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Appendix 10B: First-Order Coupled Differential Equations for Waves in Inhomogeneous Warm Magnetoplasmas*

Appendix 10B: First-Order Coupled Differential Equations for Waves in Inhomogeneous Warm Magnetoplasmas *

Dikshitulu K. Kalluri and H. Unz

The first-order coupled differential equations for waves in temperate inhomogeneous magnetoplasmas are well known [1]. The extension of this method for the case of waves in inhomogeneous warm magnetoplasmas is presented here. In this form, they are well suited for solution on a digital computer by the Runge–Kutta method. The plasma is assumed to be neutral and in equilibrium, and the motion of the ions is neglected. Taking ${\bar{E}}_{0}$ ${\bar{E}}_{0}$ , ${\bar{H}}_{0}$ ${\bar{H}}_{0}$ , p₀, N₀, and $T_{0} ({\bar{u}}_{0} = 0)$ $T_{0} ({\bar{u}}_{0} = 0)$ as the stationary values of the plasma which are given functions of position, and ${\bar{E}}_{1}$ ${\bar{E}}_{1}$ , ${\bar{H}}_{1}$ ${\bar{H}}_{1}$ , p₁, N₁, T₁, and ${\bar{u}}_{1}$ ${\bar{u}}_{1}$ as the harmonic time-varying (e^iωt) small components of the wave in the plasma, and defining the plasma parameters

\begin{matrix} X (\bar{r}) & = & \frac{e^{2} N_{0} (\bar{r})}{ω^{2} ε m}, Y (\bar{r}) = \frac{|e| μ}{m ω} {\bar{H}}_{0} (\bar{r}), U (\bar{r}) = 1 - \bar{i} \frac{ν (\bar{r})}{\bar{ω}}, \\ δ (\bar{r}) & = & \frac{a^{2} (\bar{r})}{c^{2}} = \frac{μ ε γ K T_{0} (\bar{r})}{m}, \end{matrix}

$\begin{matrix} X (\bar{r}) & = & \frac{e^{2} N_{0} (\bar{r})}{ω^{2} ε m}, Y (\bar{r}) = \frac{|e| μ}{m ω} {\bar{H}}_{0} (\bar{r}), U (\bar{r}) = 1 - \bar{i} \frac{ν (\bar{r})}{\bar{ω}}, \\ δ (\bar{r}) & = & \frac{a^{2} (\bar{r})}{c^{2}} = \frac{μ ε γ K T_{0} (\bar{r})}{m}, \end{matrix}$

where $a (\bar{r})$ $a (\bar{r})$ is the acoustic velocity in the electron gas, by taking a new set of dependent variables

(10B.1)

\bar{E} = {\bar{E}}_{1}, \bar{H} = {[\frac{μ}{ε}]}^{\frac{1}{2}} {\bar{H}}_{1}, \bar{u} = \frac{ω m}{e} {\bar{u}}_{1}, p = \frac{e}{ω m} {[\frac{μ}{ε}]}^{\frac{1}{2}} p_{1},

$\bar{E} = {\bar{E}}_{1}, \bar{H} = {[\frac{μ}{ε}]}^{\frac{1}{2}} {\bar{H}}_{1}, \bar{u} = \frac{ω m}{e} {\bar{u}}_{1}, p = \frac{e}{ω m} {[\frac{μ}{ε}]}^{\frac{1}{2}} p_{1},$

one obtains [2] for small-signal theory, where k₀ = ω(με)^1/2 = ω/c:

(10B.2a)

\nabla \times \bar{E} = - i k_{0} \bar{H},

$\nabla \times \bar{E} = - i k_{0} \bar{H},$

(10B.2b)

\nabla \times \bar{H} = i k_{0} \bar{E} - k_{0} X \bar{u},

$\nabla \times \bar{H} = i k_{0} \bar{E} - k_{0} X \bar{u},$

(10B.2c)

