Wave Motion ( ) –

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Wave Motion

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The two-dimensional free-space Green’s function and itsderivatives in a tangent-normal systemJohn A. DeSanto ∗

Emeritus and Department of Physics, Colorado School of Mines, Golden, CO 80401, United States

h i g h l i g h t s

• Two-dimensional free-space Green’s function.• Non-Cartesian geometries, representations in tangent-normal system.• Derivatives of single-layer acoustic potentials.• Double-layer acoustic potentials.• Derivatives of double-layer acoustic potentials.

a r t i c l e i n f o

Article history:Received 2 January 2013Received in revised form 9 March 2014Accepted 25 March 2014Available online xxxx

Keywords:Green’s functionScatteringNon-Cartesian geometries

a b s t r a c t

The two-dimensional free-space Green’s function, G(2), and its derivatives, are usedextensively in the formulation of scattering and diffraction problems through its presencein single- and double-layer potentials, and their use in integral equations. The vastmajorityof the results from elementary classical mathematical physics for G(2) is based on Cartesiancoordinate-space, either directly as a Hankel function in coordinate-space or through atransform, such as theWeyl transform, also based on Cartesian coordinate-space. However,if the geometry of the problem is not Cartesian, for example in scattering from a roughsurface, there are difficulties in using a transform representation for G(2) which depends onCartesian geometry, as the standard Weyl transform does. Here we formulate transform-space representations using a tangent-normal coordinate system. The result for G(2) is anew Weyl-type tangent-normal transform representation from which the results for thevector derivatives of the single-layer potential, the double-layer potential, and the vectorderivatives of the double-layer potential follow quite simply. The latter three results can beexpressed in terms of two new spectral functions in tangent-normal space, S1 and S2. Theoverall results are new representations for G(2) and its derivatives which may be useful inintegral equation formulations of scattering problems for non-Cartesian geometries.

© 2014 Elsevier B.V. All rights reserved.

1. The free-space Green’s function in Cartesian coordinates

The free-space Green’s function G(2)(x′, x) with the source (x = (x, z)) and field (x′= (x′, z ′)) coordinates satisfies the

two-dimensional inhomogeneous Helmholtz equation

(∂2x + ∂2

z + k20)G(2)(x′, x) = −δ(x′

− x), (1)

∗ Tel.: +1 303 979 8988; fax: +1 303 273 3919.E-mail addresses: bajdesanto@mac.com, jdesanto@mines.edu.

http://dx.doi.org/10.1016/j.wavemoti.2014.03.0070165-2125/© 2014 Elsevier B.V. All rights reserved.

2 J.A. DeSanto / Wave Motion ( ) –

where k0 is the wavenumber. Fourier transforming the equation yields the two-dimensional representation [1]

G(2)(x′, x) =1

(2π)2

G(2)(k) exp(ik · (x′

− x))dkxdkz, (2)

where G(2)(k) = (k2 − k20)−1, k2 = k2x + k2z , and the integrations are from −∞ to ∞. If k0 has a small positive imaginary

part, k0 + iϵ, integration in say the complex kz-plane yields the one-dimensional Weyl representation

G(2)(x′, x) =iπ

(2π)2

1Kx

exp(i(kx(x′− x) + Kx|z ′

− z|))dkx, (3)

with

Kx =

k20 − k2x , |kx| 6 k0,

= ik2x − k0, |kx| > k0. (4)

These representations are just those of the Hankel function of first kind and zero order

G(2)(x′, x) =i4H(1)

0 (k0|x′− x|), (5)

which, with an implicit time factor exp(−iωt) (where ω is circular frequency), behaves like an outgoing cylindrical wavefor large argument.

