underwater-acoustics

name	underwater-acoustics
description	Underwater acoustics — sound propagation in seawater (speed, absorption, spreading loss), sonar equation (SL-TL-NL+DI≥DT), passive and active sonar, transmission loss (spherical/cylindrical/ray tracing), transducer design (piezoelectric, BVD model, directivity, TVR/OCV), ambient noise (Wenz curves), target strength, sound speed profiles (SOFAR channel), and NATO/ISO standards for underwater acoustic systems.
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Underwater Acoustics — Complete Skill

Sound Speed in Seawater

Empirical Speed Equations

Mackenzie (1981) sound speed formula: c = 1448.96 + 4.591T − 0.05304T² + 0.0002374T³ + 1.340(S−35) + 0.01630D + 1.675×10⁻⁷D² − 1.025×10⁻²T(S−35) − 7.139×10⁻¹³TD³ Where: T = temperature [°C]; S = salinity [PSU]; D = depth [m] Simplified: c ≈ 1449 + 4.6T − 0.055T² + 0.00029T³ + (1.34−0.01T)(S−35) + 0.016D [Urick approximation]

Approximate sensitivities: ∂c/∂T ≈ +4.6 m/s per °C [dominant effect in upper ocean] ∂c/∂S ≈ +1.34 m/s per PSU ∂c/∂D ≈ +0.016 m/s per m depth (∂c/∂z from pressure increase)

Typical values: Surface (15°C, 35 PSU): c ≈ 1507 m/s Deep ocean (2°C, 35 PSU, 4000 m): c ≈ 1517 m/s Minimum at SOFAR channel: ~1480 m/s at ~1000 m depth (mid-latitude)

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Sound Propagation and Transmission Loss

Spreading Loss

Spherical spreading (no boundaries): TL_sph = 20 × log₁₀(R/R₀) [TL in dB; R = range [m]; R₀ = 1 m reference; power ∝ 1/R²] Applies: short range (R < water depth); omnidirectional source

Cylindrical spreading (deep channel, long range): TL_cyl = 10 × log₁₀(R/R₀) [power ∝ 1/R; energy trapped in channel] Applies: beyond range where sound has bounced multiple times (R >> 2H, H = water depth)

Combined (practical rule): TL = 20 log₁₀(R₁) + 10 log₁₀(R/R₁) + α×R [R₁ = transition range ≈ water depth; α = absorption coefficient]

Absorption Loss

Attenuation coefficient α [dB/km] — Thorp formula (f in Hz): α = 0.11 × f² / (1 + f²) + 44 × f² / (4100 + f²) + 2.75×10⁻⁴ × f² + 0.003 [f in kHz; valid 0.1–100 kHz]

Approximate values: 1 kHz: α ≈ 0.07 dB/km (long-range sonar frequency) 10 kHz: α ≈ 1.0 dB/km (medium range) 100 kHz: α ≈ 40 dB/km (short range, AUV obstacle avoidance) 500 kHz: α ≈ 150 dB/km (very short range; ADCP, imaging sonar)

Total transmission loss: TL = TL_spreading + α × R/1000 [TL_spreading in dB; α in dB/km; R in m]

SOFAR Channel (Sound Fixing And Ranging)

SOFAR channel: minimum sound speed at ~1000 m depth in mid-latitudes Sound rays refracted back into channel axis → trapped → propagates thousands of km with cylindrical spreading only Used by: SOSUS (US Navy hydrophone array); whale communication; T-phase earthquake detection Application: RAFOS floats transmit acoustics → tracked globally

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The Sonar Equation

Active Sonar

Active sonar equation: SE = SL − 2×TL + TS − NL + DI − DT ≥ 0 [Signal Excess ≥ 0 for detection] Where: SL = Source Level [dB re 1 μPa @ 1 m] — transmitter output TL = Transmission Loss [dB] one way; ×2 for round trip (target and back) TS = Target Strength [dB re 1 m²] — target's acoustic scattering cross-section NL = Noise Level [dB re 1 μPa] at receiver DI = Directivity Index of receiving array [dB] — array gain vs. omnidirectional DT = Detection Threshold [dB] — minimum signal excess for prescribed P_d, P_fa

Active sonar detection range: SE = 0: TL = (SL + TS − NL + DI − DT) / 2 → find R from TL formula

Example: SL = 200 dB; TS = −15 dB; NL = 70 dB; DI = 20 dB; DT = 15 dB; frequency 3 kHz TL_max = (200 − 15 − 70 + 20 − 15) / 2 = 120/2 = 60 dB Spherical: 60 = 20 log₁₀(R) → R = 10^3 = 1000 m; absorption: α(3 kHz) ≈ 0.2 dB/km More precisely: TL = 20 log(R) + 0.2R/1000 = 60 → R ≈ 920 m

Passive Sonar

Passive sonar equation: SE = SL − TL − NL + DI − DT ≥ 0 SL = Radiated Noise Level of target No TS term (one-way; no active pulse) Passive sonar: long range; covert; detects submarine radiated noise

Target Strength

Target Strength definition: TS = 10 × log₁₀(I_reflected at 1m from target / I_incident) [dB re 1 m²]