- i U X \bar{u} = \frac{1}{k_{0}} \nabla p + X \bar{E} + \frac{δ}{γ k_{0}^{2}} \frac{\nabla p_{0}}{p_{0}} (\nabla \cdot \bar{E}) + X \bar{u} \times \bar{Y},

$- i U X \bar{u} = \frac{1}{k_{0}} \nabla p + X \bar{E} + \frac{δ}{γ k_{0}^{2}} \frac{\nabla p_{0}}{p_{0}} (\nabla \cdot \bar{E}) + X \bar{u} \times \bar{Y},$

(10B.2d)

- \nabla \cdot \bar{u} = \frac{i k_{0}}{δ X} p + \bar{u} \cdot \frac{\nabla p_{0}}{γ p_{0}} .

$- \nabla \cdot \bar{u} = \frac{i k_{0}}{δ X} p + \bar{u} \cdot \frac{\nabla p_{0}}{γ p_{0}} .$

For the case of a free-space wave of arbitrary polarization with direction cosines S₁, S₂, and C, obliquely incident on the plasma medium, one [1] obtain the following:

(10B.3)

\frac{\partial}{\partial x} = - i k_{0} S_{1}, \frac{\partial}{\partial y} = - i k_{0} S_{2} .

$\frac{\partial}{\partial x} = - i k_{0} S_{1}, \frac{\partial}{\partial y} = - i k_{0} S_{2} .$

Assuming the plasma parameters vary in the z-direction only such that N₀(z), T₀(z), v(z), and H₀(z), and substituting Equation 10B.3 into Equation 10B.2, one obtains 10 equations with 10 unknowns $\bar{E}, \bar{H}, \bar{u}, and p$ $\bar{E}, \bar{H}, \bar{u}, and p$ as functions of z only. Eliminating E_z, H_z, u_x, and u_y from these equations, one obtains the following six first-order linear coupled differential equations with six unknowns, where primes denote derivatives with respect to z:

(10B.4a)

\frac{- 1}{i k_{0}} E_{x}^{'} = S_{1} S_{2} H_{x} + (1 - S_{1}^{2}) H_{y} - i S_{1} X u_{z},

$\frac{- 1}{i k_{0}} E_{x}^{'} = S_{1} S_{2} H_{x} + (1 - S_{1}^{2}) H_{y} - i S_{1} X u_{z},$

(10B.4b)

\frac{- 1}{i k_{0}} E_{y}^{'} = - (1 - S_{2}^{2}) H_{x} - S_{1} S_{2} H_{y} - i S_{2} X u_{z},

$\frac{- 1}{i k_{0}} E_{y}^{'} = - (1 - S_{2}^{2}) H_{x} - S_{1} S_{2} H_{y} - i S_{2} X u_{z},$

(10B.4c)

\begin{matrix} \frac{- 1}{i k_{0}} H_{x}^{'} & = & - (S_{1} S_{2} + \frac{i X Y_{z}}{U^{2} - Y_{z}^{2}}) E_{x} + (S_{1}^{2} - 1 + \frac{U X}{U^{2} - Y_{z}^{2}}) E_{y} \\ + (\frac{U X Y_{x} + i X Y_{y} Y_{z}}{U^{2} - Y_{z}^{2}}) u_{z} - (\frac{S_{1} Y_{z} + i U S_{2}}{U^{2} - Y_{z}^{2}}) p, \end{matrix}

$\begin{matrix} \frac{- 1}{i k_{0}} H_{x}^{'} & = & - (S_{1} S_{2} + \frac{i X Y_{z}}{U^{2} - Y_{z}^{2}}) E_{x} + (S_{1}^{2} - 1 + \frac{U X}{U^{2} - Y_{z}^{2}}) E_{y} \\ + (\frac{U X Y_{x} + i X Y_{y} Y_{z}}{U^{2} - Y_{z}^{2}}) u_{z} - (\frac{S_{1} Y_{z} + i U S_{2}}{U^{2} - Y_{z}^{2}}) p, \end{matrix}$

