Underwater Acoustic Communication Range Calculator

Close an acoustic link budget before you go to sea. Enter a modem source level, the receiver noise and detection threshold, the spreading law that matches your water column, and the temperature, salinity, depth and pH of the site; the calculator returns the absorption coefficient, the allowable transmission loss, the range at which the link closes, and the acoustic latency at that range.

Introduction to underwater acoustic link budgets

Seawater is an excellent conductor, which makes it close to opaque at radio frequencies: a 100 MHz carrier is attenuated by hundreds of decibels within a few metres. Optical links do better but are defeated by turbidity and biofouling beyond a few tens of metres in most coastal water. Sound is the only carrier that reaches useful distances, which is why every deep-water telemetry system, acoustic release, USBL positioning head and AUV command link operates somewhere between roughly 1 kHz and 100 kHz.

The engineering question is always the same: given a transducer that can radiate a certain acoustic power, and a receiver that needs a certain signal-to-noise ratio to demodulate a packet, how far apart can the two ends be? The answer is bookkeeping in decibels, and the ledger is the sonar equation. Every gain and every loss along the path is expressed logarithmically, summed, and compared against the margin the demodulator needs. This page implements the one-way (communications) form of that ledger and then inverts it for range.

One convention deserves emphasis before anything else. Underwater sound pressure levels are referenced to 1 micropascal, not to the 20 micropascals used in airborne acoustics, and source levels are additionally referenced to a nominal distance of 1 m from the acoustic centre. Because the two references differ by a factor of 20 in pressure, the same physical pressure reads 26 dB higher when quoted re 1 µPa than when quoted re 20 µPa; comparing acoustic intensities between the two media adds a further 36 dB from the difference in characteristic impedance. A commercial modem quoted at 190 dB is 190 dB re 1 µPa at 1 m. Drop an in-air figure into the source level box and the whole budget is wrong. Check the suffix every time.

How to use the estimator before a sea trial

Work through the form from the top. Source level comes from the modem datasheet; it is normally quoted as dB re 1 µPa at 1 m and often given as a range because many modems have selectable power steps. Carrier frequency is the centre frequency of the acoustic band, in kilohertz, and it drives absorption far more strongly than anything else on the form.

Ambient noise can be entered two ways. If you have a band level — the total noise power integrated across the receiver bandwidth, in dB re 1 µPa — choose the band-level mode and enter it directly. If you are reading a value off a Wenz-style spectrum plot, that number is a spectrum level in dB re 1 µPa²/Hz, and the calculator will integrate it over the bandwidth you supply. Receiver bandwidth is used for that conversion and is also reported back so you can sanity-check the implied spectrum level.

Directivity index is the array gain of the receiving transducer against isotropic noise; a single omnidirectional hydrophone is 0 dB, a modest transducer with a 30° beam is roughly 10 dB, and long towed or bottom-moored arrays reach 20 dB and beyond. Detection threshold is the signal-to-noise ratio the demodulator needs in the receiver band for the target bit error rate; incoherent FSK modems typically want 8–12 dB, while coherent PSK with strong forward error correction can work near 0 dB or below.

The spreading exponent selects the geometry of the wavefront. Choose spherical in deep water at ranges short compared with the depth, cylindrical once the field is trapped between the surface and the seabed or inside a sound channel, and the practical intermediate value when you are in shallow water at an intermediate range and want a defensible compromise. Finally, the environment block — temperature, salinity, mean depth and pH — feeds the absorption model; take those from a CTD cast or a climatology such as the World Ocean Atlas rather than guessing.

Results update as you type. The panel reports the absorption coefficient, the allowable transmission loss, the range, how that loss splits between geometry and chemistry, the sound speed and the one-way acoustic latency. Below the panel a frequency sweep and a link-budget chart show how the answer moves if you change carrier frequency, which is usually the single most productive design lever.

Sonar equation formula and the Francois-Garrison absorption model

The one-way sonar equation for a communications link states that the signal-to-noise ratio available at the receiver is the source level minus the transmission loss, corrected for receive array gain and the noise level:

Here $\mathrm{SL}$ is the source level in dB re 1 µPa at 1 m, $\mathrm{TL}$ is the one-way transmission loss in dB, $\mathrm{DI}$ is the receiver directivity index in dB, and $\mathrm{NL}$ is the ambient noise level in the receiver band in dB re 1 µPa. The link closes when the available SNR equals the detection threshold $\mathrm{DT}$, so the maximum transmission loss the budget can absorb is

Because noise is quoted per hertz on the Wenz curves, a spectrum level $\mathrm{NSL}$ must be integrated over the receiver bandwidth $B$ in hertz before it enters the equation:

Transmission loss is modelled as a geometric spreading term plus a chemical absorption term that grows linearly with range:

The spreading exponent is $N=20$ for spherical spreading, because intensity falls as the inverse square of range; $N=10$ for cylindrical spreading in a waveguide, where intensity falls only as the inverse of range; and Urick recommends the empirical compromise $N=15$ for shallow water at intermediate ranges.

