error

SS 128 by 128 4788 0.24 -14.8

SS 138 by 138 6272 0.74 -15.3

SS 148 by 148 7930 0.88 -15.4

SC 128 by 128 2.0 0.64 0.62 -15.7

SC 138 by 138 2.2 0.74 0.72 -15.7

SC 148 by 148 2.3 0.82 0.91 -15.1

CC1 64 by 64 1.0 501 0.11 -0.2

CC1 128 by 128 2.0 2039 0.61 -1.6

CC1 256 by 256 4.0 8165 4.18 -2.7

CC2 64 by 64 1.0 808 0.10 0.1

CC2 128 by 128 2.0 3283 0.62 -5.7

CC2 256 by 256 4.0 13201 4.04 -8.0

CG 65 by 65 1.0 515 0.10 -0.2

CG 129 by 129 2.0 2133 0.61 -1.3

CG 257 by 257 4.0 8401 4.15 -3.4

Table 1. Example 1, Case A, no grazing incidence and 128 real Bragg modes. Theerror for SS and SC was extremely small and the fill time for the SC matrix elementsnearly negligible. The only way a reasonably small error could be attained for thecoordinate based methods was to increase the matrix size. Fill time plus solution timefor the SC method was negligible compared to the other methods.

17

Surface and Incoming Waves Scattered Energy Distribution

−0.5 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 0.5

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

−0.06 −0.04 −0.02 0 0.02 0.04 0.060

0.02

0.04

0.06

0.08

0.1

(a) (b)

Real(N(x)) vs x |N(x)| vs x

−0.5 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 0.5−2.5

−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

2.5

−0.5 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 0.51.7

1.75

1.8

1.85

1.9

1.95

2

2.05

(c) (d)

Figure 2. Example 1, Case A: (a) Surface and incoming waves, (b) Scattered energydistribution, and surface current ((c) real part (d) modulus) for the CC2 formalism,matrix size 256 by 256.

18

Case B, Near-Grazing Incidence/Reflection


d/L 0.075

λ/L 0.01566499626662

θi 75◦




SS 128 by 128 1305 0.24 -0.9

SS 138 by 138 2058 0.76 0.1

SS 148 by 148 2901 0.86 -0.6

SC 128 by 128 2.0 0.76 0.59 -0.5

SC 138 by 138 2.2 0.74 0.73 -0.9

SC 148 by 148 2.3 0.83 0.88 -3.1

CC1 64 by 64 1.0 541 0.10 -0.5

CC1 128 by 128 2.0 2194 0.60 -0.4

CC1 256 by 256 4.0 8781 4.11 -4.7

CC2 64 by 64 1.0 848 0.11 -0.3

CC2 128 by 128 2.0 3450 0.59 -0.8

CC2 256 by 256 4.0 13851 3.98 -4.9

CG 65 by 65 1.0 574 0.11 -0.5

CG 129 by 129 2.0 2296 0.62 -0.5

CG 257 by 257 4.0 9070 4.12 -3.9

Table 2. Example 1, Case B, near-grazing incidence and reflection with 128 real Braggmodes. The error for all methods (especially SS and SC) increased considerably. Errorsfor the coordinate based methods were adequate only for large matrix size. Again, filltime plus solution time for SC was negligible compared to all the other methods.

19


−0.5 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 0.5

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

−0.15 −0.1 −0.05 0 0.05 0.1

−0.05

0

0.05

0.1

(a) (b)


−0.5 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 0.5−1.5

−1

−0.5

0

0.5

1

1.5

−0.5 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 0.50

0.2

0.4

0.6

0.8

1

1.2

(c) (d)

Figure 3. Example 1, Case B: (a) Surface and incoming waves, (b) Scattered energydistribution, and surface current ((c) real part (d) modulus) for the CC1 formalism,matrix size 256 by 256.

