An engineer wants to transmit a 5 GHz microwave signal through a rectangular waveguide whose dominant mode has a cutoff frequency of 6 GHz. What will happen to the signal?
AThe signal propagates normally because 5 GHz is close enough to the cutoff
BThe signal propagates but at reduced speed compared to free space
CThe signal decays exponentially along the waveguide and does not propagate
DThe signal reflects back toward the source and sets up a standing wave
Below the cutoff frequency ω_c = k_c · c, the dispersion relation k_z² = (ω/c)² − k_c² gives a negative value for k_z², making k_z imaginary. An imaginary k_z means the field varies as e^(−|k_z|z) — exponential decay, not propagation. This is an evanescent mode. The signal does not travel through the waveguide; it is attenuated exponentially from the input. This is the key engineering constraint: waveguide dimensions must be chosen so the operating frequency sits above the dominant-mode cutoff.
Question 2 Multiple Choice
Why can a single hollow rectangular waveguide (one metal tube, no inner conductor) not support a TEM mode?
ABecause the rectangular geometry forces the fields to be purely transverse
BBecause TEM modes require a second conductor to complete the return current path
CBecause TEM modes have zero cutoff frequency, which conflicts with the waveguide's boundary conditions
DBecause the metal walls absorb transverse field components
A TEM (transverse electromagnetic) mode has both E and B purely transverse to the propagation direction. By Ampere's law, a purely transverse B field requires a longitudinal current, which must flow on a conductor. In a coaxial cable, the inner conductor provides this return path. A hollow waveguide with only one conductor (the outer tube) has no inner conductor, so TEM cannot exist. Instead, waveguides support TE modes (E_z = 0, B_z ≠ 0) or TM modes (B_z = 0, E_z ≠ 0), where one field component is longitudinal and drives the transverse fields.
Question 3 True / False
In a waveguide, once you solve the 2D eigenvalue problem for the single longitudinal field component (E_z for TM or B_z for TE), all transverse field components can be determined from it algebraically.
TTrue
FFalse
Answer: True
This is correct and is one of the most powerful structural features of waveguide analysis. After separating longitudinal and transverse dependencies, Maxwell's equations reduce to a 2D Helmholtz equation for the longitudinal component. Once that equation is solved (giving the mode shape and the cutoff wavenumber k_c), the transverse components E_x, E_y, B_x, B_y follow directly from algebraic relationships involving k_z, k_c, and the longitudinal component and its derivatives. This is why the mode is fully characterized by solving a single scalar PDE.
Question 4 True / False
The cutoff frequency of a waveguide mode is determined primarily by the length of the waveguide rather than its cross-sectional dimensions.
TTrue
FFalse
Answer: False
The cutoff frequency is determined entirely by the transverse geometry — the cross-sectional shape and dimensions of the waveguide. The cutoff wavenumber k_c is the eigenvalue of the 2D Helmholtz equation solved on the cross-section with boundary conditions on the walls. For a rectangular waveguide of width a and height b, the TE_{mn} cutoff wavenumber is k_c = π√((m/a)² + (n/b)²). Length affects the longitudinal standing wave structure in a cavity resonator, but not the cutoff frequencies of propagating modes.
Question 5 Short Answer
What physically happens to an electromagnetic wave whose frequency is below the cutoff frequency of all modes in a waveguide, and why does the dispersion relation predict this?
Think about your answer, then reveal below.
Model answer: The wave does not propagate — it decays exponentially along the waveguide length (evanescent behavior). The dispersion relation k_z² = (ω/c)² − k_c² gives a negative value when ω < ω_c = k_c·c, so k_z is imaginary. Writing k_z = iα (α real and positive), the longitudinal dependence becomes e^(ikzz) = e^(−αz) — exponential decay rather than oscillation. No energy is transmitted along the waveguide; the field is localized near the input and falls off on a scale of 1/α.
The evanescent nature below cutoff is a direct consequence of the dispersion relation. The transverse eigenvalue k_c² comes from the geometry and boundary conditions; it is fixed. The longitudinal propagation constant k_z must then satisfy k_z² = (ω/c)² − k_c². When the wave frequency ω is too low, (ω/c)² < k_c², and k_z must be imaginary. This is not attenuation due to absorption — a perfect conductor waveguide has no resistive loss — but a geometric constraint: the transverse standing wave pattern requires a minimum frequency to exist as a propagating mode.