In this sound propagation study of the effect of Doppler on source and/or receiver motion has been developed. It is based on modifying the inputs to existing normal-mode software so that the source and/or receiver trajectory and consequent Doppler effects of the given ocean experiment are properly modeled without requiring direct changes in the static softward. From this the effects of the received signal after long-range, multipath sound propagation from a moving source can be modeled and interpreted.

The basic stochastic wave equation for sound propagation through a turbulent field is derived from first principles as represented by the continuity equation, the Navier-Stokes equations, and an appropriate equation of state. It is shown that, for very low turbulent Mach numbers and monochromatic transmissions sound propagation in a turbulent field can be adequately represented by the stochastic Helmholtz equation, ∇2p+k02μ2p = 0, if the acoustic wavelength does not exceed the Taylor microscale λg divided by [(σκ)12(u/c)]12, where σκ is the Prandtl number and u/c is the turbulent Mach number. In addition, the comparison of similar turbulent flows in air and in water, with respect to estimating their acoustic frequency limitations, is illustrated by contrasting: (1) the Baerg and Schwartz experiments with the Stone and Mintzer experiments, and (2) the turbulent lower atmosphere with the turbulent upper ocean.

A wide-aperture, horizontal line array receiving a low-frequency, long-range signal is considered. It is shown that the sound field structure as manifested by amplitude nonhomogeneity and the wave-front corrugation can reduce the performance of a conventional linear beamformer. Beamformer expressions that explicitly show the effects of amplitude nonhomogeneity and wave-front corrugation in a multipath environment are given. The limitations of some array algorithms that are inappropriate for a multipath environment are indicated. Array signal gain and side-lobe suppression relations for an ocean multipath environment are generalized from similar relations that are not valid in an ocean multipath environment.

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.

By considering the stochastic nature of the phase fluctuations in the ocean, the conventional ray theory intensity relaxation was extended in an earlier paper [J. A. Neubert, J. Acoust. Soc. Am. 51, 310–322 (1972)] to permit consideration of partial coherence in multipath problems. Although this relation worked well in the open ocean [J. A. Neubert, J. Acoust. Soc. Am. 62, 326–334 (1977)], it proves to be incomplete for sound propagation through a random ocean front. By also considering the amplitude fluctuations, a mean multipath intensity relation (as well as its standard deviation σI) is found that takes into consideration the strong horizontal sound-speed gradients that occur in certain important ocean frontal regions.