20

9.2. Example 2, λ/L ≈ 1

Case A, No Grazing


d/L 0.25

λ/L 0.95

θi 20◦

Error = log10 |1 − Normalized Energy|Matrix Linear Solution

Formalism Size λ/∆x Fill Time Time ErrorSS 2 by 2 0.29 0.01 -0.4SS 6 by 6 1.63 0 -1.7SS 10 by 10 4.7 0 -2.2SS 14 by 14 10.1 0 -2.3SS 18 by 18 18.2 0 -2.4SS 22 by 22 29 0.01 -2.3SC 2 by 2 1.9 0.11 0 −∞SC 6 by 6 5.7 0.02 0 -2.5SC 10 by 10 9.5 0.02 0.01 -2.7SC 14 by 14 13.3 0.05 0.01 -2.7SC 18 by 18 17.1 0.06 0.01 -2.7CC1 64 by 64 60.8 298 0.11 -4.7CC1 128 by 128 122 1224 0.63 -5.3CC1 256 by 256 243 4993 4.2 -5.9CC2 64 by 64 60.8 478 0.10 -5.5CC2 128 by 128 122 1959 0.60 -6.4CC2 256 by 256 243 8126 4.1 -7.3CG 65 by 65 61.8 310 0.11 -8.3CG 129 by 129 123 1231 0.61 -9.5CG 257 by 257 244 4898 4.05 -10.7

A−1 = −0.17963135247142− 0.65697150178152i

A0 = −0.75962914626508 + 0.17615097897977i

Table 3. Example 2, Case A, no grazing incidence or reflection. Only two real Braggmodes are present. Again, fill time plus solution time for SC were negligible comparedto other methods. Accuracy for the coordinate methods increased as the matrix sizeincreased.

21


−0.5 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 0.5

−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5

−0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.30

0.1

0.2

0.3

0.4

0.5

0.6

(a) (b)


−0.5 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 0.5−3

−2.5

−2

−1.5

−1

−0.5

0

0.5

−0.5 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 0.51

1.5

2

2.5

3

3.5

(c) (d)


22


S(x) −(d/2) cos(2πx/L)d/L 0.25λ/L 0.95θi 75◦


Formalism Size λ/∆x Fill Time Time ErrorSS 3 by 3 0.38 0 -1.1SS 7 by 7 2.1 0 -1.9SS 11 by 11 5.7 0 -2.0SS 15 by 15 11.9 0 -2.0SS 19 by 19 20.2 0.01 -1.9SS 23 by 23 31.4 0.01 -1.8SC 3 by 3 2.9 0.01 0 -1.4SC 7 by 7 6.7 0.03 0 -2.8SC 11 by 11 10.5 0.04 0.01 -3.4SC 15 by 15 14.3 0.05 0 -3.6SC 19 by 19 18.1 0.07 0.01 -3.7SC 23 by 23 21.9 0.07 0.02 -3.7SC 27 by 27 25.7 0.09 0.02 -3.6CC1 64 by 64 60.8 303 0.11 -5.7CC1 128 by 128 122 1182 0.58 -6.3CC1 256 by 256 243 4799 4.27 -6.9CC2 64 by 64 60.8 469 0.12 -6.1CC2 128 by 128 122 1928 0.61 -7.0CC2 256 by 256 243 7787 4.06 -7.9CG 65 by 65 61.8 305 0.11 -9.7CG 129 by 129 123 1216 0.62 -9.8CG 257 by 257 244 4824 4.14 -9.7

A−2 = 0.07250263029849 + 0.00497683319193i

A−1 = −0.01402209947690 − 0.18205512555385i

A0 = −0.91996242949398 + 0.13259130973127i

Table 4. Example 2, Case B, near-grazing incidence and reflection with three modes.Note a general increase in accuracy from Case A. The SC method was the fastest.

23


−0.5 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 0.5

−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5

−0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8

−0.4

−0.2

0

0.2

0.4

0.6

(a) (b)


−0.5 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 0.5−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

−0.5 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 0.50.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

(c) (d)