A single-layer acoustic potential with density u, evaluated on a surface z = h(x), defined, using G(2), as

(Su)(x′) =

G(2)(x′, xh)u(xh)ds, (6)

where xh = (x, h(x)), ds =√gdx, where g = 1 + [h′(x)]2 is the determinant of the metric tensor, (h′

= dh/dx), alongwith a double layer potential (defined later), contributes to integral equation formulations of scattering and diffraction fromsurfaces through Green’s theorem [2,3]. The vector derivative ∂ ′

j , j = 1, 2, of Eq. (6) using the vector derivative of Eq. (3) (seton the surface) in Cartesian coordinates produces a term of the form

δj2sgn(z ′− h(x)), (7)

(sgn is the sign function) which is discontinuous across the surface h(x) as z ′ approaches the surface from above and below.However, the surface normal nj = δj2 − h′δj1 and the tangent tj = δj1 + h′δj2 both produce discontinuities using thisrepresentation, which leads to an incorrect tangential discontinuity of the derivative of the single-layer potential. With thisrepresentation the tangential discontinuity vanishes only for a flat surface.

We thus must change the spectral analysis to retain the correct normal discontinuity and to get zero for the tangentialdiscontinuity in the general case of a rough surface. The approach here is to redefine the Fourier transform in Cartesiancoordinates into a tangent-normal transform system (Section 2). The resulting vector derivative has the correct discontinuityresult as a distribution (Section 4) and the approach can be extended to the vector derivative of a double-layer acousticpotential (Section 7).

We refer to other derivations in several books [2,4–6]. These latter treat the problem in coordinate space, and webelieve the development can sometimes be cumbersome. Our previous paper [7] treated the problem in transform space aswell as a tangent-normal system, but still focused more on results in coordinate-space. Here the methodology is to focusimmediately on transform-space, and this yields a simpler derivation as well as new representations for single- and double-layer potentials and all their derivatives both on and off the surface.

We also note that the methodology we develop is not a function-space based method as is common in the moremathematical literature. The work is Fourier analysis in a tangent-normal system. The mathematical foundation is thusthe existence of these Fourier integrals as well as distribution theory. This is implicit throughout the paper. The results arevalid for an unbounded smooth surface.

2. The free-space Green’s function in tangent-normal transform space

There are two tangent-normal systems, one associated with the source coordinate x and the second with the fieldcoordinate x′. In single and double layer potentials, the source coordinate is always on the surface, and we choose thissystem for the development. The unit tangent (t) and normal (n) vectors for a surface z = h(x) are

t =1

√g(e1 + h′(x)e2), (8)

J.A. DeSanto / Wave Motion ( ) – 3

and

n =1

√g(e2 − h′(x)e1), (9)

in terms of unit vectors e1 = (1, 0) and e2 = (0, 1). Both t and n are defined on the whole space. Inverting the system yields

e1 =1

√g(t − h′(x)n), (10)

and

e2 =1

√g(h′(x)t + n). (11)

Any vector k can be expanded on this basis

k = k1e1 + k2e2, (12)

or on the tangent-normal basis

k = kt t + knn, (13)

and we have the obvious relations

kt =1

√g(k1 + h′(x)k2), (14)

and

kn =1

√g(k2 − h′(x)k1), (15)

where k2 = k21 + k22 = k2t + k2n. Note that while kt and kn are functions of x, k2t + k2n is not. The transformation from (k1, k2)to (kt , kn) has Jacobian equal to one (it is just a rotation) and so we can write the two-dimensional spectral representationfor G(2) in tangent-normal coordinates following Eq. (2) as

G(2)(x′, x) =1

(2π)2

exp(i[ktU + knW ])

k2t + k2n − k20dktdkn, (16)

where

U = t · (x′− x) =

1√g(x′

− x + h′(x)(z ′− z)), (17)

and

W = n · (x′− x) =

1√g(z ′

− z − h′(x)(x′− x)). (18)

We can also integrate Eq. (16) in the complex kn-plane. The poles are at kn = ±Kt where

Kt =

k20 − k2t , |kt | 6 k0

= ik2t − k20, |kt | > k0. (19)

The result is a Weyl-type representation in tangent-normal coordinates

G(2)(x′, x) =iπ

(2π)2

1Kt

exp(i[ktU + Kt |W |])dkt . (20)

This representation, although straightforward to derive, does not appear to occur in the literature. It can be used in single-layer potentials, and is used here, with Eq. (16), as the basis for all the layer-potential results.

