I am currently studying the textbook Microwave Engineering, fourth edition, by David Pozar. Chapter 1.4 THE WAVE EQUATION AND BASIC PLANE WAVE SOLUTIONS says the following:
The Helmholtz Equation
In a source-free, linear, isotropic, homogeneous region, Maxwell's curl equations in phasor form are $$\nabla \times \bar{E} = -j \omega \mu \bar{H} \tag{1.41a}$$ $$\nabla \times \bar{H} = j \omega \epsilon \bar{E}, \tag{1.41b}$$ and constitute two equations for the unknowns, $\bar{E}$ and $\bar{H}$. As such, they can be solved for either $\bar{E}$ or $\bar{H}$. Taking the curl of (1.41a) and using (1.41b) gives $$\nabla \times \nabla \times \bar{E} = - j\omega \mu \nabla \times \bar{H} = \omega^2 \mu \epsilon \bar{E},$$ which is an equation for $\bar{E}$. This result can be simplified through the use of vector identity (B.14), $\nabla \times \nabla \times \bar{A} = \nabla (\nabla \cdot \bar{A}) - \nabla^2 \bar{A}$, which is valid for the rectangular components of an arbitrary vector $\bar{A}$. Then, $$\nabla^2 \bar{E} + \omega^2 \mu \epsilon \bar{E} = 0, \tag{1.42}$$ because $\nabla \cdot \bar{E} = 0$ in a source-free region. Equation (1.42) is the wave equation, or Helmholtz equation, for $\bar{E}$. An identical equation for $\bar{H}$ can be derived in the same manner: $$\nabla^2 \bar{H} + \omega^2 \mu \epsilon \bar{H} = 0. \tag{1.43}$$ A constant $k = \omega \sqrt{\mu \epsilon}$ is defined and called the propagation constant (also known as the phase constant, or wave number), of the medium; its units are $1/m$.
Plane Waves in a Lossless Medium
In a lossless medium, $\epsilon$ and $\mu$ are real numbers, and so $k$ is real. A basic plane wave solution to the above wave equation can be found by considering an electric field with only an $\hat{x}$ component and uniform (no variation) in the $x$ and $y$ directions. Then, $\partial/\partial{x} = \partial/\partial{y} = 0$, and the Helmholtz equation of (1.42) reduces to $$\dfrac{\partial^2{E_x}}{\partial{z}^2} + k^2 E_x = 0. \tag{1.44}$$
It then later says the following:
Plane Waves in a General Lossy Medium
Now consider the effect of a lossy medium. If the medium is conductive, with a conductivity $\sigma$, Maxwell's curl equations can be written, from (1.41a) and (1.20) as $$\nabla \times \bar{E} = - j \omega \mu \bar{H}, \tag{1.50a}$$ $$\nabla \times \bar{H} = j \omega \epsilon \bar{E} + \sigma \bar{E}. \tag{1.50b}$$ The resulting wave equation for $\bar{E}$ then becomes $$\nabla^2 \bar{E} + \omega^2 \mu \epsilon \left( 1 - j \dfrac{\sigma}{\omega \epsilon} \right) \bar{E} = 0, \tag{1.51}$$ where we see a similarity with (1.42), the wave equation for $\bar{E}$ in the lossless case. The difference is that the quantity $k^2 = \omega^2 \mu \epsilon$ of (1.42) is replaced by $\omega^2 \mu \epsilon[1 - j(\sigma/\omega \epsilon)]$ in (1.51). We then define a complex propagation constant for the medium as $$\gamma = \alpha + j \beta = j \omega \sqrt{\mu \epsilon} \sqrt{1 - j \dfrac{\sigma}{\omega \epsilon}}, \tag{1.52}$$ where $\alpha$ is the attenuation constant and $\beta$ is the phase constant. If we again assume an electric field with only an $\hat{x}$ component and uniform in $x$ and $y$, the wave equation of (1.51) reduces to $$\dfrac{\partial^2{E_x}}{\partial{z}^2} - \gamma^2 E_x = 0, \tag{1.53}$$
Notice that both these sections discuss an "electric field with only an $\hat{x}$ component" and "uniform in $x$ and $y$." This is written as $E_x$. But, beyond just being written as $E_x$, what precisely is this (mathematically)? Is this just an electric field with only an $x$-component, so that $E_x = E(x)$? But, then, if there's no $y$ component, then how does it make sense to say "uniform in $x$ and $y$"? Is the basis vector $\hat{i} = (1, 0, 0)$ included somehow, and is it somehow related to $\hat{x}$? Furthermore, a subscript often denotes the partial derivative, so is $E_x$ somehow the partial derivative of $E$ with respect to $x$? Overall, I do not think the author has phrased this well, or done enough to clarify his phrasing (using mathematics). What, precisely, using mathematics, is meant by "electric field with only an $\hat{x}$ component"?