(10B.4d)

\begin{matrix} \frac{- 1}{i k_{0}} H_{y}^{'} = (1 - S_{2}^{2} - \frac{U X}{U^{2} - Y_{z}^{2}}) E_{x} + (S_{1} S_{2} - \frac{i X Y_{z}}{U^{2} - Y_{z}^{2}}) E_{y} \\ + (\frac{U X Y_{y} - i X Y_{x} Y_{z}}{U^{2} - Y_{z}^{2}}) u_{z} + (\frac{i S_{1} U - S_{2} Y_{z}}{U^{2} - Y_{z}^{2}}) p, \end{matrix}

$\begin{matrix} \frac{- 1}{i k_{0}} H_{y}^{'} = (1 - S_{2}^{2} - \frac{U X}{U^{2} - Y_{z}^{2}}) E_{x} + (S_{1} S_{2} - \frac{i X Y_{z}}{U^{2} - Y_{z}^{2}}) E_{y} \\ + (\frac{U X Y_{y} - i X Y_{x} Y_{z}}{U^{2} - Y_{z}^{2}}) u_{z} + (\frac{i S_{1} U - S_{2} Y_{z}}{U^{2} - Y_{z}^{2}}) p, \end{matrix}$

(10B.4e)

\begin{matrix} \frac{- 1}{i k_{0}} u_{z}^{'} = - (\frac{S_{2} Y_{z} + i U S_{1}}{U^{2} - Y_{z}^{2}}) E_{x} + (\frac{S_{1} Y_{z} - U S_{2}}{U^{2} - Y_{z}^{2}}) E_{y} \\ + (\frac{S_{1} Y_{x} Y_{z} + S_{2} Y_{y} Y_{z} + i U S_{1} Y_{y} - i U S_{2} Y_{x}}{U^{2} - Y_{z}^{2}} - \frac{i}{γ k_{0}} \frac{p_{0}^{'}}{p_{0}}) u_{z} \\ + \frac{1}{X} (\frac{1}{δ} - \frac{U S_{1}^{2} + U S_{2}^{2}}{U^{2} - Y_{z}^{2}}) p, \end{matrix}

$\begin{matrix} \frac{- 1}{i k_{0}} u_{z}^{'} = - (\frac{S_{2} Y_{z} + i U S_{1}}{U^{2} - Y_{z}^{2}}) E_{x} + (\frac{S_{1} Y_{z} - U S_{2}}{U^{2} - Y_{z}^{2}}) E_{y} \\ + (\frac{S_{1} Y_{x} Y_{z} + S_{2} Y_{y} Y_{z} + i U S_{1} Y_{y} - i U S_{2} Y_{x}}{U^{2} - Y_{z}^{2}} - \frac{i}{γ k_{0}} \frac{p_{0}^{'}}{p_{0}}) u_{z} \\ + \frac{1}{X} (\frac{1}{δ} - \frac{U S_{1}^{2} + U S_{2}^{2}}{U^{2} - Y_{z}^{2}}) p, \end{matrix}$

(10B.4f)

\begin{matrix} \frac{- 1}{i k_{0}} p^{'} = (\frac{U X Y_{y} + i X Y_{x} Y_{z}}{U^{2} - Y_{z}^{2}}) E_{x} + (\frac{i X Y_{y} Y_{z} - U X Y_{x}}{U^{2} - Y_{z}^{2}}) E_{y} \\ - i S_{2} X H_{x} + i S_{1} X H_{y} \\ + {U X - X^{2} - U X \frac{Y_{x}^{2} + Y_{y}^{2}}{U^{2} - Y_{z}^{2}} + \frac{δ X}{k_{0}^{2} γ^{2}} (\frac{p_{0}^{'}}{p_{0}}) - \frac{X^{'} δ}{γ k_{0}^{2}} \frac{p_{0}^{'}}{p_{0}}} u_{z} \\ + (\frac{S_{2} Y_{y} Y_{z} + S_{1} Y_{x} Y_{z} + i U S_{2} Y_{x} - i U S_{1} Y_{y}}{U^{2} - Y_{z}^{2}} + \frac{i}{k_{0} γ} \frac{p_{0}^{'}}{p_{0}}) p . \end{matrix}

