Sonar Range Equation Calculator
Compute signal excess, figure of merit and the maximum detection range of an active or passive sonar from the sonar equation in decibels.
📡 What is the Sonar Range Equation?
The sonar range equation, usually just called the sonar equation, is a running tally of decibels that tells you whether a sonar can hear its target. It starts with the sound the source puts into the water, subtracts what the sea takes away on the way, adds what the target and the receiving array contribute, subtracts the background noise and the margin needed to call a detection, and checks whether anything is left over. That leftover is the signal excess SE. If SE is zero or positive the target is detected; if it is negative the signal is buried. Because every term is a level in decibels relative to 1 micropascal, the whole calculation is simple addition and subtraction.
There are two versions. An active sonar, such as a naval echo-ranging set or a fish-finding echo sounder, sends a ping and listens for the echo, so the sound pays the transmission loss twice and the target reflects it with a target strength TS. A passive sonar, such as a hydrophone on a buoy or a towed array, only listens to sound the target radiates itself, so the loss is paid once and there is no target strength. Both are used to find submarines and ships, to survey fish, to track whales and to monitor underwater noise.
The most useful answer is usually not the signal excess at one range but the range at which signal excess reaches zero. Since transmission loss climbs steadily with distance, that maximum detection range is unique, and it depends on the figure of merit, the one-way loss the sonar can afford. A frequent mistake is to expect range to double when the source level rises by 6 dB. With spherical spreading it does, but absorption makes the gain smaller at high frequency, and in active mode the same 6 dB buys only a 41 percent range increase, because the loss is doubled.
This calculator handles both modes with sliders and defaults, finds the maximum detection range by bisection, reports the figure of merit and the detection verdict at your range, flags the case where no detection is possible even at 1 m, and draws signal excess against range on a logarithmic axis. It is a budget tool, not a propagation model, so treat results as a first estimate.
📐 Formula
Passive: SE = SL − TL − NL + DI − DT
📖 How to Use This Calculator
Steps
💡 Example Calculations
Example 1 — Active ASW Sonar
SL = 220, TS = 15, NL = 65, DI = 20, DT = 10 dB, α = 0.06 dB/km, spherical spreading, r = 10 km
Example 2 — Fish Finder Echo Sounder
SL = 220, TS = −40 (single fish), NL = 50, DI = 25, DT = 10 dB, α = 55 dB/km (200 kHz), r = 0.5 km
Example 3 — Passive Hydrophone Listening to a Ship
SL = 170, NL = 75, DI = 10, DT = 10 dB, α = 0.01 dB/km, spherical spreading, r = 20 km
Example 4 — Marine Mammal Call
Low frequency whale call: SL = 188, NL = 85, DI = 0 (single hydrophone), DT = 5 dB, α = 0.001 dB/km, r = 50 km
❓ Frequently Asked Questions
🔗 Related Calculators
What is the sonar equation?
The sonar equation is a decibel energy budget that says whether a sonar can detect a target. For active sonar it is SE = SL - 2TL + TS - NL + DI - DT, and for passive sonar SE = SL - TL - NL + DI - DT. A signal excess SE of zero or more means the signal is at or above the detection threshold.
How do I calculate the maximum detection range of a sonar?
Set the signal excess to zero and solve for range. That is the same as finding the range at which the one-way transmission loss equals the figure of merit. Because TL = 20 log10 r + alpha r / 1000 increases steadily with range, bisection finds the single root quickly. This calculator does exactly that.
What is the figure of merit of a sonar?
The figure of merit FOM is the largest one-way transmission loss the sonar can tolerate and still detect the target. For passive sonar it is SL - NL + DI - DT. For active sonar on a one-way budget it is (SL - NL + DI - DT + TS) / 2, because the loss is paid twice.
What do SL, TL, TS, NL, DI and DT stand for?
SL is the source level (dB re 1 uPa at 1 m), TL the transmission loss, TS the target strength, NL the noise level, DI the directivity index of the receiving array, and DT the detection threshold, the signal to noise ratio needed for a given probability of detection.
What is the transmission loss formula for sonar?
Here TL(r) = 20 log10(r) + alpha r / 1000 for spherical spreading, with r in metres and alpha in dB/km. In shallow water with cylindrical spreading the first term becomes 10 log10(r). TL is referenced to 1 m, so it is zero at 1 m apart from absorption.
Why is the active sonar equation 2TL and the passive one TL?
An active sonar pings and listens for the echo, so the sound crosses the water twice and loses TL on the way out and again on the way back. A passive sonar only receives sound that the target itself radiated, so it pays the transmission loss once.
What absorption coefficient should I use for sonar?
Absorption depends strongly on frequency. Typical seawater values are about 0.07 dB/km at 1 kHz, about 1 dB/km at 10 kHz, about 35 dB/km at 100 kHz and about 50 dB/km at 200 kHz. Below 1 kHz it is under 0.1 dB/km, which is why low frequency sonar can reach so far.
What does a negative signal excess mean?
A negative SE means the received echo or signal is below the level needed to meet the detection threshold, so the target is not reliably detected at that range. A positive SE is margin. For example, SE = +10 dB means the signal can fall 10 dB, or in an active sonar the one-way loss can rise 5 dB, before detection is lost.
When should I use cylindrical spreading?
Use 10 log10 r when the sound is confined between the sea surface and a reflecting bottom or trapped in a duct, usually at ranges well beyond a few times the water depth. Near the source the wave spreads spherically at 20 log10 r. Real propagation often lies between the two.
Why does the calculator report no detection?
It reports no detection when the signal excess is negative even at 1 m, the reference distance of the source level. That happens when the figure of merit is smaller than the loss at 1 m, for example with a very weak source, a very noisy sea or a high detection threshold. No range can fix that.
How accurate is the sonar equation?
It is a budget, not a propagation model. Real transmission loss depends on sound speed profile, refraction, bottom type, surface roughness and the sonar geometry, and noise and target strength vary with aspect and sea state. Treat the result as an estimate good to several dB, then refine it with a propagation model.