Application of the normal wave method to calculations of VLF...

5. Main Formulae for the Eigenfunctions, Excitation Coefficients, and the Field at the Observational Point

[33] We write down the normalized elements of six vectors Y_m (eigenfunctions of the lateral operator K) in the vacuum cavity, neglecting the terms of the order of 1/ n_m.

(52)

(53)

(54)

(55)

(56)

(57)

and elements Y_m⁺ of the eigenfunctions of the adjoint operator K⁺

We find the normalizing factor n_m with the help of formula (34), first differentiating it with respect to l₁, and then equalizing l₁=l₂.

(58)

The derivatives of the function F_ik with respect to the spectral parameter l are expressed via functions F_ik by formulae (38), the derivatives vanishing at x = 0. The eigenfunctions V satisfy the boundary conditions, therefore y₁ = - (x₁/T₁), y₂ = -(x₂/ T₂), and T₁ = ( D₁₁/D₀₁), T₂ = D₁₀/ D₀₀). The eigenfunctions at any x are expressed by the following formulae

(59)

(60)

(61)

(62)

The functions F_ik(l,x) and D_ik are determined by formulae (35) and (40' ). On the Earth surface x = l, taking into account F₀₀F₁₁ - F₀₁F₁₀ = 1, the above indicated formulae are transformed to the form

(63)

(64)

where t_1,2 are determined in section 4. At an effective height x = 0, we have

(65)

(66)

In order to obtain the functions of the conjugated operator V⁺ and dV⁺/ dx, one should substitute x₁ to x₁⁺ and x₂ to x₂⁺ in formulae (59)-(66). Equation (41) we rewrite in the form

and from this obtain a characteristic equation

If for the "m" mode ( l_m enters as a parameter into D_ik )

we take

then

(67)

In the opposite case

If for the "m" mode

we take

then

(68)

In the opposite case

[34] According to (1), (3) and (9) we write the field in a regular waveguide in terms of Y_m

(69)

Comparing (69) to formulae (46)-(51), we obtain the relation between L_m and L_m

(70)

We will name L_m a modified excitation coefficient.

[35] To find the excitation coefficients of modes by the antenna located at a height (b - a) over the Earth surface and oriented in an arbitrary way we use the generalized reciprocity theorem for anisotropic media [Felsen et al., 1973]. It follows from this theorem

(71)

where p_k = j_k^edV (k = 1,2), and in the case of short linear antenna considered here

where J is the current at the antenna base, l_p is the antenna virtual height, l_k is a unit vector directed along the antenna, and E₁ is the field of the antenna with the moment p₁ in the waveguide filled by the medium with the dielectric permittivity e(z), in the observation point coinciding to the position of the auxiliary source p₂. On the Earth surface, the impedance d_e is given and p₁ is oriented in an arbitrary way. E₂ is the field of the source p₂ of a vertical short antenna in the point coincided with the source position p₁. The waveguide is filled by the medium with a transposed tensor of the dielectric permittivity e^T(z) and the same impedance on the Earth.

[36] Let the current momenta of both sources p₁ = p₂ and heights (b-a) of their position over the Earth surface coincide by magnitude. We present the field in the form of a sum

(72)

over normal waves of the lateral operator K, (formula (2)). One has to determine L_m.

[37] In the second problem we present in the same way

(73)

E_m⁽²⁾ being the eigenfunctions of the operator K⁽²⁾ which is different from K⁺ (formula (5' )) in the following way: (1) the sign at nabla _t is changed and (2) the sign in the boundary conditions is changed. These differences are compensated by the changes in the sign of l_Q in the second problem. The eigenvalues n_m in (72) and (73) coincide. Formulating the problem, we have noted that the receiver and transmitter are located in the near-ground layer of the atmosphere below the ionosphere, therefore

then

Let

Using equality (71) we obtain

(74)

or, equalizing to zero each term in the sum, we obtain

the following formula being valid

(75)

[38] The modified excitation coefficients L_m are presented in millivolts, if the current in the antenna J, antenna length l_p, and source coordinate b are expressed in amperes, meters, and kilometers, respectively.

[39] In the real conditions, the Earth-ionosphere waveguide appears irregular because of the inhomogeneity of geophysical conditions (conductivity of the Earth surface, illumination of the path, and magnetic field of the Earth). In the model we use, the real waveguide is presented as a piecewise-homogeneous one.

[40] At the homogeneous piece with number 0 in the vicinity of the transmitter we find the eigenvalues n_m^(o) and eigenfunctions Y_m^(o), and take L_m^(o) = L_m. Using formula (12) we determine the matrix of transformation of normal waves P_nm⁽¹⁾ = (1/2)(Y_n⁽¹⁾⁺, GY_m^(o)) at the joint boundary of homogeneous pieces, and then calculate the amplitudes of the normal waves falling onto the boundary of the next to number N homogeneous piece by the formula

(76)

[41] The field in the irregular waveguide we find using the formula

(77)

[42] Thus for calculation of any component of the field at any height in the vacuum cavity of the irregular waveguide using formula (77), we have formulae for L_m⁽⁰⁾; L_m^(N) is calculated using formula (76) with the help of the transformation matrix P_nm^(N). Elements of the latter matrix may be calculated approximately by the formula

The eigenfunctions (52)-(57) are recalculated to any height by the formulae (59)-(62). The approximated value of the normalized factor is calculated by formula (58).

[43] The relative error D of calculation of the normalizing factor is estimated by the formula

(78)

for every mode m.

[44] If the observation point is located below the surface level, the field first is calculated on the Earth surface by the above described formulae E(a), and then the components E_Q and E_j are multiplied to exp(-ik_Ed_a), where d_a < 0 is the depth of the receiver location, k_E = k₀(e_E)^1/2, e_E = e + (is/ we₀), e is the relative dielectric permittivity, and s is the conductivity of the medium where the receiver is located,

(79)

(80)

We do the same for three components of the magnetic field

(81)

The phase of the components of the field F is determined relative to the phase of the current J at the antenna input.