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Hydrostatic pressure
Pressure grows with depth as dP/dz = ρ(P) g. Seawater is slightly compressible: ρ = ρ0 / (1 - β ΔP) with β the (effective, constant) compressibility. Substituting gives ΔP - β ΔP²/2 = ρ0 g z, so ΔP = (1 - sqrt(1 - 2 β ρ0 g z)) / β. The page integrates the ODE with RK4 and compares it with this closed form.
Temperature and sound speed
Temperature follows T(z) = Tdeep + (Ts - Tdeep) e-z/h, a one-parameter thermocline. Sound speed uses the Mackenzie (1981) nine-term formula c(T, S, D). The echo time is t = 2 ∫ dz / c(z), which differs from the naive 2z/1500 because c is not constant.
Honest simplifications: constant g, constant effective β (real β falls with pressure), exponential thermocline, no currents or internal waves, and a vertical sounding path with no ray bending. Values are teaching estimates, not survey-grade.