24

9.3. Example 3, λ/L 1

Case A, No Grazing

S(x) −(d/2) cos(2πx/L)d/L 2.5λ/L 100θi 20◦


Formalism Size λ/∆x Fill Time Time ErrorSS 1 by 1 0.36 0 -2.0SS 5 by 5 1.27 0 -4.7SS 9 by 9 3.9 0.01 -7.5SS 13 by 13 8.5 0.01 -6.2SS 17 by 17 15.6 0.01 -7.7SS 21 by 21 25.5 0.01 -8.2SS 29 by 29 55.4 0.01 -5.3SC 1 by 1 100 0 0 -15.7SC 5 by 5 500 0.02 0.01 −∞SC 9 by 9 900 0.02 0.01 -15.4SC 13 by 13 1300 0.04 0.01 −∞SC 17 by 17 1700 0.05 0.01 -13.1SC 21 by 21 2100 0.07 0 -10.3CC1 64 by 64 6400 149 0.1 -15.4CC1 128 by 128 12800 526 0.6 -15.4CC1 256 by 256 25600 2042 4.1 -15.1CC2 64 by 64 6400 267 0.1 -6.7CC2 128 by 128 12800 893 0.6 -8.7CC2 256 by 256 25600 3317 3.9 -9.0CG 65 by 65 6500 159 0.11 -15.1CG 129 by 129 12900 542 0.63 −∞CG 257 by 257 25700 2091 4.16 -14.8

A0 = −0.99185964722787 + 0.12733593444511i

Table 5. Example 3, Case A, no grazing incidence or reflection with only the specularmode. All the methods were highly accurate with the SC as the fastest. The surfacehas a very large maximum slope (πd/L).

25


−4 −3 −2 −1 0 1 2 3 4

−2

−1

0

1

2

3

4

5

−0.6 −0.4 −0.2 0 0.2 0.4 0.60

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

(a) (b)


−0.5 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 0.5−9

−8

−7

−6

−5

−4

−3

−2

−1

0

−0.5 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 0.50

1

2

3

4

5

6

7

8

9

(c) (d)


26


S(x) −(d/2) cos(2πx/L)d/L 2.5λ/L 100θi 75◦


Formalism Size λ/∆x Fill Time Time ErrorSS 1 by 1 0.24 0 -3.1SS 5 by 5 1.25 0 -5.8SS 9 by 9 3.9 0.01 -8.6SS 13 by 13 8.5 0.01 -8.0SS 17 by 17 15.5 0.01 -8.8SS 21 by 21 25.2 0 -8.2SS 29 by 29 54.7 0.02 -5.6SC 1 by 1 100 0 0 -15.2SC 5 by 5 500 0.01 0.01 -15.7SC 9 by 9 900 0.03 0.01 -15.2SC 13 by 13 1300 0.05 0.01 -15.7SC 17 by 17 1700 0.05 0 -13.7SC 21 by 21 2100 0.07 0.01 -10.2CC1 64 by 64 6400 154 0.1 -12.7CC1 128 by 128 12800 539 0.6 -12.9CC1 256 by 256 25600 2108 4.2 -13.2CC2 64 by 64 6400 277 0.1 -6.7CC2 128 by 128 12800 917 0.6 -8.7CC2 256 by 256 25600 3442 3.9 -9.0CG 65 by 65 6500 165 0.1 -12.7CG 129 by 129 12900 557 0.6 -12.9CG 257 by 257 25700 2145 4.0 -13.2

A0 = −0.99938168553634 + 0.03516029884120i

Table 6. Example 3, Case B, near grazing incidence and reflection with one mode.All methods highly accurate with SC the fastest. The maximum slope (πd/L) is quitelarge.

27


−4 −3 −2 −1 0 1 2 3 4

−2

−1

0

1

2

3

4

5

−1 −0.5 0 0.5

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

(a) (b)


−0.5 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 0.5−2.5

−2

−1.5

−1

−0.5

0

−0.5 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 0.50

0.5

1

1.5

2

2.5

(c) (d)


28

9.4. Example 4, Very Rough Surface

This section presents results for a surface with extremely large slopes. We show that

our code still converges for such cases. The maximum slope, πd/L, is about 25. λ

is relatively small here. It is expected that better results should be attainable with

larger values of λ. Times are reported in seconds of CPU time required by a SPARC 20

workstation with 32 MB of memory.

Interface Perfectly Reflecting


d/L 8

λ/L 0.05

θi 20◦




CC2 512 25.6 9870 28 1.9

CC2 1024 51.2 39488 250 -3.9

CG 513 25.7 7288 29 -0.2

CG 1025 51.3 28872 250 -0.2

Table 7. Example 4 for a very rough surface with a maximum slope about 25 andmany modes. Only CC2 converged with small errors and then only for a very largematrix size. The convergence was very slow.

