Rectangular Waveguide Propagation
Interactive Simulation & Transverse Field Visualizer
Propagation in a waveguide viewed as ray reflections or distinct transverse field distributions down the guide (\(z\)-axis).
Mode: Zig-Zag Wave Ray Path Propagation
1. Introduction & Geometry
A waveguide consists of a hollow conducting metallic tube of uniform cross-section. Field modes (\(\text{TE}_{mn}\) or \(\text{TM}_{mn}\)) dictate how fields distribute inside the cross-section.
Waveguide dimensions:
- a: Width along the x-axis (\(x = 0\) to \(x = a\))
- b: Height along the y-axis (\(y = 0\) to \(y = b\))
- z: Axis of propagation (wave travels along \(+a_z\))
Boundary Conditions:
\[E_x = E_z = 0 \quad \text{at } y = 0 \text{ and } y = b\]
\[E_y = E_z = 0 \quad \text{at } x = 0 \text{ and } x = a\]
2. Maxwell's Equations in Non-Conducting Medium
For a non-conducting medium (\(\sigma = 0\)), Maxwell's equations in phasor form are:
\[\nabla \times \mathbf{H} = j\omega\varepsilon \mathbf{E}\]
\[\nabla \times \mathbf{E} = -j\omega\mu \mathbf{H}\]
Expanding the curl equations into scalar components assuming \(z\)-variation of \(e^{-\gamma z}\) (\(\frac{\partial}{\partial z} = -\gamma\)):
Curl Equations Expanded:
\[\frac{\partial H_z}{\partial y} + \gamma H_y = j\omega\varepsilon E_x \quad \text{--- (1a)}\]
\[\frac{\partial H_z}{\partial x} + \gamma H_x = -j\omega\varepsilon E_y \quad \text{--- (1b)}\]
\[\frac{\partial H_y}{\partial x} - \frac{\partial H_x}{\partial y} = j\omega\varepsilon E_z \quad \text{--- (1c)}\]
\[\frac{\partial E_z}{\partial y} + \gamma E_y = -j\omega\mu H_x \quad \text{--- (2a)}\]
\[\frac{\partial E_z}{\partial x} + \gamma E_x = j\omega\mu H_y \quad \text{--- (2b)}\]
\[\frac{\partial E_y}{\partial x} - \frac{\partial E_x}{\partial y} = -j\omega\mu H_z \quad \text{--- (2c)}\]
3. Transverse Field Components in terms of \(E_z\) and \(H_z\)
By solving the linear system of equations, all transverse field components (\(E_x, E_y, H_x, H_y\)) are expressed in terms of the longitudinal axial components (\(E_z, H_z\)):
\[E_x = -\frac{\gamma}{h^2} \frac{\partial E_z}{\partial x} - \frac{j\omega\mu}{h^2} \frac{\partial H_z}{\partial y}\]
\[E_y = -\frac{\gamma}{h^2} \frac{\partial E_z}{\partial y} + \frac{j\omega\mu}{h^2} \frac{\partial H_z}{\partial x}\]
\[H_x = -\frac{\gamma}{h^2} \frac{\partial H_z}{\partial x} + \frac{j\omega\varepsilon}{h^2} \frac{\partial E_z}{\partial y}\]
\[H_y = -\frac{\gamma}{h^2} \frac{\partial H_z}{\partial y} - \frac{j\omega\varepsilon}{h^2} \frac{\partial E_z}{\partial x}\]
Where the cutoff wavenumber parameter \(h^2\) is defined as:
\[h^2 = \gamma^2 + \omega^2\mu\varepsilon\]