4 J.A. DeSanto / Wave Motion ( ) –

3. Vector Derivatives of G(2)

We discuss the vector derivatives of Eq. (16). There are two vector derivatives, the first with respect to the field variablex′ used in the derivative of a single layer potential, and the second with respect to the source variable x used in double layerpotentials discussed later. They are obviously the negative of each other. Direct differentiation of Eq. (16) with respect tothe field variable using the relation

∂ ′

j [ktU + knW ] = kt tj + knnj, (21)

yields tangent and normal integrals

∂ ′

jG(2)(x′, x) =

i(2π)2

tj

kt exp(i[ktU + knW ])

k2t + k2n − k20dktdkn +

i(2π)2

nj

kn exp(i[ktU + knW ])

k2t + k2n − k20dktdkn. (22)

The integrals can be integrated in the complex kn-plane to yield

∂ ′

jG(2)(x′, x) = −tjS1(x′, x) −

14π

njsgn(W )

exp(i[ktU + Kt |W |])dkt , (23)

where

S1(x′, x) =14π

ktKt

exp(i[ktU + Kt |W |])dkt . (24)

In the limit asW → 0, the normal integral in Eq. (23) is a delta function inU . Subtract this term off andwe canwrite (settingthe source variable on the surface h)

∂ ′

jG(2)(x′, xh) = −tjS1(x′, xh) − njS2(x′, xh) −

12njsgn(W )δ(U), (25)

where

S2(x′, x) =sgn(W )

4π

exp(iktU)[exp(iKt |W |) − 1]dkt . (26)

The vector derivative of G(2) thus has obvious tangent and normal components. Both S1 and S2 have the same form if thevariables x′ or x are off or on the surface. S2 can also be expressed as a principal value integral in two-dimensions

S2(x′, x) =−i

(2π)2

K 2t exp(iktU + iknW )

k2n − K 2t

P1kn

dktdkn (27)

where P stands for the principle value of the kn = 0 singularity. S2 is thus the tangent-normal spectral integral version of theprincipal value of the normal derivative used in surface integral equations. So far as we know, these specific representationsdo not occur in the literature.

Using the definitions ofW and U on the surface, the distribution in Eq. (25) can be rewritten as

sgn(W )δ(U) = sgn(z ′− h(x))δ(U), (28)

and in the limit as z ′→ h(x′) as

sgn(W )δ(U) → sgn(z ′− h(x′))δ(x′

− x)/√g. (29)

The distributional discontinuity of this exterior vector derivative as z ′→ h(x′) from above (+) and below (−) is

[∂ ′

jG(2)(x′

h, xh)]+

−= lim

z′→h(x′)+∂ ′

jG(2)(x′, xh) − lim

z′→h(x′)−∂ ′

jG(2)(x′, xh), (30)

and, using Eq. (25), is thus in the normal direction

[∂ ′

jG(2)(x′

h, xh)]+

−= −njδ(x′

− x)/√g. (31)

(Note that the integral S2 is not distributionally discontinuous). The normal derivative discontinuity is thus

[n′

j∂′

jG(2)(x′

h, xh)]+

−= −δ(x′

− x)/√g, (32)

and the tangential derivative is continuous

[t ′j ∂′

jG(2)(x′

h, xh)]+

−= 0. (33)

The results stand in contrast to the standard Weyl results in Cartesian coordinates in Section 1.

J.A. DeSanto / Wave Motion ( ) – 5

4. Vector derivatives of single-layer acoustic potentials

For a single layer acoustic potential Eq. (6), we can write the surface limits for its vector derivative using Eq. (25) as

[∂ ′

j (Su)(x′

h)]±

= −

tjS1(x′

h, xh)u(xh)ds −

njS2(x′

h, xh)u(xh)ds ∓12n′

ju(x′

h). (34)

The vector derivatives of the single layer potential thus include two surface integrals whose densities are in the tangent andnormal directions. (Note, obviously, the tangent and normal parts of the densities are functions of the integration variable,not the exterior variable). They also contain both integral functions S1 and S2 from Eqs. (24) and (26). These latter functionsare defined on both the exterior surface point x′

h, and the interior surface point xh, andwe should comment on this. S1(x′

h, xh)is the tangential derivative of G(2) (see Eq. (44)), and when the exterior and interior arguments coincide (x′