29


−10 −8 −6 −4 −2 0 2 4 6 8 10−150

−100

−50

0

50

100

150

200

250

300

−0.2 −0.15 −0.1 −0.05 0 0.05 0.10

0.05

0.1

0.15

0.2

0.25

(a) (b)


−10 −8 −6 −4 −2 0 2 4 6 8 10−100

−80

−60

−40

−20

0

20

40

60

80

100

−10 −8 −6 −4 −2 0 2 4 6 8 100

50

100

150

200

250

(c) (d)

Figure 8. (a) Surface and incoming waves (carefully note the scale), (b) Scatteredenergy distribution, and surface current ((c) real part (d) modulus) for the CC2formalism, matrix size 1024 by 1024.

30

9.5. Example 5, Highly Oscillatory Surface with Continuous Derivative

Due to the roughness of the surface, it is sampled uniformly in arc length (instead of

x).

S(x)

{ −(d/2) cos (2πx/L+ 10π(2x/L)3) , |x| ≤ L/2

S(x+ nL) = S(x), elsewhere

d/L 0.15

λ/L 0.05

θi 20◦




SS 40 by 40 1316 0.01 -0.2

SS 48 by 48 1887 0.05 1.2

SS 56 by 56 2482 0.08 1.7

SC 40 by 40 2.0 0.26 0.05 1.4

SC 48 by 48 2.4 0.16 0.05 1.5

SC 56 by 56 2.8 0.23 0.10 0.7

CC1 512 by 512 25.6 22406 28.0 -0.6

CC1 1024 by 1024 51.2 89674 247 -0.8

CC2 512 by 512 25.6 36635 28.3 -2.0

CC2 1024 by 1024 51.2 146970 246 -3.1

CG 513 by 513 25.7 22111 30.2 -2.4

CG 1025 by 1025 51.3 88367 249 -4.4

Table 8. Example 5, highly oscillatory surface with continuous derivative at the endpoints. The surface was sampled uniformly in arc length. The lack of convergence inboth spectral related methods is noted. The coordinate-related methods required avery large size to get good convergence.

31


−10 −8 −6 −4 −2 0 2 4 6 8 10

−6

−4

−2

0

2

4

6

8

−0.05 0 0.05 0.10

0.05

0.1

(a) (b)


−10 −8 −6 −4 −2 0 2 4 6 8 10−30

−20

−10

0

10

20

30

−10 −8 −6 −4 −2 0 2 4 6 8 100

5

10

15

20

25

30

35

40

45

(c) (d)

Figure 9. Example 5: (a) Surface and incoming waves, (b) Scattered energydistribution, and surface current ((c) real part (d) modulus) for the CG formalism,matrix size 1025 by 1025.

10. Conclusions

Computational results have been presented for scattering from various periodic surfaces.

The results include examples with grazing incidence, a very rough surface and a

highly oscillatory surface. The examples vary over a parameter set which includes the

geometrical optics regime, physical optics and resonance regime, and a renormalization

regime.

32

The main objective of this study was to determine the best computational method

for these problems. Briefly, the SC method was the fastest but did not converge

for large slopes or very rough surfaces. The topological basis used in the method

was not a complete set, and computationally, the dynamical range in the matrix

increased exponentially with surface height. The SS method was slower and had the

same convergence difficulties as SC. The CC methods were extremely slow but always

converged. The simplest approach is to try the SC method first. Convergence, when

the method works, is very fast. If convergence does not occur then try SS and finally

CC. Results for the remaining mixed representation (CS) can be found in the literature

[16].

Acknowledgments

Effort sponsored by the Air Force Office of Scientific Research, Air Force Materials

Command, USAF, under the Multi-University Research Initiative (MURI) program

Grant #F49620-96-1-0039.

The US Government is authorized to reproduce and distribute reprints for

governmental purposes notwithstanding any copyright notation thereon. The views and

conclusions contained herein are those of the authors and should not be interpreted as

necessarily representing the official policies or endorsements, either expressed or implied

of the AFOSR or the US Government.

Erdmann’s research was supported in part by an Undergraduate Research Grant

from the Colorado Advanced Software Institute (CASI) and a Grant-in-Aid of Research

from Sigma Xi, The Scientific Research Society.

Discussions with Dr. Gary Brown and Capt. Jeff Boleng were very helpful.

