Propagation of electromagnetic waves in a longitudinally...

3. Solution and Results

[7] It is easy to see that the Maxwell equations are reduced to the Helmholtz equations for H_z of the following form:

with the boundary condition partial

H_z/

n|_G, where G is the outline of the region and n is a normal to it.

[8] The point ( 0,h ) is a singular point of the Helmholtz operator. In order to get rid of the singularity we perform a conform transformation of coordinates transforming the lower and upper walls of the waveguide into planes. It is easy to see that the transformation w = (t²+h²)^1/2 (where t=x+iy are the physical coordinates, w=x + i h are the conform coordinates, and the branch of the root is taken in such a way that Im(t²+h²)^1/2>0 ) satisfies this requirement.

[9] It follows from the condition x₀ d that x₀ d. So we find

and then the initial equation takes the form

(1)

Here

is the Lame coefficient of the performed transformation. Inverting (1) by the Kirchoff's formula, we obtain the integral equation

(2)

where G(x, h; x',h') is the Green function for the Neumann problem and

(3)

The solution for the vacuum having a correct behavior at the infinity is taken as H_z⁽⁰⁾. Under this choice and taking into account that h_x

1 if |x|

, the limitation of the integral operator in L₂ is provided.

[10] We will solve equation (2) by the sequential approximations method. For its application one has to be sure that the norm of the integral operator is less than unity:

Using asymptotics for the Lame coefficient, one can show that under the condition k< p/d (a low-frequency case, i.e., a one-mode waveguide) the operator norm has the form:

Therefore, if the conditions kh(h/d)<1 and k<p/d are fulfilled, one can apply the sequential approximations method. We take H_z⁽⁰⁾ (3) as a zero approximation. Then we find the first approximation using

where

Here A_nq and B_nq are the coefficients of propagation and reflection of the q th wave caused by its interaction with the n th wave.

[11] Since all waves (except the zero one) are local waves, x₀ d, and |x| d, the only significant input into the field is provided by the A₀₀ and B₀₀ coefficients. Therefore the further solution is reduced to finding these coefficients.

[12] Neglecting by the terms of infinitesimal of higher orders of the kh(h/d) parameters we obtain

under x<-d and

under x>d.

[13] Since all local waves are exponentially small (because x₀ d and |x| > d ), one can write an expression for the field taking into account only the zero modes. Taking into account that the norm K has an order of kh(h/d), one has in the first approximation to exclude all the terms of the higher infinitesimal degree. This leads to a fact that in the first approximation one can obtain a correction to the zero approximation only in phase.

[14] To the left from the barrier (that is in the opposite relative to the source part of the waveguide) the expression for the field has the form

(4)

whereas to the right from the barrier it is written as

(5)

The electric field at the barrier in the zero approximation is written in the following way:

The behavior of the field in the vicinity of a singular point is mainly determined by the Lame coefficient. In the zero approximation in the vicinity of the singular point ( x=0, h=0 )

which corresponds to the Meixner condition for a semiplane.

[15] Transferring (4) and (5) into physical coordinates and neglecting the terms of higher orders by kh(h/d) we obtain

under x<-d and

under x>d.

[16] Taking into account that kh(h/d) 1 one can rewrite (4a) and (5a) in the form

under x<-d and

under x>d, where F_1,2 are the corrections to the phase to the left and to the right from the barrier, respectively.

[17] Thus the obtained solution does not contain terms with the propagation and reflection coefficients depending on x. Both, the reflected and falling waves have constant amplitudes (at least with the accuracy to the second order of infinitesimal) also at |x|>d. Therefore the local irregularity (i. e., the barrier) provides the main influence on the propagation, but not the distributed one (curved upper wall).

Figure 2

[18] Now we come to a consideration of the problem of diffraction in a similar waveguide but with a plain upper wall (Figure 2). Let us assume that from the left a normal wave falls on the barrier. The wave is a zero one for the TM field ( U₀=e^ikx ) and a first one for the TE field ( U₀ = e^il₁x sinpy/d, where U corresponds to H_z and E_z for the transversal magnetic (TM) and transversal electric (TE) fields, respectively). This is equivalent to consideration of the field of a dipole located far from the obstacle in a one-mode waveguide. According to the general scheme of MQGF we present the field in the waveguide as a sum of normal waves:

[19] TE case: \newline In the W₁ region

In the W₂ region

(6)

TM case: \newline In the W₁ region

in the W₂ region

(7)

Then at the boundary of the regions we will join solutions assuming equality of the fields and their normal derivatives and also their being equal to zero at the barrier.

