Mathematical Expression of the Plane Wave Solution

Within the framework of classical electromagnetism, Maxwell's equations serve as the fundamental laws governing all electromagnetic phenomena. From these equations, the wave equation is derived, and its most essential particular solution is the plane wave. Plane waves are not merely theoretical simplifications; they are the building blocks of electromagnetic theory. Through the principle of Fourier analysis, any complex electromagnetic field—regardless of its radiation or scattering pattern—can be decomposed into a superposition of infinite plane waves with varying frequencies and propagation directions.
Intuitively, a plane wave is an electromagnetic wave whose wavefronts (surfaces of constant phase) are infinite parallel planes. In such a wave, the electric and magnetic field intensities and phases are perfectly uniform across any plane perpendicular to the direction of propagation.

For a wave propagating in a boundless, linear, isotropic, and homogeneous medium, the direction of travel is defined by a unit vector $\mathbf{n}$. If we define a position vector $\mathbf{r} = x\mathbf{e}_x + y\mathbf{e}_y + z\mathbf{e}_z$, the spatial phase of a harmonic plane wave is characterized by the relation:
$$\mathbf{k} \cdot \mathbf{r} = \text{constant}$$
Here, $\mathbf{k}$ represents the wave vector, defined as $\mathbf{k} = k \mathbf{n}$, where $k = |\mathbf{k}| = \frac{2\pi}{\lambda}$ is the phase constant (or wavenumber) and $\lambda$ is the wavelength.

Derivation from Maxwell’s Equations

In a source-free region—where the charge density $\rho = 0$ and the conduction current density $\mathbf{J} = 0$—Maxwell's equations in differential form are:

1. $\nabla \cdot \mathbf{D} = 0$
2. $\nabla \cdot \mathbf{B} = 0$
3. $\nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}}{\partial t}$
4. $\nabla \times \mathbf{H} = \frac{\partial \mathbf{D}}{\partial t}$

For a simple medium characterized by the constitutive relations $\mathbf{D} = \epsilon \mathbf{E}$ and $\mathbf{B} = \mu \mathbf{H}$, we can derive the wave equation by taking the curl of the curl of the electric field. Utilizing the vector identity $\nabla \times (\nabla \times \mathbf{A}) = \nabla(\nabla \cdot \mathbf{A}) - \nabla^2 \mathbf{A}$, we find that both the electric field $\mathbf{E}$ and the magnetic field $\mathbf{H}$ satisfy the homogeneous vector wave equation in the time domain, or the vector Helmholtz equation in the frequency domain.

Mathematical Representations: Time-Domain vs. Phasors

To facilitate analysis, electromagnetic fields are typically expressed in either instantaneous time-domain forms or complex frequency-domain (phasor) forms.

1. Instantaneous Time-Domain Expression

Consider a plane wave propagating in the positive $z$-direction and polarized along the $x$-axis. The instantaneous electric and magnetic fields are expressed as:

• $\mathbf{E}(z, t) = E_0 \cos(\omega t - kz + \phi_0) \mathbf{e}_x$
• $\mathbf{H}(z, t) = H_0 \cos(\omega t - kz + \phi_0) \mathbf{e}_y$

In these expressions, $E_0$ and $H_0$ denote the amplitudes, $\omega = 2\pi f$ is the angular frequency, and $\phi_0$ is the initial phase.

2. Complex Phasor Expression

In engineering and physics, the use of Euler's formula allows us to represent harmonic fields as complex phasors, which transforms differential operations into simpler algebraic ones. The phasor forms of the fields mentioned above are:

• $\mathbf{E}_s(z) = E_0 e^{-jkz} \mathbf{e}_x$
• $\mathbf{H}_s(z) = H_0 e^{-jkz} \mathbf{e}_y$

The relationship between the phasor $\mathbf{E}_s$ and the physical instantaneous field is given by $\mathbf{E}(z, t) = \text{Re}{\mathbf{E}_s(z) e^{j\omega t}}$.

Core Mathematical and Physical Properties

The constraints imposed by Maxwell's equations lead to three defining characteristics of plane wave solutions:

• Transversality: Plane waves are Transverse Electromagnetic (TEM) waves. This means the electric field $\mathbf{E}$, the magnetic field $\mathbf{H}$, and the propagation vector $\mathbf{k}$ are mutually orthogonal. Mathematically:
  $$\mathbf{k} \cdot \mathbf{E} = 0, \quad \mathbf{k} \cdot \mathbf{H} = 0, \quad \text{and} \quad \mathbf{E} \cdot \mathbf{H} = 0$$
• Wave Impedance: The ratio of the electric field amplitude to the magnetic field amplitude is determined by the intrinsic properties of the medium, defined as the wave impedance $Z$:
  $$Z = \frac{E_x}{H_y} = \sqrt{\frac{\mu}{\epsilon}}$$
  In a vacuum, this characteristic impedance is approximately $377,\Omega$ (or $120\pi,\Omega$).
• Phase Velocity: The speed at which the phase of the wave propagates through the medium is governed by the permittivity $\epsilon$ and permeability $\mu$:
  $$v_p = \frac{\omega}{k} = \frac{1}{\sqrt{\mu\epsilon}}$$

Generalization to Arbitrary Propagation

When a plane wave propagates in an arbitrary direction defined by the wave vector $\mathbf{k} = k_x \mathbf{e}_x + k_y \mathbf{e}_y + k_z \mathbf{e}_z$, the general complex phasor expression is:
$$\mathbf{E}(\mathbf{r}) = \mathbf{E}_0 e^{-j \mathbf{k} \cdot \mathbf{r}}$$
$$\mathbf{H}(\mathbf{r}) = \frac{1}{\omega \mu} \mathbf{k} \times \mathbf{E}(\mathbf{r}) = \mathbf{H}_0 e^{-j \mathbf{k} \cdot \mathbf{r}}$$

This generalized form is indispensable for analyzing complex phenomena such as spatial scattering, diffraction, and waveguide modes. By mastering the mathematical expression of the plane wave, one gains the necessary tools to transition from ideal vacuum propagation to the analysis of lossy media, anisotropic materials, and the application of boundary conditions in real-world electromagnetic systems.