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[3] D. Colton and R. Kress, Integral Equation Methods in Scattering Theory, Wiley, New York (1983).[4] J. A. DeSanto, “Scattering from a perfectly reflecting arbitrary periodic surface: An exact theory,”

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2202–2207 (1985).

33

[6] J. A. DeSanto, Scalar Wave Theory: Green’s Functions and Applications, Springer Verlag, NewYork (1992).

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[8] J. A. DeSanto and P. A. Martin, “On the derivation of boundary integral equations for scatteringby an infinite one-dimensional rough surface,” J. Acoust. Soc. Am. 102, 67-77 (1997).

[9] P. A. Martin and F. J. Rizzo, “On boundary integral equations for crack problems,” Proc. R. Soc.Lond. A 421, 341-355 (1989).

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[12] N. Morita, N. Kumagai and J. R. Mautz, Integral Equation Methods for Electromagnetics, ArtechHouse, Boston (1990).

[13] M. Nieto-Vesperinas, Scattering and Diffraction in Physical Optics, Wiley, New York (1991).[14] F. Oberhettinger and L. Badii, Tables of Laplace Transforms, Springer Verlag, Berlin (1973).[15] D. Porter and D. S. G. Stirling, Integral Equations, Cambridge University Press, Cambridge (1990).[16] M. Saillard and J. A. DeSanto, “Coordinate-spectral method for rough surface scattering,” Waves

in Random Media 6, 135-150 (1996).[17] L. Schwartz, Mathematics for the Physical Sciences, Addison-Wesley, Reading MA (1966).[18] M. E. Veysoglu, H. A. Yueh, R. T. Shin and J. A. Kong, “Polarimetric passive remote sensing of

periodic surfaces,” J. Elect. Waves & Appls. 5, 267-280 (1991).

Appendix A. Periodic Green’s Function

In Section 4 we derived the first (or spectral) representation of Gp1 given by

Gp1(x, x′) =

i

4π

λ

L

∞∑j=−∞

1

βjeik1[αj(x−x′)+βj |s(x)−s(x′)|]. (A1)

A second representation follows from using Eqs. (4.1) to (4.3). We have

∫ ∞

−∞G(2)(xs,x

′s)N

T(x′)dx′ =∞∑

n=−∞In, (A2)

where now we use the Hankel function representation for G(2) (from Eq. (1.3) )

In =∫ (2n+1)L

2

(2n−1)L2

i

4H

(1)0 (k1[(x− x′)2 + (s(x) − s(x′))2]

12 )NT(x′)dx′. (A3)

Shift the integration variable (x′′ = x′ − nL), and use the periodicity of s(x) and the

Floquet periodicity of the boundary unknown Eq. (4.5). The result is

∫ ∞

−∞G(2)(xs,x

′s)N

T(x′)dx′ =∫ L

2

−L2

Gp1(x, x′)NT(x′)dx′, (A4)

34

where now

Gp1(x, x′) =

i

4

∞∑n=−∞

eik1α0nLH(1)0 (k1{[(x− (x′ + nL))]

2+ [s(x) − s(x′)]2}

12 ), (A5)

which is the representation ofGp1 in terms of a phased periodic array of Hankel functions.

A third representation can also be derived using Eq. (A5) [18]. From tables [14] we

have the Laplace transform representation

H(1)0 (

√s2 + a2) = −2i

πeis

∫ ∞

0e−st cos[a(t2 − 2it)

12 ]

(t2 − 2it)12

dt. (A6)

Transforming this equation using t = u2 we have

H(1)0 (

√s2 + a2) = −4i

πeis

∫ ∞

0e−su2

D(a, u)du, (A7)

where

D(a, u) =cos[au(u2 − 2i)

12 ]

(u2 − 2i)12

. (A8)

Rewrite the sum in Eq. (A5) in three parts, the n = 0 term, a sum from 1 to ∞, and a

sum from −1 to −∞. Let n → −n in the latter sum. If we define as the s−variable in

Eq. (A7)

s± = k1(±(x′ − x) + nL), (A9)

and use

a = k1[s(x) − s(x′)], (A10)

then Eq. (A5) can be written using Eq. (A7) as

Gp1(x, x′) =

i

4H

(1)0 (k1[(x− x′)2 + (s(x) − s(x′))2]

12 )

+1

π

∞∑n=1

eik1α0nLeis+

∫ ∞

0e−s+u2

D(a, u)du

+1

π

∞∑n=1

e−ik1α0nLeis−∫ ∞

0e−s−u2

D(a, u)du. (A11)

The summations can be performed. Using Eq. (A9) define the coefficients of n in Eq.

