RF Theory › Transmission Lines
Standing Waves on a Transmission Line
VSWR, reflection coefficient and node/antinode positions are usually taught as static formulas. This page animates the actual physics — two traveling waves summing in real time — so you can see why the pattern stands still even though both waves keep moving.
01 — The Physics
Why Two Moving Waves Look Still
When a transmission line is terminated in anything other than its characteristic impedance Z₀, part of the signal reflects back toward the source. At every point on the line you now have two waves travelling in opposite directions at the same time — the incident wave heading toward the load, and the reflected wave heading back toward the generator.
Add them together at any fixed point in space and watch that sum over time, and something surprising happens: the combined wave doesn't appear to travel anywhere. Its amplitude at each position oscillates up and down in place, but the pattern — where the swings are big and where they're near zero — stays fixed. That fixed pattern is the standing wave.
Voltage on the Line, Distance d from the Load
V(d) = V⁺·[e+jβd + ΓL·e−jβd]The first term is the incident wave, the second is the reflected wave scaled by the load's reflection coefficient ΓL = |Γ|∠θ. Take the real, time-varying part of each and you get two counter-propagating sinusoids:
v(d,t) = cos(ωt + βd) (incident, moving toward the load)
+ |Γ|·cos(ωt − βd + θ) (reflected, moving toward the generator)
The key idea: wherever the two waves are always exactly out of phase, no matter what instant you look at, the sum is always zero — that point is a node. Wherever they're always exactly in phase, the sum swings between the largest possible positive and negative values — that's an antinode. Everywhere in between oscillates with some intermediate, fixed amplitude. Nothing here is moving sideways — only up and down, in place.
02 — See It Live
Interactive Visualizer
The solid cyan trace is the actual instantaneous voltage on the line, animating in real time. The dashed amber lines are the envelope — the maximum swing the solid trace can ever reach at each position. Adjust the reflection coefficient below, or jump straight to a textbook case with the preset buttons.
03 — Design Numbers
Reading Node & Antinode Positions
The envelope's shape depends only on |Γ| and θ_Γ. Its peak-to-trough ratio is the VSWR you already know; its peak and trough positions along the line are just as important, but rarely visualized.
Envelope, VSWR and Position Formulas
|V(d)| = |V⁺|·√(1 + |Γ|² + 2|Γ|·cos(θ − 2βd))VSWR = |V|max / |V|min = (1+|Γ|) / (1−|Γ|)
First voltage maximum at distance dmax from the load (smallest non-negative solution):
dmax = (θ / 4π) · λ (mod λ/2)
First voltage minimum is always exactly λ/4 away from the nearest maximum:
dmin = dmax ± λ/4
| Load | Γ_L | Vmax at d = | Vmin at d = | Why |
|---|---|---|---|---|
| Matched (Z_L = Z₀) | 0 | no standing wave — flat line, VSWR = 1 | nothing reflects | |
| Open circuit | +1 (θ=0°) | 0 (at the load) | λ/4 | current must be zero at an open — voltage is maximum there |
| Short circuit | −1 (θ=180°) | λ/4 | 0 (at the load) | voltage must be zero at a short — that's the boundary condition |
| Z_L > Z₀ (real) | real, positive | 0 (at the load) | λ/4 | same sense as open, just less extreme |
| Z_L < Z₀ (real) | real, negative | λ/4 | 0 (at the load) | same sense as short, just less extreme |
This is exactly how a slotted-line VSWR meter measures an unknown load — before vector network analyzers existed, engineers slid a probe along an air line and physically measured where the minima fell. The minimum position gives you θ_Γ (hence the sign/type of the reactance), and the min/max ratio gives you |Γ| — together, the complete load impedance, with no phase-sensitive receiver required.
04 — Practical Consequence
Why This Matters for Power Handling
A mismatched line doesn't just waste power in reflection — it also concentrates voltage at the antinodes. The peak voltage anywhere on the line is |V⁺|·(1+|Γ|), which can be far higher than the matched-line voltage for the same forward power.
Voltage Derating from VSWR
Vpeak = Vmatched · (1 + |Γ|) = Vmatched · VSWR+1 (scaling relative to a matched line carrying the same forward power)At VSWR = 3:1 (|Γ| = 0.5): peak voltage is 1.5× the matched-line value → peak power handling capability drops to 1/1.5² ≈ 44% of the matched rating.
At VSWR = 5:1 (|Γ| = 0.667): peak voltage is 1.67× → power handling drops to ~36%.
Real-world failure mode: a poorly matched high-power transmitter line can arc or suffer dielectric breakdown exactly at a voltage antinode — even though the average forward power looks perfectly safe. This is why every high-power RF system datasheet specifies a maximum operating VSWR, not just a maximum forward power.
This same antinode/node structure is also the physical basis for the λ/4 transformer and single/double-stub matching techniques covered on the main Transmission Line Theory page — every matching network works by manipulating exactly the pattern you just animated above.