= x), it issingular, behaving like (x′

− x)−1. Care must be taken when evaluating this integral and analogous integrals in Eqs. (37)and (69) where S1(x′

h, xh) occurs. Without going into a great deal of detail, the integral can be decomposed as a principalvalue integral excluding an ϵ-neighborhood, plus the integral around the ϵ-neighborhood (x′

− ϵ, x′+ ϵ) of the point. The

latter integral can be evaluated exactly using a Taylor expansion of the density to get a result proportional to ln(−1) oriπ , independent of ϵ. An alternate method is to use the Dirac–Plemelj distributional relation on the singularity (see [1]).In addition, the normal derivative of G(2) is not singular when the exterior and interior arguments coincide, and S2(x′

h, xh),derived from the normal derivative by subtracting the discontinuity, is also not singular when the arguments coincide. Infact, it vanishes when this occurs (see Eq. (26)).

From Eq. (34), the normal derivative of the single layer potential is thus discontinuous

[n′

j∂′

j (Su)(x′

h)]+

−= −u(x′

h), (35)

and the tangential derivative is continuous

[t ′j ∂′

j (Su)(x′

h)]+

−= 0. (36)

These are the standard results, in contrast to those of the Weyl representation in Cartesian coordinates. In addition, thelimiting cases of the normal derivative of the single layer potential are

[n′

j∂′

j (Su)(x′

h)]±

= −n′

j

tjS1(x′

h, xh)u(xh)ds − n′

j

njS2(x′

h, xh)u(xh)ds ∓12u(x′

h). (37)

So far as we know, Eqs. (34) and (37) are new representations for these limiting cases in terms of the spectral integrals Eqs.(24) and (26). (Note, again obviously, that the first integral term in Eq. (37) does not vanish.)

5. Double-layer acoustic potentials

The vector derivative of G(2) with respect to the source coordinate is just the negative of the vector derivative of the fieldcoordinate

∂jG(2)(x′, xh) = −∂ ′

jG(2)(x′, xh). (38)

(To be notationally clear, the sequence of operations on the left-hand side is to first differentiate with respect of the vectorx, and then to set the result on the surface.) Then, following Eqs. (31) and (33), we have the distributional discontinuities

[nj∂jG(2)(x′

h, xh)]+

−= δ(x′

− x)/√g, (39)

and

[tj∂jG(2)(x′

h, xh)]+

−= 0. (40)

Thus for a double layer potential with density v defined as

(Dv)(x′) =

nj∂jG(2)(x′, xh)v(xh)ds, (41)

we have, using Eqs. (25) and (38), the limiting cases

[(Dv)(x′

h)]±

=

S2(x′

h, xh)v(xh)ds ±12v(x′

h), (42)

and the corresponding discontinuity

[(Dv)(x′

h)]+

−= v(x′

h). (43)

6 J.A. DeSanto / Wave Motion ( ) –

Again, the discontinuity is the standard result, and we believe that the limiting cases, in terms of the spectral integral S2, arenew results. Note that the limiting cases of the double-layer potential only contain the functional integral S2 whereas thevector derivative of the single-layer potential equation (34) contained both S1 and S2. Also, to repeat, S2 in Eq. (42) is notsingular.

In addition, using Eqs. (25) and (38) we can derive relations off the surface which we use later. S1 is the tangentialderivative

tj∂jG(2)(x′, xh) = S1(x′, xh), (44)

and S2 is that part of the normal derivative minus the discontinuity

nj∂jG(2)(x′, xh) = S2(x′, xh) +12sgn(W )δ(U). (45)

These are used later in the paper.

