# Reduction of the Helmholtz equation for an electric field with only an $\hat{x}$ component and uniform (no variation) in the $x$ and $y$ directions

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 $$\dfrac{\partial^2{E_x}}{\partial{z}^2} + k^2 E_x = 0. \tag{1.44}$$

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 $$\dfrac{\partial^2{E_x}}{\partial{z}^2} + k^2 E_x = 0. \tag{1.44}$$
If the electric field only has an $$\hat{x}$$ component, as stated, then how does it make sense to say that it has "uniform (no variation) in the $$x$$ and $$y$$ directions"? Shouldn't there not be anything in the $$y$$ direction? And how does this all reduce the Helmholtz equation to $$\dfrac{\partial^2{E_x}}{\partial{z}^2} + k^2 E_x = 0$$? I would appreciate an explanation with mathematics that illustrates this.
Equation 1.44 is for (an infinite) plane wave propagating in the $$z$$ direction. The $$\mathbf E$$ vector points in the $$x$$ direction and is assumed to have the same value for all values of $$x$$ and $$y$$. It does vary with $$z$$ and $$t$$.