The two terms carry different length units, and this is where careless implementations go wrong. The logarithmic term is referenced to the same 1 m distance at which the source level is defined, so $r$ inside the logarithm must be in metres; absorption coefficients, meanwhile, are conventionally published in dB per kilometre. Writing the equation with a range in kilometres inside the logarithm silently moves the source-level reference distance from 1 m to 1 km and inflates the predicted range by orders of magnitude. This page evaluates

so that a range of 1 m gives zero spreading loss, exactly as the source-level definition requires.

The absorption coefficient is where cruder calculators go wrong. Francois and Garrison fitted ocean measurements to a three-relaxation model that remains the reference expression today:

The first term is the boric acid relaxation, which dominates below about 1 kHz; the second is magnesium sulphate, which dominates from a few kilohertz to a few hundred kilohertz; the third is the viscous absorption of pure water, which takes over above roughly 200 kHz. With $f$ in kHz, $T$ in °C, $S$ in parts per thousand, $D$ in metres and $\theta =T+273$ in kelvin, the boric acid coefficients are

the magnesium sulphate coefficients are

and the pure-water viscous coefficients are given on two temperature branches:

for $T\le 20$ °C, and

for $T>20$ °C, with the depth correction

The sound speed $c$ that appears inside $A_1$ and $A_2$ is the simple linear expression published alongside the absorption model,

while the sound speed reported in the results panel, and used for the acoustic latency, is the more accurate nine-term equation of Mackenzie:

Solving $\mathrm{TL}\left(r\right)={\mathrm{TL}}_{\max }$ for $r$ has no closed form in elementary functions, but the left-hand side is strictly increasing on $r>0$, so the root is unique and bisection converges unconditionally. This page brackets the root by doubling an upper bound until the loss exceeds the budget, then bisects to machine precision. Naive fixed-point iteration of the form $r\leftarrow {10}^{({\mathrm{TL}}_{\max }-\alpha r)/N}$ looks tempting but is unstable: the derivative of that map at the fixed point has magnitude $\alpha r\hspace{0.17em}\ln 10/N$, so the iteration diverges as soon as the absorption loss exceeds $N/\ln 10$ decibels — about 8.7 dB for spherical spreading, which almost every real link exceeds.

Finally, the received level and the surviving margin at the solved range follow from the same ledger:

Worked example: a 10 kHz AUV telemetry link

Take a mid-frequency modem radiating $\mathrm{SL}=190$ dB re 1 µPa at 1 m on a 10 kHz carrier, received on a single omnidirectional hydrophone ($\mathrm{DI}=0$ dB) that measures a band noise level of 70 dB re 1 µPa. The demodulator needs $\mathrm{DT}=10$ dB. The site is temperate open water: 10 °C, 35 ppt, mean depth 100 m, pH 8.0, and the range is short enough relative to the depth that spherical spreading ($N=20$) is the honest choice.

The Francois-Garrison sound speed is 1487.4 m/s, the boric acid relaxation sits at 1.12 kHz and the magnesium sulphate relaxation at 75.9 kHz. Evaluating the three terms at 10 kHz gives 0.114 dB/km from boric acid, 0.805 dB/km from magnesium sulphate and 0.031 dB/km from pure water, so $\alpha =0.950$ dB/km. The allowable loss is

Bisecting $20{\log }_{10}\left(1000r_{\text{km}}\right)+0.950r_{\text{km}}=110$ gives $r=23.69$ km. At that range the spreading term contributes 87.49 dB and absorption 22.51 dB, so about a fifth of the budget goes on chemistry and the rest on geometry. The arithmetic checks itself: the received level is $190-110=80$ dB re 1 µPa, which is exactly the 70 dB noise level plus the 10 dB threshold, as it must be at the marginal range. Mackenzie sound speed at these conditions is 1491.4 m/s, so a packet takes 15.9 s to travel one way — which is why acoustic network protocols are designed around latency rather than throughput.

Interpreting the range figure this page returns

The number is the range at which the received signal-to-noise ratio equals the detection threshold you supplied — the point at which the link is exactly marginal, not the range at which it is comfortable. Because the absorption term grows linearly with range while the spreading term grows only logarithmically, the value of an extra decibel collapses as frequency rises. Adding six decibels of budget (four times the transmitted power) takes the worked example from 23.69 km to 28.36 km, a gain of under 20 percent. The same six decibels at 1 kHz, where absorption is only 0.060 dB/km, moves the range from 129.4 km to 181.0 km — a gain of 40 percent.