6. Second derivatives of G(2) and representations

The second vector derivatives of G(2) are used when differentiating the double layer potential. In this case one derivativeis with respect to the source coordinate and one with respect to the field coordinate. Using Eq. (21) and noting that

∂m(kt tj + knnj) = 0, (46)

we can differentiate Eq. (20) directly to yield

∂ ′

j ∂mG(2)(x′, x) = tj tmD1(x′, x) + (tjnm + tmnj)D2(x′, x) + njnmD3(x′, x), (47)

where

D1(x′, x) =1

(2π)2

k2t exp[i(ktU + knW )]

k2t + k2n − k20dktdkn, (48)

D2(x′, x) =1

(2π)2

ktkn exp[i(ktU + knW )]

k2t + k2n − k20dktdkn, (49)

and

D3(x′, x) =1

(2π)2

k2n exp[i(ktU + knW )]

k2t + k2n − k20dktdkn. (50)

They are related

D3(x′, x) + D1(x′, x) = k20G(2)(x′, x) + δ(U)δ(W ). (51)

We also have a symmetry relation

∂ ′

j ∂mG(2)(x′, x) = ∂ ′

m∂jG(2)(x′, x), (52)

and, if both derivatives are with respect to the same variable

∂ ′

j ∂mG(2)(x′, x) = −∂j∂mG(2)(x′, x). (53)

The latter relation is useful in deriving specific relations for the Di functions in terms of derivatives of G(2). With x on thesurface, first rewrite Eq. (47) as

∂m∂jG(2)(x′, xh) = −tj tmD1(x′, xh) − (tmnj + tjnm)D2(x′, xh) − njnmD3(x′, xh). (54)

Multiply Eq. (54) by tj, and bring the tangent under them-differentiation to get

∂m tj∂jG(2)(x′, xh) = (∂m tj)∂jG(2)(x′, xh) − tmD1(x′, xh) − nmD2(x′, xh). (55)

Use the result

∂m tj = δm1njC(x), (56)

where C(x) is the curvature

C(x) =h′′

(1 + (h′)2)3/2. (57)

J.A. DeSanto / Wave Motion ( ) – 7

Multiply the resulting equation by tm to get a relation for D1

D1(x′, xh) = C(x)nj∂jG(2)(x′, xh) − tm∂m(tj∂jG(2)(x′, xh)). (58)

(The second tangential derivative termwill be integrated by parts later in the double-layer derivative).Multiply the resultingequation by nm to get a relation for D2

D2(x′, xh) = −h′(x)C(x)nj∂jG(2)(x′, xh) − nm∂m(tj∂jG(2)(x′, xh)). (59)

Multiply Eq. (54) by nj and bring the normal under them-derivative. Use

∂mnj = −δm1tjC(x). (60)

The result is

∂m(nj∂jG(2)(x′, xh)) = −δm1C(x)tj∂jG(2)(x′, xh) − tmD2(x′, xh) − nmD3(x′, xh). (61)

Multiply Eq. (61) by tm to yield a second relation for D2

D2(x′, xh) = −C(x)tj∂jG(2)(x′, xh) − tm∂m(nj∂jG(2)(x′, xh)), (62)

(the latter term of which can be integrated by parts, unlike the second term in Eq. (59)), and multiply Eq. (61) by nm to yielda relation for D3

D3(x′, xh) = h′(x)C(x)tj∂jG(2)(x′, xh) − nm∂m(nj∂jG(2)(x′, xh)). (63)

The functions Di in the second vector derivatives of G(2) in Eq. (47) can thus be defined as various combinations of normal,tangential, and second normal and tangential derivatives of G(2) in coordinate space via Eqs. (58), (59), (62) and (63), or asspectral integrals in two-dimensions via Eqs. (48)–(50). Eqs. (48)–(50) can also be integrated in the complex kn-plane toyield one-dimensional Weyl-type integrals.

7. Derivatives of the double-layer acoustic potential

The double-layer potential equation (41) can be written as

(Dv)(x′) =

∂mG(2)(x′, xh)nmv(xh)ds. (64)

Take the vector derivative with respect to the exterior variable

∂ ′

j (Dv)(x′) =

∂ ′

j ∂mG(2)(x′, xh)nmv(xh)ds. (65)

Use Eq. (47) to replace the derivative of G(2) to get

∂ ′

j (Dv)(x′) =

[tjD2(x′, xh) + njD3(x′, xh)]v(xh)ds. (66)

Replace D3 in Eq. (66) using Eq. (51) to get

∂ ′

j (Dv)(x′) = δ(z ′− h(x′))n′

jv(x′

h) + k20

G(2)(x′, xh)njv(xh)ds +

[tjD2(x′, xh) − njD1(x′, xh)]v(xh)ds. (67)