$\begin{matrix} \frac{- 1}{i k_{0}} p^{'} = (\frac{U X Y_{y} + i X Y_{x} Y_{z}}{U^{2} - Y_{z}^{2}}) E_{x} + (\frac{i X Y_{y} Y_{z} - U X Y_{x}}{U^{2} - Y_{z}^{2}}) E_{y} \\ - i S_{2} X H_{x} + i S_{1} X H_{y} \\ + {U X - X^{2} - U X \frac{Y_{x}^{2} + Y_{y}^{2}}{U^{2} - Y_{z}^{2}} + \frac{δ X}{k_{0}^{2} γ^{2}} (\frac{p_{0}^{'}}{p_{0}}) - \frac{X^{'} δ}{γ k_{0}^{2}} \frac{p_{0}^{'}}{p_{0}}} u_{z} \\ + (\frac{S_{2} Y_{y} Y_{z} + S_{1} Y_{x} Y_{z} + i U S_{2} Y_{x} - i U S_{1} Y_{y}}{U^{2} - Y_{z}^{2}} + \frac{i}{k_{0} γ} \frac{p_{0}^{'}}{p_{0}}) p . \end{matrix}$

Equations 10B.4 may be written in the matrix form as

(10B.5)

\frac{- 1}{i k_{0}} \frac{d}{d z} [e] = [T] [e],

$\frac{- 1}{i k_{0}} \frac{d}{d z} [e] = [T] [e],$

where [e] is a (6 × 1) column matrix with the elements E_x, E_y, H_x, H_y, u_z, and p.

For the particular case of a homogeneous plasma, the elements of the [T] matrix are constants. The characteristic equation of the coupled differential equation (10B.5) obtained by setting

(10B.6a)

\frac{d}{d z} = - i k_{0} q

$\frac{d}{d z} = - i k_{0} q$

is given by the determinant

(10B.6b)

det ([T] - q [I]) = 0.

$det ([T] - q [I]) = 0.$

Equation 10B.6b is a sixth-order algebraic equation and it is found to be in agreement with an earlier result by Unz [3].

From the above definitions, one has [2]

(10B.7)

\frac{\nabla p_{0}}{p_{0}} = \frac{\nabla X}{X} + \frac{\nabla δ}{δ} .

$\frac{\nabla p_{0}}{p_{0}} = \frac{\nabla X}{X} + \frac{\nabla δ}{δ} .$

In a recent communication, Burman [4] has suggested that ∇p₀ = 0. By substituting $p_{0}^{'} = 0$ $p_{0}^{'} = 0$ into Equation 10B.4, one obtains our corresponding result.

Acknowledgment

This work was supported in part by the National Science Foundation.

References

1.Budden, K. G., Radio Waves in the Ionosphere, Cambridge University Press, Cambridge, MA, 1961.
2.Unz, H., Wave propagation in inhomogeneous gyrotropic warm plasmas, Am. J. Phys., 35, 505–508, 1967.
3.Unz, H., Oblique wave propagation in a compressible general magneto-plasma, Appl. Sci. Res., 16, 105–120, 1966.
4.Burman, R., Coupled wave equations for propagation in generally inhomogeneous compressible magnetoplasmas, Proc. IEEE (Lett.), 55, 723–724, 1967.
5.Kalluri, D. and Unz, H., The first-order coupled differential equations for waves in inhomogeneous warm magnetoplasmas, Proc. IEEE, 55(9), 1620–1621, September 1967.

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Table of Contents for Appendix 10B: First-Order Coupled Differential Equations for Waves in Inhomogeneous Warm Magnetoplasmas

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Acknowledgment

References

Table of Contents for
Appendix 10B: First-Order Coupled Differential Equations for Waves in Inhomogeneous Warm Magnetoplasmas