Acoustic Propagation Theory
Sound Speed in Seawater
The speed of sound in seawater depends on temperature (T), salinity (S), and pressure (P).
We use the Mackenzie empirical formula:
c = 1448.96 + 4.591×T - 5.304×10⁻²×T² + 2.374×10⁻⁴×T³
+ 1.340×(S - 35) + 1.630×10⁻²×D + 1.675×10⁻⁷×D²
- 1.025×10⁻²×T×(S - 35) - 7.139×10⁻¹³×T×D³
where:
- T = temperature (°C)
- S = salinity (PSU, parts per thousand)
- D = depth (m)
Typical Ocean Sound Speed Profiles
Ocean sound speed profiles exhibit characteristic layers:
- Surface Layer: Mixed by wind and waves, nearly isothermal, speed ~1520 m/s
- Thermocline: Rapid temperature decrease with depth, speed decreases
- Deep Isothermal Layer: Cold water (4°C), speed increases slowly due to pressure
- Sound Channel Axis: Minimum speed (~1490 m/s), traps acoustic energy
Snell's Law for Ray Bending
Acoustic rays bend according to Snell's Law as they pass through layers with different sound speeds.
Rays refract toward regions of lower sound speed:
c₁ / cos(θ₁) = c₂ / cos(θ₂) = constant (ray parameter)
where:
- c = sound speed at depth
- θ = ray angle from horizontal
Ray Tracing Algorithm
We trace acoustic rays by numerically integrating the ray equations:
dx/ds = cos(θ)
dz/ds = sin(θ)
dθ/ds = (1/c) × (dc/dz) × cos(θ)
where s is arc length along the ray
Sound Channels and Shadow Zones
Sound speed gradients create acoustic ducts and shadow zones:
- SOFAR Channel: Deep sound channel where low-frequency sound travels thousands of km
- Surface Duct: Traps sound between surface and thermocline, enhances detection ranges
- Convergence Zones: Rays refract back to surface every ~30-60 km in deep water
- Shadow Zones: Regions with no direct acoustic path, limited detection
Implications for Underwater Navigation
Sound velocity variations affect acoustic positioning systems:
- LBL/USBL range measurements must be ray-traced, not straight-line
- Uncorrected ranges can have 1-5% errors (5-50m at 1km range)
- SVP calibration improves positioning accuracy significantly
- Real-time SVP updates enable adaptive beamforming and positioning