Using Eqs. (58) and (62), the latter integral in Eq. (67) can be integrated by parts with the integrated term vanishing. Inparticular, the second terms in Eqs. (58) and (62), the tangential derivatives, are integrated by parts. The curvature termscancel and we get

∂ ′

j (Dv)(x′) = δ(z ′− h(x′))n′

jv(x′

h) + k20

G(2)(x′, xh)njv(xh)ds

+

[tjnl∂lG(2)(x′, xh) − nj tl∂lG(2)(x′, xh)]tm∂mv(xh)ds. (68)

To get the surface limiting cases, replace the normal and tangential derivatives of G(2) using Eqs. (44) and (45) in Eq. (68) toyield

[∂ ′

j (Dv)(x′

h)]±

= k20

G(2)(x′

h, xh)njv(xh)ds +

[tjS2(x′

h, xh) − njS1(x′

h, xh)](tm∂mv(xh))ds ±12t ′j (t

′

m∂ ′

mv(x′

h)). (69)

8 J.A. DeSanto / Wave Motion ( ) –

The delta term in Eq. (68) does not contribute in the limit. The S1 term in Eq. (69) contains the only singularity and it can beevaluated following the discussion in Section 4. The result for the vector derivative of the double-layer potential evaluatedon the surface is the sum of a single-layer potential with vector density, an integral involving the tangential derivative ofthe density, and a discontinuous term in the tangential direction

[t ′j ∂′

j (Dv)(x′

h)]+

−= t ′m∂ ′

mv(x′

h), (70)

equal to the tangential derivative of the density. The results, Eqs. (68) and (69), modulo notational differences, agreewith [2](see Theorem 2.23), as they should. They are an equivalent but different representation.

8. Summary

We have presented a new development for the spectral decomposition of the two-dimensional free-space Green’sfunction and its derivatives in a tangent-normal system of coordinates. This yielded new representations for the vectorderivatives of the single-layer acoustic potential as well as the double-layer acoustic potential and its vector derivatives.The results were presented for these potentials both when the exterior coordinate was off the surface as well as the limitingcases when the exterior variable approaches the surface from above and below the surface. The latter is the representationsused in integral equation formulations of scattering problems.

All the results could be expressed in terms of two one-dimensional spectral integrals S1 and S2 from Eqs. (24) and (26).S1 was simple in coordinate-space (Eq. (44)), but when used in integral equations has a singularity when exterior andintegration variables coincide. We briefly described methods to handle this for computational purposes in Section 4. S2contained a subtraction (Eq. (45)) related to the normal derivative discontinuity, and it was not singular when the exteriorand integration arguments coincided. Derivatives of the single-layer potential contained both S1 and S2, Eqs. (34) and (37),while the limits of the double-layer potential equation (42) contained only S2. The limits of the vector derivatives of thedouble-layer potential contained both S1 and S2 (Eq. (69)). All the classical results for these potentials can thus be expressedin a unified manner using only the two spectral integrals S1 and S2, with care taken when evaluating S1.

We also argue for the simplicity of all our derivations above compared to the corresponding derivations in the literature.Our derivations and equations present an equivalent but different representation of the layer potential results.

References

[1] J.A. DeSanto, Scalar Wave Theory, Springer-Verlag, Berlin-Heidelberg-New York, 1992.[2] D. Colton, R. Kress, Integral Equation Methods for Scattering Theory, Wiley-Interscience, New York, 1983.[3] J.A. DeSanto, Scattering from rough surfaces, in: R. Pike, P. Sabatier (Eds.), Scattering, Academic, New York, 2002, pp. 15–36.[4] O.D. Kellogg, Foundations of Potential Theory, Dover, New York, 1953.[5] W.D. MacMillan, Theoretical Mechanics, The Theory of the Potential, McGraw-Hill, New York, 1930.[6] N.M. Günter, Potential Theory and Its Applications to Basic Problems of Mathematical Physics, F. Unger, New York, 1967.[7] J.A. DeSanto, k-space properties of classical single- and double-layer potentials and their derivatives, Wave Motion 17 (1993) 143–159.