Example 5.3 – Waveguide

Visualization of wave in a waveguide for the first five modes independent of x coordinate.

The electric field in the waveguide is therefore dependent only on coordinates (y, z, t). Boundary conditions then force that the only non-zero component of the electric field intensity vector is E_x, thus E = (E_x, 0, 0).

In the following animated graphs, the value of electric field intensity is displayed on the plane x = x₀ (where x₀ is arbitrary, since the electric field does not depend on the x coordinate).

Electric field intensity for progressive waves has the form: E_x(y, z, t) = sin(mπ/b) cos(ωt - kz), for damped waves then E_x(y, z, t) = sin(mπ/b) e^-κz cos(ωt). b is the dimension of the waveguide, m is the mode number. The relationships between ω, k and κ are given by the dispersion relation, see below.

For progressive waves (here for the first four modes) the dispersion relation ω² = c² ( (mπ/b)² + k²) applies; for damped waves (here the fifth and higher modes) ω² = c² ( (mπ/b)² – κ²).

For given ω we can express from the dispersion relation for individual modes the wavelength λ = 2π/k and phase velocity v_φ = ω/k. We arrive at the relations λ₁ < λ₂ < λ₃ < λ₄ and v_φ1 < v_φ2 < v_φ3 < v_φ4.

First mode m = 1:

Second mode m = 2:

Third mode m = 3:

Fourth mode m = 4:

Multiple modes at once – here "1st mode + 1/2 * 4th mode"

Fifth mode – standing exponentially damped wave:

For the fifth mode, ω < ω_min(m = 5); and we have the "dispersion" relation ω² = c² ( (mπ/b)² – κ²), where κ determines the degree of wave attenuation in the waveguide (E_x ~ e^-κz). The wave does not propagate through the waveguide, but forms a standing damped wave.

Source code

Source code that generated the gifs displayed above: nb