Math Bridge: Two Equal Mesh Hops, 18 dB Apart

A field team faces an unresolved physical question: Why can two equal-length mesh hops differ by 18 dB? They must answer it before changing cluttered mesh path exponent on the real device. Predict the direction first.

See the relationship before changing it

The figure reads from left to right. The blue card is cluttered mesh path exponent. The middle card applies this page's relationship. The green card is open path loss. Walk the arrows once: set the input, apply the rule, then read the result with its unit.

The retained audit below checks several chapter fixtures. This added model holds every other chapter fixture fixed, so the numeric fixture does not switch without explanation.

Walk the arrows. The exponent multiplies log distance. RPL sees the result through link-quality evidence, not through geometry alone.

Derive the baseline in four named moves

1
Name the input. The chapter baseline for cluttered mesh path exponent is 3.6.
2
Name the relationship. PLopen=PL1m+10(2.2)log10(20); PLclutter=PL1m+10nlog10(20)
3
Substitute the chapter fixture. Set cluttered mesh path exponent to 3.6. The page ledger gives open path loss as 68.85 dB.
4
Read the result. Keep dB beside the value. Use it only inside the technical boundary on this page.

Predict, then change cluttered mesh path exponent

Try Predict the direction of open path loss. Move one control, calculate, then check your prediction.

Observe The exponent multiplies log distance. RPL sees the result through link-quality evidence, not through geometry alone. Reset the control to 3.6 and compare open path loss.

Explain Only cluttered mesh path exponent moves here. The other chapter fixtures remain fixed.

Check yourself

What should you do before trusting a moved-control result?

Answer: Predict its direction, apply the shown relationship, keep the units, and reset to the worked baseline.

What does this small model leave out?

Answer: Only cluttered mesh path exponent moves. Field effects named in the page's technical boundary stay fixed.

1. Equal distance does not mean equal path

Two radios can be 20 m apart in both cases. One path crosses open air; the other stays near soil, damp walls, furniture, and metal. The environment changes how quickly power falls with distance.

Pete: Hop count tells you how many links. It does not tell you what each link crosses.

2. Give the environment an exponent

Anchor at 1 mCompute PL(d0)=20log10(4πd0/λ).

Grow the lossAdd 10nlog10(d/d0).

Compare with sensitivityM=Pt−PL−Srx.

3. Compare paths without changing distance

PL(d,n)=PL(d0)+10nlog10(d/d0); M=Pt−PL−Srx

Holding radio, distance, transmit power, and receiver sensitivity fixed isolates what the environmental exponent does.

4. Try the cluttered-path exponent

PLopen=PL1m+10(2.2)log10(20); PLclutter=PL1m+10nlog10(20)

TryMove only the cluttered path's exponent. The open path stays at n=2.2 so the missing margin remains visible.

ObserveAt n=3.6, the open path is about 68.8 dB with 36.2 dB margin; the cluttered path is about 87.1 dB with 17.9 dB margin. The same distance loses about 18.2 dB more.

ExplainThe exponent multiplies log distance. RPL sees the result through link-quality evidence, not through geometry alone.

Technical boundaries.

The exponents, radio power, sensitivity, and 1 dB RSSI step are explicitly catalog-typical examples.

ETX: Needs separate evidence
rank: Needs separate evidence
energy: Needs separate evidence
hysteresis: Needs separate evidence
parent stability: Needs separate evidence
interference: Needs separate evidence
retries: Needs separate evidence
policy: Needs separate evidence

Use field evidence or a deeper model before release.

5. Reproduce the chapter values

At 2.45 GHz, λ≈0.122 m and 1 m FSPL≈40.2 dB. At 20 m, n=2.2 gives 68.8 dB and 36.2 dB margin. n=3.6 gives 87.1 dB and 17.9 dB margin. Their loss gap is about 18.2 dB, far larger than a 1 dB RSSI quantisation step.

6. Prove the parent choice

Keep RSSI/SNR distributions, retries or ETX, parent rank, battery state, route-repair time, packet freshness, latency budget, antenna placement, and results across representative locations and traffic loads.

7. Check yourself

Why can a longer route be safer?

Answer: Its individual hops may have stronger margin, fewer retries, and healthier parent energy even if hop count or latency is larger.

Does n=3.6 describe distance?

Answer: No. Distance is held at 20 m; n describes how the environment makes loss grow with distance.

Does an 18 dB RSSI gap prove application delivery?

Answer: No. It is strong link evidence, but routing state, interference, retries, queues, policy, and receiver handling still matter.

Honesty boundary.

These are the chapter inputs, worked results, and named teaching assumptions.

2.45 GHz: Frequency, sample rate, or event rate
20 m: Distance, wavelength, or size
+8 dBm: Radio power level
−97 dBm: Radio power level
n=2.2: Named physical or model constant
n=3.6: Named physical or model constant
40.2 dB: Gain, loss, margin, or level ratio
68.8 dB: Gain, loss, margin, or level ratio
87.1 dB: Gain, loss, margin, or level ratio
36.2 dB: Gain, loss, margin, or level ratio
17.9 dB: Gain, loss, margin, or level ratio
18.2 dB: Gain, loss, margin, or level ratio
1 dB: Gain, loss, margin, or level ratio

They do not certify the game's route or a real mesh deployment.