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Atmosphere, antenna and target
Radio refractivity N = 77.6 P/T + 3.73e5 e/T^2 depends on pressure, temperature and water vapour. Near the sea the vapour pressure drops quickly with height, so N falls steeply; when dM/dh = dN/dh + 157 per km becomes negative the rays bend down faster than the Earth curves and are trapped in a duct.
Model checks
The science
What is modelled
A fan of rays is traced from the antenna through a layered atmosphere. Temperature, pressure and water vapour set the radio refractivity N; the ray equation in a flat-earth frame uses the modified refractivity M = N + 106 h / RE, which includes Earth curvature. Rays that reach the sea stop there, with no reflection.
Equations
N = 77.6 P / T + 3.73×105 e / T²(dry plus wet term).- Ray curvature:
dθ/dx = 10-6 (dN/dh) / n, integrated with fourth-order Runge-Kutta. - Trapping:
dM/dh < 0; the duct top is where dM/dh = 0. - Geometric horizon:
√(2 R h1) + √(2 R h2).
Checks run in the page
- Standard-atmosphere gradient and effective Earth radius factor near 4/3.
- Earth curvature drop sqrt(2Rh).
- Duct height against an independent scan for the minimum of M.
- Step-size convergence of the ray integrator.
Honest simplifications
A geometric-optics ray model with a smooth exponential profile: no waveguide modes, no wave diffraction or sea-surface reflection and no measured profiles. Real propagation prediction uses parabolic-equation codes and measurements.
What to try
- Press Evaporation duct, then Standard atmosphere, and compare how many rays reach 65 km.
- Raise the humidity decay height and watch the duct disappear.
- Press Warm inversion and move the target beyond the geometric horizon.