(A11)

b± = k1L(u2 − i[1 ± α0]), (A12)

and then the sums are∞∑

n=1

e−nb± =e−b±

1 − e−b±. (A13)

35

If we further define

p± = eik1L(1±α0), (A14)

then Eq. (A11) can be written as

Gp1(x, x′) =

i

4H

(1)0 (k1[(x− x′)2 + (s(x) − s(x′))2]

12 )

+1

πp+eik1(x−x′)

∫ ∞

0

e−u2k1(x′−x+L)

1 − p+e−k1Lu2D(a, u)du

+1

πp−eik1(x′−x)

∫ ∞

0

e−u2k1(x−x′+L)

1 − p−e−k1Lu2D(a, u)du, (A15)

which is the third representation for Gp1 on the surface.

The same analysis follows for the function off the surface. Extend a to b where

b = k1(z − z′), (A16)

and we have the general off-the-surface representation

Gp1(x,x′) =

i

4H

(1)0 (k1[(x− x′)2 + (z − z′)2]

12 )

+1

πp+eik1(x−x′)

∫ ∞

0

e−u2k1(x′−x+L)

1 − p+e−k1Lu2D(b, u)du

+1

πp−eik1(x′−x)

∫ ∞

0

e−u2k1(x−x′+L)

1 − p−e−k1Lu2D(b, u)du, (A17)

which is used to compute the normal derivative of Gp1 as in Eq. (4.14).

To compute the impedance kernel in Section 4 we use the Green’s function

representation in various ways. To compute Gp1(x, x′) for example consider the following

cases:

(a) For x �= x′ and s(x), s(x′) far apart we use the spectral sum Eq. (A1).

(b) For x �= x′ but s(x) close to s(x′) use Eq. (A15) and directly evaluate it. We

approximate the integrals using piecewise Gaussian quadrature. The integrands

decay rapidly, and we determine where the integrand is negligible and approximate

the number of integration intervals to achieve good accuracy.

(c) For x = x′ we again use the representation Eq. (A15) as follows. In the two integral

terms set x = x′ and evaluate as in case (b). The Hankel function term in Eq. (A15)

must be treated as described in Appendix B by evaluating the self-cell integral.

Although Eq. (A5) is the canonical representation for the periodic Green’s function in

two-dimensions, we generally do not use it for evaluation purposes since the convergence

is slow and evaluation time per term is long.

To evaluate G′p1

we have corresponding cases:

(a) For x �= x′ and s(x), s(x′) far apart take the normal derivative of Eq. (A1).

36

(b) For x �= x′ but s(x) close to s(x′) take the normal derivative of Eq. (A15).

(c) For x = x′ the normal derivative of the Hankel function H(1)0 is

np∂pH(1)0 (k1|xs − x′

s|) = −np|xs − x′s|pH(1)

1 (k1|xs − x′s|)

|xs − x′s|

, (A18)

which is finite in the limit as xs → x′s

limxs→x′

s

np∂pH(1)0 (k1|xs − x′

s|) = − i

π

s′′(x′)1 + [s′(x′)]2

, (A19)

so the self-cell evaluation is not a problem.

Although we do not use it for our calculations, we can also define the Fourier transform

of the periodic Green’s function. It is given by

Gp1(kx, kx′) =∫ L

2

−L2

∫ L2

−L2

e−i(kx x + kx′ x′)Gp1(xs, x′s) dx dx

′

≈ ∑j

∑m

e−i(kx xj + kx′ xm)Gp1(xj, xm)∆xj∆xm (A20)

Several examples are presented in Section 10.7.

