Rectangular Waveguide: Complete Theory from First Principles

We consider a rectangular metallic waveguide of cross-section

\[ 0 \le x \le a, \quad 0 \le y \le b \]

with propagation along the z-direction.
The walls are assumed to be perfect electric conductors (PEC) and the interior is vacuum (\(\varepsilon_0, \mu_0\)).

1. Governing Equations in the Waveguide

Inside the waveguide there are no free charges and no currents. Maxwell’s equations reduce to:

\[ \nabla \times \mathbf{E} = -\mu_0 \frac{\partial \mathbf{H}}{\partial t} \] \[ \nabla \times \mathbf{H} = \epsilon_0 \frac{\partial \mathbf{E}}{\partial t} \]

Assume monochromatic time dependence:

\[ \mathbf{E}, \mathbf{H} \propto e^{-j\omega t} \]

Then:

\[ \nabla \times \mathbf{E} = -j\omega \mu_0 \mathbf{H} \] \[ \nabla \times \mathbf{H} = j\omega \epsilon_0 \mathbf{E} \]

Each field component satisfies the Helmholtz equation:

\[ \left(\nabla^2 + \frac{\omega^2}{c^2}\right)\mathbf{E} = 0, \quad \left(\nabla^2 + \frac{\omega^2}{c^2}\right)\mathbf{H} = 0 \]

2. Propagation Ansatz and Separation

We seek waves propagating along z, so we assume:

\[ \mathbf{F}(x,y,z) = \mathbf{F}(x,y)e^{-jk_z z} \]

Substituting into the Helmholtz equation gives:

\[ \left(\frac{\partial^2}{\partial x^2} + \frac{\partial^2}{\partial y^2} + h^2 \right)F(x,y) = 0 \]

where

\[ h^2 = \frac{\omega^2}{c^2} - k_z^2 \]

This is a 2D eigenvalue problem in the transverse plane.

3. Longitudinal vs Transverse Components

Fields decompose naturally into:

• Longitudinal: \(E_z, H_z\)
• Transverse: \(E_x, E_y, H_x, H_y\)

From Maxwell’s equations, all transverse components can be expressed entirely in terms of \(E_z\) and \(H_z\). The key relations (your eqs. 57–61) are:

\[ E_x = \frac{j}{h^2}\left(k_z\frac{\partial E_z}{\partial x} + \omega\mu_0\frac{\partial H_z}{\partial y}\right) \] \[ E_y = \frac{j}{h^2}\left(k_z\frac{\partial E_z}{\partial y} - \omega\mu_0\frac{\partial H_z}{\partial x}\right) \] \[ H_x = \frac{j}{h^2}\left(k_z\frac{\partial H_z}{\partial x} - \omega\epsilon_0\frac{\partial E_z}{\partial y}\right) \] \[ H_y = \frac{j}{h^2}\left(k_z\frac{\partial H_z}{\partial y} + \omega\epsilon_0\frac{\partial E_z}{\partial x}\right) \]

Physical meaning:
Once you know the longitudinal field, the entire electromagnetic structure is fixed. The waveguide does not allow arbitrary field polarization.

4. Mode Classification: TE and TM

Two mutually exclusive cases arise.

Case I: TE (Transverse Electric) Modes

\[ E_z = 0, \quad H_z \neq 0 \]

Electric field is entirely transverse to propagation.

Case II: TM (Transverse Magnetic) Modes

\[ H_z = 0, \quad E_z \neq 0 \]

Magnetic field is entirely transverse.

There is no TEM mode in a hollow waveguide because both \(E_z\) and \(H_z\) cannot vanish simultaneously while satisfying boundary conditions.

