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Underwater Acoustics

Underwater Acoustic Sonar Equation Range Simulator

A real-time calculator for the active and passive sonar equations (Urick 1983). Sweep source level, target strength, ambient noise, directivity index, detection threshold, frequency, depth and water temperature, and watch sound speed, absorption coefficient, allowable transmission loss, detection range and wavelength update instantly — useful for ASW, hydrographic survey, fisheries acoustics and AUV design.

Parameters
Sonar Type
Active: round-trip 2·TL. Passive: one-way TL
Source Level SL
dB re 1μPa @ 1m
Target Strength TS
dB
Large submarine ≈ +25, small mine ≈ −20
Ambient Noise NL
dB re 1μPa/√Hz
Directivity Index DI
dB
Receive array directivity gain
Detection Threshold DT
dB
Frequency f
kHz
Higher f → absorption α grows as f²
Water Depth D
m
Water Temperature T
°C
Results
—
Sound speed c (m/s)
—
Absorption α (dB/km)
—
Allowable TL_max (dB)
—
Max range (m)
—
Max range (km)
—
Wavelength λ (m)
—
Sonar propagation concept view

Concentric blue wavefronts radiate from the sonar transducer; orange wavefronts are the echo returning from the submarine target. Top edge is the sea surface, bottom edge the seabed.

Transmission loss TL vs range R
Absorption α vs frequency f
Theory & Key Formulas

$$SL - 2\,TL + TS - (NL - DI) \geq DT,\qquad TL = 20\log_{10}R + \frac{\alpha\,R}{1000}$$

Active sonar equation (round-trip). SL = source level, TL = transmission loss (spherical spreading + absorption), TS = target strength, NL = noise, DI = directivity index, DT = detection threshold. R in m, α in dB/km.

$$SL - TL - (NL - DI) \geq DT$$

Passive sonar equation (one-way, no TS). Listens directly to noise radiated by the target.

$$\alpha \approx 0.106\,\frac{f^{2}}{f^{2}+1} + 0.52\,\frac{f^{2}}{f^{2}+4100} + 4.9\!\times\!10^{-4}\,f^{2}\;[\text{dB/km}]$$

Absorption α (simplified Francois-Garrison, f in kHz): boric-acid relaxation, magnesium-sulphate relaxation, pure-water viscosity.

$$c \approx 1449.2 + 4.6\,T - 0.055\,T^{2} + 0.016\,D\;[\text{m/s}]$$

Sound speed (UNESCO simplified, salinity 35‰). T in °C, D in m.

Underwater Acoustics — Sonar Equation, Range & Transmission Loss

🙋
So "sonar" is the "ping… ping…" thing submarines use to find each other, right? How is the detection range actually determined?
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Right, the one from the movies. The textbook answer is the sonar equation, codified by Urick back in 1983. For active sonar it reads SL − 2·TL + TS − (NL − DI) ≥ DT. Plain English: take the level you transmit, subtract the round-trip transmission loss, add the target's echo, subtract the noise minus your directivity gain, and if what's left exceeds the detection threshold — you see it. It is honestly just additions and subtractions in dB.
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Only additions! So if I know how TL grows with distance, the range basically falls out?
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Exactly. TL is spherical spreading 20·log10(R) plus absorption α·R/1000. At short range the log term dominates and the curve rises quickly; further out the linear absorption term kicks in. Look at the TL-vs-R chart — there is a clear "elbow" where the curve bends. The range where this curve crosses the horizontal TL_max line is your maximum detection range R_max.
🙋
So drop the frequency and absorption almost vanishes — why isn't every sonar low-frequency?
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Good question. Between 1 kHz and 100 kHz α changes by roughly 400×. That is why ASW sonars sit at a few kHz to a dozen kHz to chase convergence zones at 30 km, while a fish finder or a multibeam (Kongsberg EM124, Furuno) sits at 100–400 kHz because they need centimeter-class resolution and only care about a few hundred meters. Long range needs low frequency; high resolution needs high frequency — and you have to pick.
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What about NL and DI? Can't I just keep cranking SL?
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Cranking SL is the obvious move, but there is a ceiling — NOAA and IMO MEPC restrict source levels for marine mammal protection. So the three real levers are DI (bigger arrays, sharper beams), NL (quieter platforms, smarter filtering) and DT (matched filtering, beamforming, longer integration). Modern naval sonars buy 20+ dB without raising SL. Drop NL by 5 dB in the slider and watch R_max jump.
🙋
Last one — the movies always mention "hiding under the thermocline". Where does that show up in the equation?
🎓
It actually does not in this simple form. Sound speed varies with T, S and depth and the resulting profile bends rays — creating shadow zones below the thermocline and convergence zones around 30 km. To do this properly you need a ray code (Bellhop) or a parabolic-equation code (RAM). This tool is a first-order approximation, but it is still very useful for sensitivity sweeps when you change SL or NL by a few dB.

