Asymptotic Solution of the Stochastic Helmholtz Equation for Turbulent Water

Abstract

It is concluded that a two-variable expansion is sufficient for representing sound propagation through a continuous weakly inhomogeneous medium and that the Debye approximation is a proper two-variable expansion. Its application to the stochastic Helmholtz equation yields, to order ao',•(Rt)l/Xa•ko •, the eikonal equation and the transport equation; a is the rms refractive index variation, •rK is the Prandtl number, Rt is the turbulent Reynolds number, Xo is the Taylor microscale, and k0 is the wavenumber. The resultant acoustic frequency limitations for the Stone and Mintzer experiment and for the turbulent upper ocean are developed and compared. The solution of the eikonal and transport equations in a continuous fluid renders a Lagrangian pressure relation. Finally, the Debye and Born approximations are compared for onedimensional sound propagation.It is mathematically convenient and physically accurate •-5 to assume that •<<•; t•(x) is then only weakly inhomogeneous. The major objective of this paper is the determination of a proper asymptotic expansion for the solution of Eq. 1 with a continuous weakly inhomogeneous refractive index t•(x).This is in direct contrast with many treatments of wave propagation in statistically homogeneous media where discrete single-scatterers are assumed ;=.6-•,• in these latter studies, the single-scattering condition is usually imposed by the use of the Born approximation. •a Since turbulence in fluids represents a continuous weakly inhomogeneous disturbance, •-•6 it is felt that a method of solution for Eq. 1 that does not invoke a discretescattering assumption is required for studying sound propagation through a turbulent fluid. The present paper is primarily concerned with developing and discussing an acceptable and useful asymptotic expression for p when t• is continuous and weakly inhomogeneous.

Aberrations in materials with random inhomogeneities

“…This approximation is only valid for weak inhomogeneities or for short path lengths. As a consequence of this approximation the amount of phase change that an incident field v to a layer of thickness ␦z would experience is given 8,19,20 …”