Influence of internal gravity waves on the reflection of...

3. Determination of the Hertz Potential

Figure 2

[6] To solve the above formulated problem, we use the conformal mapping x(u) transforming the z leq

h region of the complex plane u = x + i z into some one cohesive region z₁ leq

f(x₁) of the x = x₁ + i z₁ plane (see Figure 2). Under such transformation, the z = h plane is transformed into a curved surface z₁ = f(x₁), and the boundary problem (2)-(3) is rewritten in the (x₁, z₁) coordinate system in the form

(4)

(5)

Here

n is the derivative along the normal to the line of the curved boundary, and the values

and

are, respectively, modulus of the derivative of the mapping x(u) and of the inverse to it transformation u(x) (it is evident that b = 1/n ).

[7] Using the uncertainty in the b(x₁, z₁) function, we choose it in such a way that the equality

(6)

(where g = i k₀ h₀ = const ) would be fulfilled. Therefore, at the boundary

Finally we come to the problem on electromagnetic wave propagation in the semispace with the known inhomogeneous refraction index and one-dimensionally curved boundary having a constant impedance.

[8] To solve the boundary problem (4), (6), we apply the operator

(7)

to the right-hand and left-hand sides of the equation (4). Here partial

s is the derivative along the direction of the tangent line to the s(x₁,z₁) curve (see Figure 2). We put on the family of curves s the following requirements:

[9] 1. Only one line s passes through every point of the (x₁,z₁) plane.

[10] 2. In the point of the boundary z₁ = f(x₁) the direction of the tangent line to the corresponding curve of the s family coincides with the direction of the normal n along which the differentiation in the boundary condition (6) is performed (see Figure 2 where the s(x₁,z₁) curves are shown by dashed curves). Therefore at the points of the curved boundary, the following equality partial / partial s = partial / partial n is true.

[11] For the sake of certainty, we choose particular family s as a set of curves that are transformed into the lines x = const under the inverse conformal mapping u(x), which transforms the considered curved semispace z₁ leq f(x₁) of the complex plane x&nbsp;=&nbsp;x₁ + i z₁ into the z leq h region of the u = x + i z plane. Under this transformation, the curve z₁ = f(x₁) is transformed into the line z = h.

[12] In an analogy with Zaboronkova et al. [2003] we perform conformal transformation in variables (x₁, z₁) and then come back to the initial coordinates (x,z). Zaboronkova et al. [2003] demonstrated that for the function

satisfying the Dirichlet's zero condition at the upper boundary, it is true

(8)

Therefore we come to the following equation in the initial variables (x,z):

(9)

In the same way as Al'pert et al. [1953] and Bezrodny et al. [1984], we will consider the case of a grazing propagation when the horizontal distance x from the source to the observational point is much larger than the vertical distance h from this point to the disturbed boundary. In this case the grazing angle of the falling wave is q_s

1. In such case, with the accuracy of the terms sim

q_s², the following relation is true:

Then we are able to rewrite the equation (9) in the form

(10)

where

Here and further the function 1+N(x, z) is interpreted as the dielectric permeability of the effective "medium". Such "medium" is interpreted as an internal factor connected with the method of solution of the problem. To each type of the impedance, inhomogeneity corresponds its own dependence N(x, z). To solve the auxiliary boundary problem (8) and (10), we apply the geometric optic method (because l/l_g

1 ). Using the methodology of Al'pert et al. [1953], we assume that in the z geq

h region there exists an effective medium with the dielectric permeability e₊ = 1 + N(x,z - h) symmetric relative to the permeability e_- = 1 + N(x,z) of the medium at z leq

h. We designate the corresponding refraction indices as n = (e_<img src="latex006.gif" alt="pm">)^1/2. We assume that in the point M₀'(0, 2h) symmetrical to M₀(0,0) (the place of the location of the source (1) relative to the plane z = h, there is an imaginary source j', the auxiliary function G=[n ( partial

(- z)) + g ]P corresponding to this source [see Al'pert et al., 1953]. It is evident that at the z = h level the sum of the fields of these two sources is equal to zero. So the boundary condition (8) is automatically satisfied.

[13] We designate as G₁^fal and G₁^ref the auxiliary function for the wave from the source (1) falling from below onto the boundary z =h and for the corresponding reflected wave, respectively. Then

(11)

Here M₁ = M₁(x/2,h) is the central point of the region important for reflection from the weak boundary z = h [see Ginzburg, 1960], R₁ = R₁ (x/2,h), and R_h = R_h (x,0). The value A is the amplitude of the function G₁^fal, multiplier r and multiplicand T are the refraction and transmission indices, respectively, and the factor A determines the divergence of the ray reflected from the boundary [see Kravtsov and Orlov, 1980].

[14] In a similar way we introduce an auxiliary function G₂^tr for the waves arriving from above from the source j':

(12)

Here R₀' = R₀'(0, 2h) is the radius vector of the point M₀' determining the coordinates of the symmetrical source j'. We can also write A( R_h - R₀' ) =A( R₁ - R₀' ) A( R_h - R₁). It follows from the problem symmetry that

(that is why, in particular, the phases of the functions G₁^ref and G₂^tr coincide) and A( R₁ - R₀' ) = A( R₁ - R₀).

[15] Finally, we have

(13)

Here it is taken into account that 1 + r = T. Thus the formulated problem is reduced to a calculation of the amplitude and phase of the ray propagating in a smoothly inhomogeneous medium with the dielectric permeability e.

[16] Taking into account grazing propagation, one can easily obtain from the expression G= gP an approximate formula for calculation of the Hertz potential P with the accuracy to the terms of the order of q_s²:

(14)

[17] On the basis of the derived formulae (13) and (14), we will study the influence of different kinds of IGW on propagation of electromagnetic waves. Let us consider two cases that correspond to the models with imaginary and real impedance values at the upper reflecting boundary. In particular, we will consider the impact of the IGW of periodical and soliton types.