Frequently Asked Questions

The sonar equation, codified by Urick (1983), is the foundational form for underwater acoustic detection. The active version reads SL − 2·TL + TS − (NL − DI) ≥ DT and the passive version reads SL − TL − (NL − DI) ≥ DT, where SL is source level (dB re 1μPa @ 1m), TL is transmission loss (spherical spreading 20·log10(R) plus absorption α·R/1000), TS is target strength, NL is ambient noise, DI is array directivity index and DT is detection threshold. This tool finds the range R that maximizes the left-hand side using iterative root-finding.
Seawater absorption comes from boric acid B(OH)3 and magnesium sulphate MgSO4 molecular relaxation plus pure water viscosity, all of which contain f² terms. Molecular relaxation dominates at low frequency and viscous absorption dominates above 100 kHz. The Francois-Garrison (1982) formula is the standard, giving roughly 0.07 dB/km at 1 kHz, 1 dB/km at 10 kHz and more than 30 dB/km at 100 kHz. The tool implements the three main terms.
Active sonar transmits and listens for the echo, so the transmission loss is doubled to 2·TL, halving the allowable TL. Passive sonar only listens to the noise the target radiates one way, and there is no TS term. With the same assumed SL passive usually reaches further, but for quiet modern submarines the radiated SL is too low for this to hold. The active/passive toggle in this tool switches the divisor (SE_factor) between 2 and 1 and recomputes TL_max accordingly.
This tool uses only spherical spreading and absorption. In real oceans the sound-speed profile creates a SOFAR channel (minimum near 1000 m depth) and convergence zones (re-focusing roughly every 30 km), causing TL to vary by ±20 dB over distance. Long-range (>30 km) studies require ray or PE solvers such as Bellhop or RAM. Use this tool as a first-order estimate for system sizing and sensitivity sweeps.

Real-world Applications

Anti-submarine warfare and surface combatant sonar: Hull-mounted sonars such as the AN/SQS-53C on Arleigh Burke destroyers, dipping sonars on MH-60R helicopters, P-8 Poseidon sonobuoys and the JMSDF P-1 patrol aircraft combine a few-kHz active source with broadband passive listening, exploiting convergence zones to reach tens of kilometres. The defaults (SL 220 dB, f around 3–10 kHz, DT 5–10 dB) sit in that regime.

Bathymetry and hydrography: Multibeam echo sounders like Kongsberg EM124 (12 kHz) or EM2040 (200–400 kHz) and Edgetech side-scan systems trade range for resolution via frequency. Slide f up to 100–200 kHz in the tool and watch the absorption rocket so that detection range collapses to a few hundred metres — a perfect picture of what a hydrographic echo sounder actually sees.

Fisheries and aquaculture: Furuno, JFE Advantech and Simrad EK80 scientific echo sounders cycle through 38 / 70 / 120 / 200 kHz depending on species and water column layer. Typical TS values: a 30 cm mackerel sits near −37 dB, a 1 m tuna around −25 dB. The simulator's TS slider lets you sweep across these target classes.

AUV and pipeline inspection: Vehicles such as Saab Sabertooth or Boston Engineering Bluefin use 300–600 kHz forward-look sonar for obstacle avoidance, 100 kHz for seabed mapping and low-frequency acoustic modems for long-haul telemetry. Subsea pipeline corrosion and buckling surveys typically combine side-scan with sub-bottom profilers. Acoustic Stealth (lowering submarine SL) and Convergence-Zone exploitation remain central tactical topics for naval architects.

Common Misconceptions and Pitfalls

The biggest trap is to assume "spherical spreading plus absorption explains real-world TL". The TL = 20·log10(R) + α·R/1000 model in this tool is the deep-water, isovelocity, free-field idealisation. Real oceans have refractive sound-speed profiles producing SOFAR channels, convergence zones and shadow zones; TL can swing by ±20 dB at 10–30 km. In shallow water (<100 m) cylindrical spreading 10·log10(R) plus surface/bottom boundary effects dominates. Treat this calculator as a first-order free-field estimate; production studies need Bellhop (ray) or RAM (PE) solvers.

Second, mixing up dB reference values. Underwater acoustics uses dB re 1μPa @ 1m, while in-air acoustics uses dB re 20μPa — a 26 dB difference. Pouring in-air SPL values into a sonar equation will give wildly wrong SNRs. Also, NL is a spectral density in dB re 1μPa/√Hz; to obtain band-integrated noise you must add 10·log10(B) where B is the bandwidth in Hz. The tool assumes B = 1000 Hz internally; rescale for your real signal bandwidth.

Finally, TS and DT are not constants. Target strength depends on aspect (broadside is strong, bow/stern can be 20 dB weaker), frequency and anti-acoustic coatings. The detection threshold DT is itself a design parameter built from pulse length, correlation time, false-alarm probability Pfa and detection probability Pd; a typical range is 0–15 dB. And remember NOAA / IMO MEPC 1/Circ.833 caps source levels in some bands for marine mammal protection. Use the numbers here for education and first-cut design, and always cross-check against ONR / Navy procedures (Wagner & Mylander, RAYMODE, etc.) for production work.

How to Use

  1. Set Source Level (dB re 1 µPa @ 1m) using the slider—typical naval sonar ranges 200–240 dB for active systems, 150–190 dB for passive.
  2. Input Target Strength (dB re 1 m²) for your object; submarines average −5 to −15 dB, surface ships +10 to +20 dB.
  3. Adjust Ambient Noise (dB re 1 µPa), Directivity Index (dB), and frequency to compute max detection range via the Urick sonar equation: SL − 2TL + TS + DI ≥ NL + DT.

Worked Example

A naval frigate operating at 10 kHz detects a submarine: Source Level = 220 dB, Target Strength = −8 dB, Ambient Noise = 60 dB, Directivity Index = 15 dB, sound speed c = 1500 m/s. Absorption coefficient α ≈ 0.045 dB/km. The allowable transmission loss TL_max = 127 dB, yielding maximum range ≈ 8.2 km (spreading loss dominates at shallow frequencies; active sonar range decreases with frequency squared).

Practical Notes

  1. Temperature and salinity stratification (thermoclines) bend sound rays in channels; adjust sound speed c downward (1480–1540 m/s) to model convergence zones where range extends 30–50 km.
  2. Reverberation from seafloor and surface (especially in 100–500 m depths) reduces effective range; increase Ambient Noise by 5–10 dB for shallow water operations.
  3. Frequency sweep from 1 kHz (long-range passive detection) to 100+ kHz (short-range active classification) reveals trade-off between detection range and target resolution.