Math Bridge: Underwater Acoustic Loss and Delay

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Math BridgeWSNStruggle-friendly runway

Why do frequency and freshness pull apart underwater?

Follow one illustrative 2 km link through absorption, spreading, travel time, and a detection check.

Packet Pete, the guidePacket Pete guides
The one targetTurn carrier frequency into an honest link-and-age record.
The chapter case12 kHz, 2.00 km, sound at 1500 m/s.
What it buys youA track that does not hide acoustic delay behind an RF habit.

A field team has a real problem to settle: Why do frequency and freshness pull apart underwater? They must decide what happens before they change acoustic carrier frequency in kilohertz on the device. Predict the direction first.

See the relationship first

The figure reads from left to right. The blue card is acoustic carrier frequency in kilohertz. The middle card uses this page's rule. The green card is spreading. Follow the arrows: set the input, use the rule, then read the result and its unit.

The audit later on checks more than one number. Here, the added model uses the baseline named below and holds every other chapter value fixed. That sentence bridges the fixtures, so the numbers do not change without a reason.

Acoustic carrier frequency in kilohertz changes spreading An input card leads through the page rule to the spreading result. SET INPUT ONE CONTROL USE RULE predict calculate check units READ RESULT
Follow the arrows. Frequency controls Thorp absorption. Distance and sound speed control delay. They belong in the same evidence record but come from different steps.

Derive the baseline in four moves

  1. 1

    Name the input. The chapter baseline for acoustic carrier frequency in kilohertz is 12.

  2. 2

    Name the rule. α=Thorp(f); TL=20log10(r)+αr; t=r/csw; SNR=SL-TL-NL

  3. 3

    Put in the chapter value. Set acoustic carrier frequency in kilohertz to 12. The page rule gives spreading as 66.02 dB.

  4. 4

    Read the result. Keep dB next to the value. Use it only within the limits on this page.

Predict, then change acoustic carrier frequency in kilohertz

Try Predict what happens to spreading. Move one control, calculate, then check your idea.

12
Chapter baseline
Spreading

Observe Frequency controls Thorp absorption. Distance and sound speed control delay. They belong in the same evidence record but come from different steps. Reset to 12 and compare spreading.

Explain Only acoustic carrier frequency in kilohertz moves here. The other chapter values stay fixed.

Check yourself

What should you do before you trust the result?
Answer: Predict its direction, use the shown rule, keep the units, and reset to the worked baseline.
What does this small model leave out?
Answer: Only acoustic carrier frequency in kilohertz moves. Field effects named in the page limits stay fixed.

Honesty note. This teaching model changes only acoustic carrier frequency in kilohertz. It replays the chapter baseline; it is not a field prediction.

1. Water carries sound, not radio

Sound spreads as it travels and seawater also absorbs some of it. Higher acoustic frequency increases that absorption. Sound moves near 1500 m/s, so kilometres create seconds of delay. A fresh reading at the sensor is already old when it reaches the buoy.

Packet Pete: Put travel time beside every underwater position label.

2. Name every algebra move

1

Square the frequencyThorp's formula uses f² in each absorption term.

2

Add the absorption termsThe result α is dB lost per kilometre.

3

Add spreading and absorptionTL = 20 log10(r metres) + αr kilometres.

4

Divide distance by speedt = r/csw.

5

Subtract losses and noiseReceived level = SL − TL; SNR = received − NL.

3. Work the illustrative link

α(12)=1.64 dB/km; TL=66.0+3.3=69.3 dB; t=2000/1500=1.33 s

Radio crosses the same range in 6.67 µs. With source level 180 dB, noise level 50 dB, and a 10 dB detection threshold, the received level is 110.7 dB, SNR is 60.7 dB, and the illustrative margin is 50.7 dB.

4. Try one controlled change

α=Thorp(f); TL=20log10(r)+αr; t=r/csw; SNR=SL−TL−NL

TryMove only acoustic frequency. Range, sound speed, source level, noise, and threshold stay fixed.

Absorption
Spreading
Absorption loss
Total loss
Acoustic delay
Radio comparison
Speed ratio
Received level
SNR
Detection margin

ObserveAt 12 kHz, absorption is 1.64 dB/km and total loss is 69.3 dB. Raising frequency increases loss and cuts margin, but the 1.33 s travel time does not change in this model.

ExplainFrequency controls Thorp absorption. Distance and sound speed control delay. They belong in the same evidence record but come from different steps.

Technical boundaries.

This is a compact empirical teaching model.

Water
Temperature, salinity, depth, and pressure change sound speed and absorption
Channel
Multipath, fading, noise spectrum, and modem bandwidth are omitted
Detection
A positive ideal margin does not prove packet delivery or tracking accuracy

Measure the site channel and carry timestamp, uncertainty, and custody through the track.

5. Do not borrow an RF answer

Radio and sound both spread, but seawater absorption and acoustic speed change the design. A lower acoustic carrier may buy range while reducing bandwidth; it cannot make a two-kilometre observation current.

6. Build the release record

Record range, carrier, bandwidth, water profile, source level, noise spectrum, travel time, modem processing time, packet age, clock basis, uncertainty, retries, and the label shown when evidence becomes stale.

7. Check yourself

Why does 12 kHz lose more than 1 kHz?
Answer: Thorp's empirical absorption coefficient rises with frequency.
Why is a 2 km acoustic reading at least 1.33 s old?
Answer: Sound needs 2000/1500 = 1.33 s just to travel.
Does 50.7 dB ideal margin guarantee a useful track?
Answer: No. Real channel variation, noise, modem behaviour, loss, and target uncertainty remain.
Honesty boundary.

The chapter fixes no modem; every numeric link input here is labelled catalog-typical and illustrative.

12 kHz, 2 km
Illustrative long-range link
180/50 dB
Illustrative source and noise levels
50.7 dB
Sanity-check margin, not a deployment promise

Go deeper in the chapter, then replace every illustrative input with measured site evidence.

Return to Underwater Acoustic WSN Tracking with the maths unpacked →