Sonar & Underwater Acoustics

Sonar applies the same estimation pattern as radar—transmit or observe a waveform, search delay/Doppler/angle hypotheses, and decide whether a target exists—but the medium changes almost every practical term. Sound speed varies with temperature, salinity, and pressure; water and boundaries refract, absorb, scatter, and reverberate; the platform and flow create self-noise; and every underwater dB quantity needs its own reference.

Active sonar transmits and listens for echoes. Passive sonar only listens to target or environmental sources. Echo sounders, multibeam bathymetry, fish finders, acoustic positioning, biological monitoring, and naval search share mechanisms while using very different frequencies, apertures, waveforms, powers, and decision criteria.

Sound propagation is a path, not one spreading law #

The rough speed scale in seawater is 1,500 m/s, but exact travel time depends on a sound-speed profile $c(z)$. Rays bend toward lower sound speed; temperature gradients, the thermocline, and pressure therefore create ducts, convergence zones, shadow zones, and multipath. Bathymetry and sediment properties determine bottom reflection, while a rough moving surface scatters and modulates energy.

For a homogeneous spherical-spreading approximation, one-way transmission loss can be written

$$TL(r,f)\approx20\log_{10}(r/r_0)+\alpha(f)r+L_{boundary}+L_{other},$$

where $r_0=1$ m and absorption coefficient $\alpha$ must use units consistent with range. Cylindrical confinement has a $10\log_{10}r$ spreading scale, but real ocean propagation is not globally one or the other. Use a measured or modeled path when range decisions matter.

Absorption rises strongly with frequency, so high frequency buys bandwidth, resolution, and smaller arrays at the cost of range. Low frequency propagates farther but needs larger apertures for the same beamwidth and encounters more shipping/flow noise.

Active and passive geometry #

flowchart TB
  subgraph PATHS["Acoustic paths"]
    direction LR
    TX["Active projector SL"] -->|"TL₁"| T["Target TS"] -->|"TL₂"| RX["Array NL, AG"]
    PS["Passive source SL"] -->|"TL"| RX
  end
  subgraph PROC["Receiver processing"]
    direction LR
    MF["Beamform + correlate"] --> D["Detect + track"]
  end
  RX --> MF

For an active monostatic approximation with equal outward and return losses,

$$SNR=SL-2TL+TS-(NL-AG),$$

where source level $SL$, target strength $TS$, noise level $NL$, and array gain $AG$ are consistently referenced and bandwidth-normalized. For unequal paths, use $TL_1+TL_2$ rather than $2TL$.

For passive reception,

$$SNR=SL-TL-(NL-AG).$$

Detection requires more than positive SNR. Processing gain, reverberation/clutter, detector loss, number of searched cells, desired detection probability, and false-alarm probability determine a detection threshold $DT$. A common operational form asks whether

$$SE=SNR-DT>0,$$

where $SE$ is signal excess. Each term's bandwidth, integration time, geometry, and statistical definition must be documented.

Worked active budget #

Use the DOSITS illustrative 8 kHz case: $SL=220$ dB re $1\ \mu$Pa at 1 m, range 10 km, spherical spreading 80 dB, absorption 5 dB, so one-way $TL=85$ dB. Let $TS=25$ dB, array gain 20 dB, and total in-band noise $NL=73$ dB re $1\ \mu$Pa. Then

$$SNR=220-2(85)+25-(73-20)=22\ \mathrm{dB}.$$

The arithmetic is exact for the stated terms; the prediction is only as good as the path and target models. Target strength changes with aspect, frequency, material, and scattering regime. Spherical spreading can miss refraction and boundary loss. Platform self-noise may dominate the nominal ambient level.

If noise is specified as a spectral level $N_0$ in dB re $1\ \mu$Pa$^2$/Hz, integrate it over effective receiver bandwidth $B_n$:

$$NL=N_0+10\log_{10}B_n.$$

Changing matched-filter bandwidth changes total noise, time resolution, and waveform processing together; it is not a free trace-floor adjustment.

