Cite this paper as: Michelitsch T.M., Wang J., Gao H., Levin V.M. (2004) On the Solutions of the Inhomogeneous Helmholtz Wave Equation for Ellipsoidal ...
ON THE SOLUTIONS OF THE INHOMOGENEOUS HELMHOLTZ WAVE EQUATION FOR ELLIPSOIDAL SOURCES T. M. Michelitsch, l J. Wang, H. Gao, and V. M. Levin 2 1 Max Planck Institute for Metals Research, Heisenbergstrasse 3 D-70569 Stuttgart, Germany Thomas Michelitsch
2Instituto Mexicano del Petroleo, Eje Central Lazaro Cardenas No 152 Apdo. Postal 14-805, Col. San Bartolo Atepehuacan, C.P.07730, Mexico, D.F., Mexico Valery Levin
Abstract
1.
The solution of the inhomogeneous Helmholtz equation (the 'dynamic' or 'Helmholtz potential') and its time domain representation (the retarded potentials) for an ellipsoidal source region is analyzed. They occur in many dynamic problems of mathematical physics such as wave propagation and scattering phenomena. From an aesthetic and practical point of view ID-integral representations for dynamic potentials are highly desirable. So far such representations seem to be absent in the literature. Here we close this gap for the internal dynamic potential of an ellipsoidal shell. The solution of the external space can be constructed by applying Ivory's theorem. Moreover we construct surface integral representations for inhomogeneous ellipsoidal sources for source densities of the form 2 2 P = e ( 1 - P ) f ( P 2) ,(P 2 = ax 22 + JS+ ~). Closed form solutions are found a2 a3 1 for the retarded potentials of inhomogeneous spherical sources. In the static limit the dynamic potentials coincide with well known classical results for the Newtonian potentials of ellipsoids of Dyson [4] and Ferrers [6].
Introduction
In the 19th century the determination of static (Newtonian) potentials of ellipsoids was a key subject in mathematical physics. They are defined by an inhomogeneous Poisson equation. Key contributions were given by several authors [1,2,4,6,7,8,10,15] [1,2,4,6-8, 10, 15] among others, where the main goal was to derive one-dimensional integral representations. Ferrers [6] and Dyson [4] have achieved this for the Newtonian potentials of inhomogeneous ellipsoids. Dyson [4] gave the solution in full generality for arbitrary inhomogeneous ellipsoidal sources. 115 D. Bergman et al. (eds.), Continuum Models and Discrete Systems, 115-122. © 2004 Kluwer Academic Publishers.
116
CONTINUUM MODELS AND DISCRETE SYSTEMS
In the second half of the 20th century the determination of Newtonian potentials attracted the interest of materials scientists due to their crucial importance for the solution of inclusion and inhomogeneity problems [5]. Whereas Newtonian potentials of ellipsoidal sources are exhaustively studied, dynamic potentials of ellipsoidal sources are only little touched [14, 9, 11-13,3]. Here we derive a 1D-integral representation for the Helmholtz potential of an homogeneous ellipsoidal shell l . Moreover, we give a surface integral representation for the retarded potential of an inhomogeneous ellipsoid.
2.
Dynamic potential of an ellipsoidal shell We consider an ellipsoid S with the semi-axes ai. The quantity p2 + + characterizes the internal and external space with P < 1 and
222 X2 Y2 Z2 a1 a2 a3
P > 1. The dynamic or Helmholtz potential is determined a inhomogeneous Helmholtz equation (1)
where 6.
=
iJ2 ox
C;-2
iJ2 iJ2 . +~ + C;-2 oz denotes the Laplacian and r = (x,' y,, z) oy~
.
.
mdlcates
Cartesian spatial coordinates, 13 = W~ir r > 0 indicates a damping parameter, p represents the source density. We consider here densities of the form p = 8(1- P)f(P 2 ) (We introduced the Heaviside step function 8(~) being defined as 8(~) = 1 if ~ > 0 and 8(~) = 0 if ~ < 0). g is determined by the Fourier integral g
where
= _1_
(21f)3
J
e
ik·r
p(k) d:3k k2 - (32
(2)
p is the Fourier transform of the density p defined by p(k)
=
J
e- ik .r p(r)d 3 r
(3)
For the the spatial coordinates we put Xi = Paini (where n 2 = 1). Then g can be rewritten as
g(r,a,(·)) =
10
1
dp'f(p,2)p'(r,p', (.))
(4)
(.) stands for either (3 in the frequency domain or time t. By using this equation we have shown recently [13] that the dynamic potential for an inhomogeneous 1For
details we refer to the recent paper [13].
SOLUTIONS OF INHOMOGENEOUS HELMHOLTZ EQUATION. . .
density of the form 8(1 - P)f(P 2) in the static limit classical results of [6,4]. From (4) we observe that
f3
117
-+ 0 reproduces
(5)
where 1'0 corresponds to the potential of an infinitely thin ellipsoid shell at r = PO. The the space-time representation yields [13]
(r,p', t)
=
8(tl:--r ~KI=l dO(K)s2(K) [6 (t -15 (t + ~(PK . n + p'))] t
where s(K) = 1/
/(2
/(2
/(2
( ---t + ---+ + ---+ ). a1 a2 a3
~(PK. n + p'))
(6)
(6) fulfills the wave equation
(~ -
12 [;~ + 1]2) (p' (r, p', t)) + r5( t)r5(P - p') = 0 (7) c ut Integration of (6) in (4) yields for the retarded potential of a source distribution ofthe form 8(1 - P)f(P 2)r5(t) the 2D-integral
xf ((cH~/.r)2) '('/2
8 (1 _(CHn/ 'r)2)
(8)
'1"2
r' = aIn~2 + a~n~2 + a§n~2 denotes a parameterization of the radius of the ellipsoidal shell. Expression (8) describes the propagation of outgoing waves. For a spherical source ai = a this expression can be evaluated in closed form and yields
(9) where F and f are related by f(A) = d~~A). Now we consider the frequency-space domain representation of (6) which yields [13] ( Paini, f3, p')
~ 47f
r
J KI=1
dO(K)s2(K)[8(PK . n
+ p')ei;5s(K) [(PK.n+p')]
1
-8 ( -(PK . n
+ pI)) e-i3s(K)[(PK.n+p')]],
(10)
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CONTINUUM MODELS AND DISCRETE SYSTEMS
=
and can be transformed into [13] (p' .f
a1 a 2 a 3
dr2(n') [8(r. n ' r'
.lln'l=l
41f
A
1)
.
+ r ' )ei .8 (r.n'+r') (11)
-8( -(r· n ' + r ' ))e- i3 (r.n'+r')] These relations yield for the internal space
iP(Paini, (3,p')p