5. Boundary Conditions (PEC Walls)

At a perfect conductor:

• Tangential E must vanish
• Normal B must vanish

This gives:

For TE modes:

\[ E_x = E_y = 0 \quad \text{on walls} \quad \Rightarrow \quad \frac{\partial H_z}{\partial n} = 0 \]

For TM modes:

\[ E_z = 0 \quad \text{on walls} \]

6. TE Mode Solution

We solve:

\[ \left(\frac{\partial^2}{\partial x^2} + \frac{\partial^2}{\partial y^2} + h^2 \right)H_z = 0 \]

Using separation of variables:

\[ H_z(x,y) = X(x)Y(y) \]

Applying boundary conditions leads to:

\[ H_z(x,y) = H_0 \cos\left(\frac{m\pi x}{a}\right) \cos\left(\frac{n\pi y}{b}\right) \]

with

\[ m,n = 0,1,2,\dots \quad \text{(not both zero)} \]

Eigenvalue:

\[ h^2 = \left(\frac{m\pi}{a}\right)^2 + \left(\frac{n\pi}{b}\right)^2 \]

All transverse field components follow by differentiation.

7. TM Mode Solution

Now solve:

\[ \left(\frac{\partial^2}{\partial x^2} + \frac{\partial^2}{\partial y^2} + h^2 \right)E_z = 0 \]

Boundary condition:

\[ E_z = 0 \quad \text{on all walls} \]

Solution:

\[ E_z(x,y) = E_0 \sin\left(\frac{m\pi x}{a}\right) \sin\left(\frac{n\pi y}{b}\right) \]

with

\[ m,n = 1,2,3,\dots \]

Eigenvalue:

\[ h^2 = \left(\frac{m\pi}{a}\right)^2 + \left(\frac{n\pi}{b}\right)^2 \]

8. Cutoff Frequency and Cutoff Wavelength

From:

\[ k_z^2 = \frac{\omega^2}{c^2} - h^2 \]

Propagation requires:

\[ k_z^2 > 0 \quad \Rightarrow \quad \omega > \omega_c \]

Cutoff frequency:

\[ \omega_c = c\sqrt{\left(\frac{m\pi}{a}\right)^2 + \left(\frac{n\pi}{b}\right)^2} \]

Cutoff wavelength:

\[ \lambda_c = \frac{2}{\sqrt{(m/a)^2 + (n/b)^2}} \]

Below cutoff: fields decay exponentially
Above cutoff: true wave propagation

9. Dominant Mode

For \(a > b\):

• TE\({}_{10}\) has the lowest cutoff frequency
• TM modes always have higher cutoff

Hence TE modes are superior for transmission, which is why real waveguides operate in TE\({}_{10}\).

10. Guide Wavelength

Define guide wavelength:

\[ \lambda_g = \frac{2\pi}{k_z} \]

Using dispersion:

\[ \frac{1}{\lambda_g^2} = \frac{1}{\lambda^2} - \frac{1}{\lambda_c^2} \]

So:

\[ \lambda_g = \frac{\lambda}{\sqrt{1 - (\lambda/\lambda_c)^2}} \]

Interpretation:
Waveguide forces the wave to “zig-zag” between walls, increasing effective wavelength along z.

11. Phase and Group Velocity

Phase velocity:

\[ v_p = \frac{\omega}{k_z} = \frac{c}{\sqrt{1-(\lambda/\lambda_c)^2}} > c \]

Group velocity:

\[ v_g = c\sqrt{1-(\lambda/\lambda_c)^2} < c \]

And:

\[ v_p v_g = c^2 \]

No violation of relativity:
Energy travels with group velocity, not phase velocity.

12. Power Flow in the Waveguide

Time-averaged power:

\[ P = \frac{1}{2}\Re\int\int (\mathbf{E}\times\mathbf{H}^*)\cdot\hat{z}\,dxdy \]

TE Modes:

\[ P_{TE} = \frac{ab}{8}\frac{\omega\mu_0 k_z}{h^2}H_0^2 \]

TM Modes:

\[ P_{TM} = \frac{ab}{8}\frac{\omega\epsilon_0 k_z}{h^2}E_0^2 \]

Interpretation:

• Power increases with frequency above cutoff
• Power vanishes at cutoff
• Finite breakdown limits maximum power

13. Final Physical Picture

• Waveguide acts as a spatial filter
• Only discrete transverse field patterns are allowed
• Cutoff arises from transverse confinement
• TE\({}_{10}\) dominates due to lowest cutoff
• Dispersion is intrinsic to guided waves
• Energy transport is constrained by geometry, not material