Inverse Scattering Fundamentals - Introduction to Imaging from Scattered Fields - page 47

so that if we now utilize the fact that as r approaches zero, 4π C = −1, which can
be used to solve for C as follows:

1
4π

C = (4.20)

This value of C can now be substituted back into Equation 4.11, which yields

e

ikr

(4.21)

G ( r =

0

4 π r

This can now be generalized by shifting the source back to r ′ which gives
Green's function in the general form for this application as

e

ik rr − ′

G ( rr ′ ) =

(4.22)

0

4 π rr − ′

Equation 4.22 gives Green's function in the form necessary to solve the
inhomogeneous Helmholtz equations. In this topic, all of the experiments are
done in 2-D; therefore, the 2-D form of Green's function can be expressed in
terms of a zero-order Hankel function of the first kind (Chew, 1995; Darling,
1984) as follows:

− i Hk

(4.23)

G (, ) rr ′ =

() 1

[ || rr − ′ ]

4

0

Now, an asymptotic assumption can be applied in that as r → ∞, the zero-
order Hankel function of the first kind becomes

41
8

1

Hk

()

[ || rr −= + ′ ] e

ikr ( π / 4 )

e

− −− ikr · | rr ′ |

(4.24)

0

i

π kr

where ˆ /| | Substituting Equation 4.24 back into Equation 4.23, the

approximated 2-D Green's function now becomes

r ′ = r r .

1
8

·

(4.25)

G (, ) rr ′ ≅− e

ikr ( + π / 4 )

e

− −− ikr | rr ′ |

π kr

Finally, a general equation for the scattered field can be obtained by sub-
stituting Equation 4.25 into Equations 4.6 and 4.7, which gives the following:

1
8

∫

··

·

·

Ψ ( rr , ) = e

ikr ( + π / 42 )

k Ve ( r ′ )

−−− ikr rr ′

Ψ ( r r ′ , ) d r ′

(4.26)

s inc inc

π kr

D

where ˆ

= φ φ is the unit vector that describes the direction of
the incident plane wave and ˆ

r inc (cos ,sin )

inc inc

r = φ φ is the unit vector that describes
the direction of the scattered wave. This equation is of the form of a Fredholm

(cos ,sin )