Sphere (radius a): TS = 10 log₁₀(a²/4) [geometric optics; for a >> λ] A sphere of radius 1 m: TS = 10 log(0.25) = −6 dB

Submarine: Aspect-dependent: bow/stern TS ≈ −25 to −15 dB; beam TS ≈ 0 to +15 dB

Fish and marine mammal: Fish swimbladder: TS ≈ 20 log₁₀(L) − 66 [L = fish length in cm; TS in dB at 38 kHz]

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Transducer Design

Piezoelectric Transducer

Resonant frequency: f₀ = 1 / (2π√(LC)) [BVD model: motional branch L-C-R in parallel with static capacitance C₀] For thickness-mode resonance: f₀ = c_piezo / (2t) [c_piezo = speed of sound in piezoceramic; t = thickness] PZT-4: c = 4000 m/s; t = 10 mm → f₀ = 200 kHz

Key piezoelectric material properties (PZT-4): d₃₃ = 289 pC/N (piezoelectric strain coefficient) k_t = 0.51 (electromechanical coupling factor) Q_m = 500 (mechanical Q; determines bandwidth) ε₃₃/ε₀ = 1300

Transmitting Voltage Response (TVR) and Omnidirectional Cavitation Limit: TVR = SL − 20 log₁₀(V_applied) [dB re 1 μPa/V @ 1 m] Bandwidth ≈ f₀/Q_m for piezoceramic; wider band with backing material (reduces Q)

Directivity Index: For piston radiator (circular flat plate diameter D): DI = 10 log₁₀(4πf²D² / (c² × 4)) [approximate for D >> λ] Or precisely: DI = 10 log₁₀(π²D²/λ²) [far-field piston; λ = c/f]

Hydrophone Sensitivity

Open-Circuit Voltage Response (OCV): OCV = 20 log₁₀(V_output / p_incident) [dB re 1 V/μPa] Typical: −185 to −200 dB re 1 V/μPa for single PZT hydrophone

Noise floor: Equivalent pressure noise = thermal noise of resistance R_element: p_noise = √(4kTR) / (OCV × bandwidth) [kT = 4×10⁻²¹ J at 20°C; in μPa/√Hz]

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Ambient Noise (Wenz Curves)

Noise Sources by Frequency

Wenz (1962) ambient noise levels:

Frequency	Dominant Source	Typical NL [dB re 1 μPa²/Hz]
< 1 Hz	Seismic, tidal	100–120
1–20 Hz	Shipping (distant)	70–90
20–500 Hz	Shipping traffic	50–80
500 Hz–50 kHz	Wind/sea state	30–70
> 50 kHz	Thermal noise	20–40

Wind-driven sea state noise (200 Hz–20 kHz): NL ≈ 44 − 17 log₁₀(f) + 6×SS [f in Hz; SS = sea state 0–6; NL in dB re 1 μPa²/Hz]

Total noise level in band: NL_band = NL_spectral + 10 log₁₀(Δf) [add bandwidth to spectral density level]

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Ray Tracing and Snell's Law

Snell's Law for sound rays: cos(θ) / c(z) = constant along ray [θ = grazing angle from horizontal; c(z) = sound speed at depth z] Rays bend toward lower sound speed (refraction toward SOFAR channel minimum)

Critical angle for reflection: θ_c = arccos(c₁/c₂) [for c₂ > c₁; ray beyond critical angle: total reflection at bottom/surface]

Convergence zones: In deep ocean with positive sound speed gradient at depth: rays refracted back to surface at ~60 km intervals (subtropical North Atlantic) → convergence zone gain ~10 dB

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Standards and References

Standard	Scope
IEC 60565	Hydrophones — calibration and measurement
IEC 60076-6	Transducer terminology (acoustic)
ISO 18405	Underwater acoustics terminology
ANSI/ASA S1.20	Hydrophone calibration
MIL-STD-1316	Sonar and underwater acoustic equipment
STANAG 4194	NATO standard for underwater acoustic measurements

Output

Provide: application (sonar type: active/passive/monostatic/bistatic; frequency range [Hz–kHz]; mission: detection/navigation/communication/imaging), sound speed profile (T profile [°C] vs. depth; c(z) [m/s] vs. depth; SOFAR axis depth [m]; surface duct [m] if present), transmission loss (spherical TL = 20log₁₀(R) [dB]; α(f) from Thorp [dB/km]; total TL at target range [dB]), sonar equation (active: SL [dB] + TS [dB] − 2TL − NL + DI − DT; passive: SL − TL − NL + DI − DT; Signal Excess [dB]; detection range [m/km]), transducer specs (frequency f₀ [kHz]; diameter D [mm]; λ = c/f [mm]; DI = 10log₁₀(πD/λ)² [dB]; TVR [dB re μPa/V]; OCV [dB re V/μPa]; material: PZT-4/PZT-5A/PVDF), ambient noise (sea state; dominant band; NL [dB re 1 μPa²/Hz]; NL_band [dB re 1 μPa] for bandwidth), and applicable standard (IEC 60565 for hydrophone calibration; ISO 18405; STANAG 4194 for NATO systems).