That asymmetry is the practical lesson. At low frequency the budget is spent almost entirely on geometry, so range responds strongly to power and array gain. As frequency rises the absorption share climbs, extra source level buys progressively less, and the effective levers become lowering the carrier frequency, adding array gain at the receiver, improving the modulation so the detection threshold falls, or inserting a relay node. The absorption-share column is the diagnostic for which regime you are in.

The acoustic latency figure deserves attention in its own right. A one-way delay of tens of seconds means that stop-and-wait protocols collapse, that a round-trip acknowledgement can cost a minute or more, and that any control loop closed over the acoustic link must tolerate delays comparable with the vehicle dynamics. Range and latency should be read together when sizing a network.

Limitations and assumptions behind this range model

This is a range-independent sonar equation, and it is deliberately optimistic. It assumes a single spreading law that holds over the whole path, an isovelocity water column with no refraction, no ducts and no shadow zones, no reflection loss at the surface or seabed, a stationary and spatially uniform noise field, a frequency-flat receiver response, and a demodulator whose performance is fully described by a single detection threshold. None of those assumptions survives contact with a real ocean.

Refraction alone can dominate. A downward-refracting summer profile bends energy into the seabed and can cut range by more than half relative to a straight-ray prediction; a surface duct or the deep sound channel can do the opposite and carry low-frequency energy far beyond anything this page predicts. If the answer matters, run a ray or parabolic-equation model such as Bellhop or RAM with a measured sound speed profile and use this calculator only to bound the search.

The channel is also time-varying. Delay spreads of tens of milliseconds, Doppler from platform motion and moving sea surfaces, and fast fading that removes 10 dB or more of margin for seconds at a time are routine. Common practice is to treat the value returned here as an upper bound, then either design for roughly half of it or add an explicit fading margin of 6–12 dB to the detection threshold before pressing calculate.

The absorption model has its own boundaries. Francois and Garrison state validity from about 200 Hz to 1 MHz, and the calculator refuses input outside that window. Accuracy within it is not uniform: later field measurements published in the Journal of the Acoustical Society of America found that the equation underestimates absorption near 333 kHz for some water conditions. The pH dependence assumes an open-ocean borate equilibrium and is unreliable in estuaries or near hydrothermal input. The Thorp alternative offered on the form is a low-frequency fit with no environmental inputs at all; it is provided for comparison and for reproducing older literature, not as a substitute.

The Mackenzie sound speed equation is stated for −2 to 30 °C, 30 to 40 ppt and 0 to 8000 m, and the page flags inputs outside that envelope. Finally, nothing here models the transmitter: cavitation limits the achievable source level near the surface, transducer efficiency varies with frequency, and battery energy per packet is often the binding constraint on an autonomous node long before acoustics is.

Questions engineers ask about acoustic modem range

Sources behind the equations on this page

Sources. Sonar equation, spreading laws and the practical shallow-water exponent: R. J. Urick, Principles of Underwater Sound, 3rd edition, McGraw-Hill, 1983 (chapters 2, 5 and 6). Absorption coefficient: R. E. Francois and G. R. Garrison, “Sound absorption based on ocean measurements. Part I: Pure water and magnesium sulfate contributions”, Journal of the Acoustical Society of America 72(3), 896–907 (1982), and “Part II: Boric acid contribution and equation for total absorption”, JASA 72(6), 1879–1890 (1982), doi:10.1121/1.388673. Independent numerical check of this implementation: G. J. Macaulay, D. Chu and E. Ona, “Field measurements of acoustic absorption in seawater from 38 to 360 kHz”, JASA 148(1), 100–107 (2020), NOAA Institutional Repository, repository.library.noaa.gov/view/noaa/53948, which quotes 0.1 dB/km (boric acid), 36.7 dB/km (magnesium sulphate) and 39.0 dB/km (pure water) at 333 kHz, 7 °C, 35 ppt and 10 m; this page reproduces 0.104, 36.73 and 39.01 dB/km at the same inputs. Sound speed: K. V. Mackenzie, “Nine-term equation for sound speed in the oceans”, JASA 70(3), 807–812 (1981). Low-frequency comparison expression: Thorp’s formula as tabulated by Urick. Ambient noise spectra referred to in the guidance above: G. M. Wenz, “Acoustic ambient noise in the ocean: spectra and sources”, JASA 34(12), 1936–1956 (1962).

One transcription caveat, stated plainly. The boric acid amplitude is written here as 8.86/c; some published implementations of Francois and Garrison use 8.696/c instead. The two differ by under two percent in a term that only matters below about 10 kHz, and the validation figures above cannot distinguish them because the reference value is quoted to one decimal place. If your application depends on sub-decibel accuracy below 1 kHz, take the constant from the original 1982 paper rather than from this page.