Two-Ray Ground Model

Back to Reality

Free space path loss assumes signals travel in a straight line through empty space.

But on Earth? There’s a giant reflector everywhere you go.

The ground.

Two Paths, One Receiver

When you transmit a signal near the ground, your receiver actually gets two copies:

1. Direct ray: straight line from transmitter to receiver
2. Reflected ray: bounces off the ground first

These two rays travel different distances and arrive at slightly different times.

The Interference Problem

When two waves meet, they combine.

If they’re in sync (peaks align with peaks):

They add together, and the signal gets stronger. This is constructive interference.

If they’re out of sync (peaks align with troughs):

They cancel out, and the signal gets weaker or disappears entirely. This is destructive interference.

Dead Spots

This is why you sometimes get dead spots in coverage.

At certain distances, the direct and reflected rays arrive exactly out of phase.

They cancel each other. No signal.

Move a few meters? The geometry changes. Signal comes back.

The Two-Ray Formula

At long distances, the two-ray model gives a simpler path loss:

$L_p = 40\log_{10}(d) - 20\log_{10}(h_t) - 20\log_{10}(h_r)$

• $d$ = distance between transmitter and receiver
• $h_t$ = height of transmitter antenna
• $h_r$ = height of receiver antenna

Notice something?

• Free space: loss grows with $d^2$ (20 log)
• Two-ray: loss grows with $d^4$ (40 log)

Ground reflections make things worse. Signal falls off much faster.

The Height Advantage

Look at the formula again. Antenna height helps.

$- 20\log_{10}(h_t) - 20\log_{10}(h_r)$

These are negative terms. Higher antennas = less path loss.

This is why cell towers are tall and why rooftop antennas work better than ground-level ones.

Which Model to Use?

Rule of thumb: If the signal can hit the ground, use two-ray.

Two-ray is still a simplification. Real environments add reflections off walls, diffraction around corners, and scattering. But two-ray captures the most important effect: ground reflection.