Delay, bandwidth, Doppler, and ambiguity #

For monostatic two-way travel time $\tau$,

$$R=\frac{c\tau}{2}.$$

A 100 ms echo at 1,500 m/s corresponds to 75 m. Sound-speed error maps directly into range error, while refracted/multipath arrivals can make geometric path length differ from straight-line range.

Effective signal bandwidth $B$ gives range-resolution scale

$$\Delta R\approx\frac{c}{2B}.$$

A 10 kHz bandwidth gives about 7.5 cm in the ideal correlation model. A long chirp or coded pulse carries more energy while matched filtering compresses it to a bandwidth-limited peak. Windowing lowers correlation sidelobes at the cost of a wider main lobe and processing loss.

For radial relative speed $v$, the monostatic Doppler scale at carrier $f_c$ is approximately

$$f_D\approx\frac{2v}{c}f_c.$$

At $f_c=30$ kHz and $v=1$ m/s, $f_D\approx40$ Hz. Platform motion, currents, and target motion all enter relative delay/Doppler; a moving array also changes self-noise and spatial coherence.

Pulse repetition creates range and Doppler ambiguities just as in radar, but much slower propagation makes long listening intervals especially expensive. Reverberation from surface, bottom, volume scatterers, and biological layers can persist into later pings.

Arrays turn aperture into angle information #

A hydrophone converts pressure to voltage; an array samples the field spatially. Delay-and-sum beamforming aligns a candidate direction and adds channels coherently. For a uniform line array with element spacing $d$, spatial aliasing (grating lobes) appears when spacing is too large relative to wavelength; $d\le\lambda/2$ is the familiar broadside-safe rule for the full visible region.

Longer aperture narrows the main beam, while shading lowers sidelobes and widens it. Array gain is not simply element count in every environment: correlated noise, calibration error, platform scattering, sound-speed mismatch, and multipath reduce usable gain.

Bearing alone does not give range. Passive localization uses multiple arrays/platform motion, time-difference of arrival, target-motion analysis, or propagation features. Active sonar obtains range from echo delay but still needs association across bearings, delays, Dopplers, and successive pings.

Transducers, cavitation, and calibration #

Piezoelectric projectors and hydrophones are reciprocal in principle but optimized designs differ. A projector's source level, efficiency, beam pattern, impedance, depth rating, and duty cycle constrain transmission. A hydrophone's sensitivity, self-noise, dynamic range, directionality, and preamplifier loading constrain reception.

Cavitation occurs when sufficiently low local pressure forms vapor/gas cavities that collapse or persist. It can create broadband noise, damage, nonlinear distortion, and loss of projector efficiency. Threshold depends on static pressure/depth, dissolved gas, nuclei, frequency, pulse duration, and geometry; one universal “cavitation power” is not defensible.

IEC 60565-1 specifies absolute three-transducer reciprocity and comparison calibration for free-field hydrophones/projectors over stated frequency/facility limits. ISO 7605:2025 treats complete ambient-sound measurement from calibrated equipment and deployment through digital acquisition, spectra, statistics, and reporting. Those boundaries matter: a calibrated bare hydrophone does not calibrate cable, preamp, recorder clock, mounting flow noise, or deployment geometry.

A defensible sonar workflow #

1. State active/passive task, geometry, range/angle/velocity resolution, probabilities of detection and false alarm, and environmental limits.
2. Measure or model sound-speed profile, bathymetry, bottom type, surface state, absorption, ambient noise, and platform self-noise.
3. Close a link/detection budget with explicit references, bandwidths, integration, target/source statistics, and threshold.
4. Select waveform, pulse interval, aperture, element spacing, and shading together; verify ambiguity and sidelobes.
5. Calibrate hydrophones/projectors and the whole acquisition path; record uncertainty and direction/frequency corrections.
6. Validate with known targets or sources across aspect, range, depth, and environmental states.
7. Inspect raw channels, beam-time data, matched-filter outputs, and tracks; a final display can hide saturation or grating lobes.
8. Treat environmental impact and permitted source exposure as design constraints, not post-processing metadata.

Sonar performance is not one source-level or range number. It is the intersection of propagation, target/source behavior, noise and reverberation, array processing, calibration, and a declared statistical decision.