Derivation of the Stochastic Helmholtz Equation for Sound Propagation in a Turbulent Fluid

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.