Appendix B. Numerical Solution of the CC Equations

In Section 4 we derived the CFIE integral equation given by

F i(x) =∫ L

2

−L2

ZDp1

(x, x′)NT(x′)dx′, (B1)

where the impedance kernel is given by

ZDp1

(x, x′) =1

2αδ(x− x′) + αPVG′

p1(x, x′) + βGp1(x, x

′), (B2)

and

F i(x) = αN i(xs) + βψi(xs). (B3)

A standard discretization of Eq. (B1) is the method-of-moments approach, which utilizes

a pulse basis and collocation in x. There are three steps in using this approach. First,

partition the interval [−L2, L

2] into a certain number (say M) subintervals, with the pth

interval called ∆p. Then Eq. (B1) can be written as

F i(x) =M∑

p=1

∫∆p

ZDp1

(x, x′)NT(x′)dx′. (B4)

Second, choose a representative point (such as the midpoint) in each subinterval, with

the pth given by xp, and approximate NT(x′) for x′ ∈ ∆p as NT(xp). Consequently,

F i(x) ≈M∑

p=1

NT(xp)∫∆p

ZDp1

(x, x′)dx′. (B5)

37

Third, collocate this equation for x equal to each of the chosen points, yielding

F i(xq) ≈M∑

p=1

NT(xp)∫∆p

ZDp1

(xq, x′)dx′. (B6)

When the kernel is not singular (i.e. when p �= q), approximate the integrals as∫∆p

ZDp1

(xq, x′)dx′ ≈ ∆p Z

Dp1

(xq, xp). (B7)

When the kernel is singular, we must analytically integrate any singular terms. As seen

in representation Eq. (A11) for Gp1, the only singular term is the Hankel function. For

x ≈ x′,

H(1)0

(k1[(x− x′)2 + (s(x) − s(x′))2]

12

)

≈ 1 +2i

π

[γ + ln

(k1

2[(x− x′)2 + (s(x) − s(x′))2]

12

)], (B8)

with Euler’s constant γ ≈ 0.5772. For small ∆q, we can therefore approximate the

integral as follows∫∆q

H(1)0

(k1 [(xq − x′)2 + (s(xq) − s(x′))2]

12

)dx′

≈∫ ∆q

2

−∆q2

{1 +

2i

π

[γ + ln(

k1|ε|√

1 + (s′(xq))2

2)]}dε

= ∆q

{1 +

2i

π

[γ + ln(

k1

√1 + (s′(xq))2

2)]}

+4i

π

∫ ∆q2

0ln ε dε

= ∆q

{1 +

2i

πln(

k1

√1 + (s′(xq))2eγ∆q

4e)}. (B9)

This is the only nonrepairable singularity present in ZDp1, so we can now form the matrix

equation we wish to solve

Fi = ZNT , (B10)

where F in = F i(xn), NT

n = NT (xn), and

Zmn = ∆n

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

α2

+ αPVG′p1

(xm, xn) + β U(xm, xn)

+ iβ4

[1 + 2i

πln

(k1

√1+(s′(xn))2eγ∆n

4e

)], m = n,

α PVG′p1

(xm, xn) + βGp1(xm, xn), m �= n,

(B11)

where U(xm, xn) is the nonsingular part of Gp1(x, x′), namely, the two integral terms in

Eq. (A15) which are evaluated at x = x′. The latter term in the m = n equation in Eq.

(B11) follows from Eq. (B9)

38

The scattered amplitudes are given by an approximation to the “−” version of Eqs.

(6.11)-(6.13)

2βnAn =1

L

M∑p=1

e−ik1[αnxp+βns(xp)]N(xp)∆p. (B12)

Appendix C. Numerical Solution of the SC Equations

In Section 6 we derived the SC integral representation:

−2Dβnδn0 =1

L

∫ −L2

−L2

e−ik1[αnx−βns(x)]N(x)dx. (C1)

We approximate the integral with the discrete quadrature rule given by

∫ −L2

−L2

f(x)dx ≈M∑

p=1

wpf(xp), (C2)

where {wp} and {xp} are the weights and sampled points, respectively. Therefore, Eq.

(C1) becomes

−2Dβnδn0 =1

L

M∑p=1

wpe−ik1[αnxp−βns(xp)]N(xp). (C3)

This can be written as the matrix equation

KN = F+, (C4)

where F+m = −2Dβ0δm0 (see Eq. (7.3), Nn = N(xn), and

Kmn =wn

Le−ik1[αmxn−βms(xn)]. (C5)

The scattering amplitudes are obtained via the following approximation of Eq. (6.12)

2βnAn =1

L

M∑p=1

wpe−ik1[αnxp+βns(xp)]N(xp). (C6)

More details on a particular numerical method for solving this system of equations can

be found in [11].

