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presentation as in (4.5) with that of a plane wave in (4.2). Consequently, (4.5) looks like a plane wave in the direction k0 kz z kr r, when r ! 1. In the spherical coordinate system fr; y; jg, the scalar wave equation, which describes propagation of spherical waves in free space, can be written as in References [1 3]   1 @ 2 @ 1 @ @ 1 @2 k2 C r 0 r 2 2 sin y 2 2 r 2 @r @r r sin y @y @y r sin y @j2 4:6
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As shown in [3], the spherical wave can be approximated by expfikrg. Thus, one can r represent the spherical wave as a plane wave, when r ! 1. 4.1.2. Green s Function Presentation Green s function is used in the description of any arbitrary source in an unbounded homogeneous medium, taking into account that each source s r can be represented as a linear superposition of point sources. Mathematically this can be expressed as s r dr0 s r0 d r r0 4:7
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The scalar wave equation with the source in the right-hand side can be presented as r2 C r k2 C r s r 4:8
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and the corresponding equation for the Green s function in an unbounded homogeneous medium can be presented as r2 G r; r0 k2 G r; r0 d r r0 : The solution of Equation (4.9) is [1 3] G r 1 expfikrg 4p r 4:10 4:9
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and the corresponding solution of (4.8) is C r dr0 G r; r0 s r0 :
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The geometry of the source s r in a space with volume V is shown in Figure 4.1. Using Equation (4.10) one can easily obtain a general solution for the inhomogeneous
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s(r) V Source
FIGURE 4.1. Geometrical presentation of a source s r inside an arbitrary volume V bounded by a surface S.
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Equation (4.8) given by expfikjr r0 jg 0 s r : C r dr0 4pjr r0 j
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This presentation is valid for any component of an EM-wave, propagating in free space, and it satis es the principle of linear superposition of point sources (4.7) for any real source of radiation. 4.1.3. Huygen s Principle This concept is based on presenting the wave eld far from any sources as shown in Figure 4.2. Here the point of observation A can be either outside the bounded surface S, as shown in Figure 4.2a, or inside, as shown in Figure 4.2b. In other words, according to Huygen s principle, each point at the surface S can be presented as an elementary source of a spherical wave, which can be observed at point A. Mathematically, the Huygen s concept can be explained by the use of Green s function. First, we multiply the homogeneous Equation (4.1) (without any source) by G r; r0 and the inhomogeneous Equation (4.9) by C r . Substracting the resulting equations from each other and integrating over a volume V containing
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0 A S V (b) (a) n A n
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FIGURE 4.2. Geometrical explanation of the Huygen s principle in a bounded surface when the receiver is located at point A outside and inside the bounded surface and when the transmitter is located at point O.
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Source S s(r)
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FIGURE 4.3. The geometry for derivation of Green s theorem for the two different boundary conditions at the bounded surface S; Neumann and Dirihlet.
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vector r0 (see Fig. 4.3), yields C r drbG r; r0 r2 C r C r r2 G r; r0 c
from which we can obtain Green s theorem or the second Green formula [1 3]: ! @C r @G r; r0 dr G r; r0 r2 C r C r r2 G r; r0 ds G r; r0 C r @n @n
4:14 This formula can be simpli ed using different boundary conditions on surface S. Using @g the relation between arbitrary scalar functions f and g: fg n f @n, we can write 0 C r dsn G r; r0 rC r C r rG r; r0 4:15
Next, if we assume that n rG r; r0 0 at the boundary surface S, de ned by the radius vector r, Equation (4.15) becomes C r0 d s G r; r0 n rC r 4:16
If the boundary condition rG r; r0 0 is applied at the surface S, then Equation (4.15) becomes C r dsC r n rG r; r0