3.2  Some Properties of the Kernel Functions

The results given in this Section are extracted mainly from Burton [16], [17]. In the following r = p - q and r = |r|, Gk = Gk(p,q), G0 = G0(p,q).

3.2.1  Derivatives of G0 with respect to r

In two dimensions we have
G0
r
 = - 1
2 p
 1
r
   ,
(3.9)
2 G0
r2
 = 1
2 p
 1
r2
   .
(3.10)
In three dimensions we have
G0
r
 = - 1
4 p
 1
r2
   ,
(3.11)
2 G0
r2
 = 1
2 p
 1
r3
   .
(3.12)

3.2.2  Derivatives of Gk (k ¹ 0) with respect to r

In two dimensions we have
Gk
r
 = - i
4
 k H1(1)  ,
(3.13)
where H1(1) is the spherical Hankel function of the first kind and of order one and
2 Gk
r2
 = i
4
 k2 ( H1(1)
kr
 -H0(1) )   .
(3.14)

In three dimensions we have

Gk
r
 = eikr
4 pr2
 ( ikr - 1 ) ,
(3.15)
2 Gk
r2
 = eikr
4 pr3
 ( 2 -2ikr -k2 r2 ) .
(3.16)

3.2.3  Expressions for the normal derivatives of Gk

The following expressions hold in both two and three dimensions and for all k:
Gk
nq
 = Gk
r
   r
nq
  ,
(3.17)
Gk
up
 = Gk
r
   r
up
  ,
(3.18)
2 Gk
up nq
 = Gk
r
   2 r
up nq
 + 2 Gk
r2
   r
up
 r
nq
  .
(3.19)

3.2.4  Expressions for the normal derivative of r

The derivatives of r with respect to up and nq may be written as follows:

r
nq
 = - r.nq
r
   ,
(3.20)
r
up
 = r.np
r
   ,
(3.21)
2 r
up nq
 = - 1
r
 ( up.nq + r
up
 r
nq
 )  .
(3.22)

3.2.5  Expressions for [(2 G0)/( up nq)]

The following results can be derived from the substitution of (3.22) and (3.9),(3.10) or (3.11),(3.12) into (3.19) with k = 0:

2 G0
up nq
 = 1
2 pr2
 ( up.nq + 2 r
up
 r
nq
 )  in two dimensions,
(3.23)
G0
up nq
 = 1
4 pr3
 ( up.nq + 3 r
up
 r
nq
 )  in three dimensions.
(3.24)

3.2.6  Asymptotic Properties

In the following results, p, q Î G where G is a surface and G is smooth at p:

lim
q ® p 
 ( Gk(p,q) -G0(p,q) ) = O(r0) ,
(3.25)

lim
q ® p 
 Gk
up
 (p,q) = O(r0) ,
(3.26)

lim
q ® p 
 Gk
nq
 (p,q) = O(r0) ,
(3.27)

lim
q ® p 
 ( 2 Gk
up nq
 (p,q) - 2 G0
up nq
 (p,q) + 1
2
 k2 Gk(p,q) ) = O(r0) .
(3.28)
Return to boundary-element-method.com/helmholtz
Return to boundary-element-method.com