Appendix D. Numerical Solution of the CC Equations: Discrete Galerkin

Method

An alternative to the method-of-moments approach in Appendix B is set forth in [1].

This method allows one to integrate the logarithmic singularity in the CC1 equation

exactly, avoiding the series approximation used in the method-of-moments approach.

39

A rigorous derivation of this method, with error analysis, can be found in [1]. We

present a more intuitive derivation here. First, we parameterize the coordinate variable,

using x(t) = Lt2π

− L2. Then Eq. (4.16) becomes

2∫ 2π

0ZD

p1(t, t′)N(x(t′))dt′ =

4π

LF i(x(t)), (D1)

where

ZDp1

(t, t′) =1

2αδ(x(t) − x(t′)) + αPVG′

p1(x(t), x(t′)) + βGp1(x(t), x(t

′)). (D2)

Now we treat the singularity in Gp1(x, x′) at x = x′. We can write

Gp1(x, x′) =

[Gp1(x, x

′) +1

2πln

([(x− x′)2 + (s(x) − s(x′))2]

12

)]

− 1

2πln

([(x− x′)2 + (s(x) − s(x′))2]

12

), (D3)

where the singularity in the bracketed expression at x = x′ is repairable. In addition,

the logarithmic term can be written as

ln([(x(t) − x(t′))2 + (s(x(t)) − s(x(t′)))2]

12

)= ln |2e− 1

2 sin(t− t′

2)| + B(t, t′), (D4)

where

B(t, t′) =

⎧⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎩

ln[ e+1

2

([(x(t)−x(t′))2+(s(x(t))−s(x(t′)))2]

12

)|2 sin( t−t′

2)|

], t− t′ �= 2mπ,

ln[e+

12

L2π

√1 + [s′(x(t))]2

], t− t′ = 2mπ.

(D5)

Since B(t, t′) is not singular, we will group it with the other nonsingular or repairable

terms of the kernel:

B(t, t′) = αδ(x(t) − x(t′)) + 2αPVG′p1

(x(t), x(t′)) + β[2Gp1(x(t), x(t

′))

+1

πln

([(x(t) − x(t′))2 + (s(x(t)) − s(x(t′)))2]

12

)− 1

πB(t, t′)

]. (D6)

Then Eq. (D1) becomes∫ 2π

0B(t, t′)N(x(t′))dt′ − β

π

∫ 2π

0ln |2e− 1

2 sin(t− t′

2)|N(x(t′))dt′ =

4π

LF i(x(t)). (D7)

Now we expand N(x) in a truncated Fourier series

N(x(t′)) ≈n∑

m=−n

Nmeimt′ , (D8)

and substitute this expression into Eq. (D7). The second integral can then be evaluated

exactly:

−βπNm

∫ 2π

0ln |2e− 1

2 sin(t− t′

2)|eimt′dt =

βNmeimt

max{1, |m|} . (D9)

40

Therefore, we can write Eq. (D7) as

n∑m=−n

Nm

[ ∫ 2π

0B(t, t′)eimt′dt′ +

βeimt

max{1, |m|}]

=4π

LF i(x(t)). (D10)

We now approximate the remaining integral in a manner which avoids redundant

computation of the complicated function B(t, t′). We divide the interval [0, 2π] into

2n+ 1 equal intervals, so that tj = 2π2n+1

(j + 12), j = 0, ..., 2n, is the midpoint of the jth

subinterval. Use the integral approximation

∫ 2π

0f(t′)dt′ ≈ 2π

2n+ 1

2n∑j=0

f(tj) (D11)

in Eq. (D10) and collocate the resulting equation at the tj points, yielding

n∑m=−n

Nm

[2π

2n+ 1

2n∑k=0

B(tj , tk)eimtk +

βeimtj

max{1, |m|}]

=4π

LF i(x(tj)), j = 0, ..., 2n.(D12)

This linear system is used to determine the expansion coefficients of N(x(t)). This

method has good convergence properties [1]. The scattering amplitudes are computed

using the approximation

2βmAm =1

2n+ 1

2n∑j=0

e−ik1[αmx(tj)+βms(x(tj ))]N(x(tj)). (D13)

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