Summary of the simulation results of the cylindrical waves

The convergence of these field components of the cylindrical waves is similar to the results we obtained with plane waves but the values of the calculated field components now depend, not only on k, a and p but also on the angle φ and the position of the source point O0 which has a distance d0 from the centre of the cylinder. A number of simulations have been carried out using the field equations in order to observe the changes in fields with respect to the positions of the field point and the source point. Here, we have not produced all the simulated results but some of the observations can be summarised as follows:

(1). Whenp < d0 , both|E1|and|E2|converge to constant values after a reasonable number of iterations.

(2). The number of iterations in (1) above is minimum when φ= 0. As φ increases in magnitude |E1| and |E2| need more iterations.

(3). As φ increases in magnitude the corresponding computed values of |E1| and |E2| decrease. (Plots are marked with the angle φ. See Figures 3.15 and 3.16.) (4). When d0 ≤p, there are no results. (This is not valid according to the forward equation.)

(5). As φ→0, the magnitude values of |E1| and |E2| are maximised.

We observed that the magnitude and phase values of E1 and E2 depend upon the position ofO and O0. These changes have been studied in detail to investigate the scattering difference between plane waves and cylindrical waves. This will be further discussed in Chapter 7.

Chapter 4

Three-dimensional scattering

problem

A three-dimensional scattering problem of a spherically-shaped object is considered in this chapter. A simple host model was used. Similar problems to this have been solved previously and the results are available in literature [7, 8, 14, 15]. We used some of those results to obtain a solution to the forward scattering problem of a uniform sphere inside our model. The application system is similar to that used in the cylindrical case (Figure 3.2). The object is illuminated by a microwave signal which is applied from the surface of the host.

The forward and inverse solutions provide the necessary base for a possible prac- tical application. First, the object is assumed to be a conducting sphere and the associated forward problem is solved. Then the result is modified for a more realistic situation using a non-conducting sphere. The inverse scattering problems in both cases are then solved to compute the unknown size and the location of the object. At the beginning of this chapter, some equations in wave theory found in Harrington [8] are used. The full derivations are given for completeness.

Z φ r X Y θ O C

Figure 4.1: The spherical coordinate system.

4.1

Wave functions at spherical boundaries

The spherical coordinate system (r, θ, φ) corresponds to the spherical wave functions and is shown in Figure 4.1. Consider a pointO with a distancer fromC the centre of the coordinate system. The Helmholtz equation in spherical coordinates which can be obtained using the Laplacian of ψ, is of the form

1 r2 ∂ ∂r r 2∂ψ ∂r ! + 1 r2_sinθ ∂ ∂θ sinθ ∂ψ ∂θ ! + 1 r2_sin2_θ ∂2_ψ ∂φ2 +k 2_{ψ = 0. (4.1)}

In order to solve equation (4.1) we use the method of separation of variables. It is assumed that the solution is a product of three functions of the form,

ψ(r, θ, φ) = R(r)Θ(θ)Φ(φ). (4.2)

Using equations (4.1) and (4.2), three separated equations can be found

d dr r 2dR dr ! +h(kr)2−n(n+ 1)iR= 0, (4.3)

1 sinθ d dθ sinθ dΘ dθ ! + " n(n+ 1)− m 2 sin2θ # Θ = 0, (4.4) d2_Φ dφ2 +m 2 Φ = 0, (4.5)

where m and n are constants. The above three equations have solutions which are each functions of one variable: r,θandφ, respectively. Equations (4.3) and (4.4) are related to Bessel’s equation and Legendre’s equation [8, 103], respectively. Equation (4.5) is the harmonic equation which has a solution of the form

Φ(φ) = Φm(φ) =ejmφ, (4.6)

where m is an integer. This ensures the solution is periodic with a period of 2π as it clearly should be since Φ(φ+ 2π) = Φ(φ).

The solutions to the Bessel’s equation are called spherical Bessel functions, de- noted by bn(kr), and n is a positive integer, since the functions in the solutions for

Θ are not finite at θ= 0, π unless this is so (this will be discussed in section 4.2.1). The spherical Bessel functions are related to the ordinary Bessel functions Bn+1/2 by [103]

bn(kr) =

r _π

2krBn+1/2(kr). (4.7)

The solutions to Legendre’s equation (4.4) are called the associated Legendre func- tions and, in general, these are denoted by Lm

n(cosθ). There are two linearly inde-

pendent families of solutions for Legendre’s equation:

• Pm

n (cosθ) which are the associated Legendre functions of the first kind,

• Qm_n(cosθ) which are the associated Legendre functions of the second kind. Now, the solutions to the Helmholtz equation can be found using the product in equation (4.2) as

ψ(r, θ, φ) =bn(kr)Lmn(cosθ)e

The appropriate function for Lm

n(cosθ) is selected from the two solutionsPnm(cosθ)

and Qm

n(cosθ) by considering the singularities in their domains. The solutions to

Legendre’s equation have singularities when θ = 0 or θ =π except Pm

n (cosθ) with

n as an integer (see Appendix D). We seek solutions for ψ to be finite in the range fromθ = 0 toθ=π (this problem will be discussed in the next section). Therefore,

Lm

n(cosθ) must bePnm(cosθ) and not theQmn(cosθ) as this is not finite atθ = 0 and

θ = π (see [8,16]). A general solution can be formed using a linear combination of all possible wave functions overn and m as

ψ(r, θ, φ) = ∞ X n=0 ∞ X m=−∞ Zn,mbn(kr)Pnm(cosθ)ejmφ, (4.9)

whereZn,m are constants. Equation (4.9) represents a linear combination of all the

possible elementary wave functions. According to equation (4.9), there are three different wave functions (of one variable) associated with the spherical coordinate system.