[20] According to the general scheme of MQGF, the Green function of the Laplace operator (satisfying the same boundary conditions as the solution) is used for the regularization of the system (6) or (7). Multiplying the fields and their derivatives by the normal derivative of the Green function and by the function itself, respectively, we integrate the fields over the boundary between regions. It can be easily shown [Konorov and Makarov, 1987] that such procedure is equivalent to a conversion of the Laplace operator.

[21] The functional equations obtained by the joining are expanded in a series about some complete system of functions and lead to a system of linear algebraic equations. According to the general MQGF theory [Konorov and Makarov, 1987; Verbitsky, 1981] the system is a regular one allowing for a truncation. Its matrix elements are double integrals of the Green function and functions used in the expansion.

[22] To find the Green function, we use a conformal transformation of coordinates

converting the initial waveguide into a plain one. The branch of the root is fixed in the following way:

Then the Green function for the Dirichlet's and Neumann problems takes, respectively, the form

and

We consider first the Dirichlet's problem. Following the general scheme we obtain the functional equation

(8)

Applying the second Green formula to the Green function and function in the and regions we find

(9)

where

Using (9), we get rid of the derivative of the Green function in (8):

(10)

Equation (10) at different x' is separated into two equations:

[23] In the W₁ region

(11)

In the W₂ region

(12)

where the denotation r_n=(pn/d) + i l_n is introduced.

[24] Expressions (11) and (12) are functional equations. To move to linear algebraic system, one has to expand these equations in some complete system of functions. It is convenient to use { sin (pp/d)y'} as such functions. Multiplying (11) and (12) by sin (pp/d)y' and integrating with respect to y' between 0 and d we obtain

(13)

and

(14)

Taking into account that (13) and (14) should be fulfilled at any x', we take x'=0 and obtain the final system of equations

(15)

in which

(16)

In a similar way we derive the system of equations in the case of the Neumann problem. Its principal difference from the Dirichlet's problem is that the Green function does not tend to zero at the infinity. So the system of functions { cos (pp/d)y'}_p0 is not complete and one should add to it [1; x']: projecting to these functions would give "zero" equations:

(17)

where

Thus regular systems of equations for the Dirichlet's and Neumann problems are obtained. Equations (15) and (17) allow for a reduction and (finding matrix elements) one can easily obtain the solution with the required accuracy, that is, in the numerical sense the problem may be considered as solved. However, if one is interested in analytical form of the solutions, one should study the B_mn dependency on the number of the element and waveguide parameters.

[25] In analytical solving the MQGF problems the main difficulty is determination of the matrix elements of the equation system, the elements being double integrals of the Green functions (16). To obtain the analytical dependence of the matrix elements on the problem parameters, we use the method suggested by Konorov and Makarov [1987].

[26] To do this, one has to obtain the expression of the matrix integrals in terms of the expansion coefficients of a conformal transformation by exponents. By rather inconvenient transformations one can obtain an expression for B_mn in the form of finite sums, the complexity of these sums growing quickly with an increase of the matrix element number.

[27] We will not describe in details the calculation of the matrix elements but present the first approximations (solutions of the first equation of the system only) for the Dirichlet's ( I₁⁽¹⁾ ) and Neumann ( I₀⁽¹⁾ ) problems:

(18)

It is worth noting that unlike the first method considered there are no limitations to the barrier height h in this solution.

[28] Analytical expressions for the second and third approximations are also obtained; however, they are rather massive. Comparing the solutions of systems containing one, two, and three equations, one can study the correction introduced into the solution by every new equation and evaluate the accuracy of the final result.

Figure 3

[29] Analyzing the one-mode waveguide one can see that to find the solution at small and large h it is enough to consider one equation. If h approx

d/2, to have more exact determination of I₁ one should take the system of two equations. The third equation introduces the correction not exceeding 10%. Figure 3 shows the behavior of the modulus of the propagation coefficient of the first mode as a function of the barrier height for d=10 and k=0.5.

Figure 4

[30] The behavior of the corrections in the first and second approximations has a similar character for the TM and TE fields. Therefore in the Neumann problem the second approximation is enough for finding the propagation coefficient of the zero wave in a one-mode waveguide (Figure 4, d=10 and k=0.2 ).

[31] Now we consider solution of the Neumann problem under small height of the barrier. In this case, expression (18) for the coefficient at the zero wave may be approximately written in the following form:

(19)

We compare it to (4a) which is a solution of the similar problem (the difference lies in a curving of the upper wall) by the semi-inversion method. In (4a) the coefficient at the propagating wave has the form

(20)

One can see that solutions (19) and (20) are almost the same. Therefore the curved upper wall gives almost no input into wave propagation. Thus one can conclude that, solving problems of propagation in the Earth-ionosphere waveguide, the main influence on the field is provided by local irregularities of the Earth surface (especially at the presence of angular points), but not smooth perturbations in the ionosphere.