Ray and Wave Invariants for SOFAR Channel Propagation

Rays and waves propagating in the SOFAR channel and encountering changes in the vertical profile of sound velocity c(z) possess certain invariants, if the changes are sufficiently gradual (adiabatic). For rays, the invariant quantity is ∮(n2 − nm2)12dz, n = 1/c, where nm denotes the value of n at maximum ray excursion, and where the integral is taken over the full ray period. The wave invariants are the individual normal modes φα (z), α = 0,1,2, …, as defined at each point of the propagation path by the local form of c(z). Adiabatic invariance of the modes will be maintained if the gradient of n2 along the path ∂n2/∂x is sufficiently small, as expressed by the condition λ/X≫2πΔn2/n2, where λ is the sound wavelength and X is the ray period, and where Δn2 = X ∫φα(∂n2/∂x)φβdz between adjacent modes α, β. For actual gradients measured off the California coast, the above condition is, numerically, λ≫2.5 m, or f≪600 Hz.