Oleg A. Godin

Reciprocity and energy theorems for waves in a compressible inhomogeneous moving fluid

Abstract A flow reversal theorem (FRT) is established in this paper for sound and acoustic-gravity waves in an arbitrary inhomogeneous moving steady ideal fluid. The theorem is an extension for moving fluid of the reciprocity principle valid in quiescent media. The FRT states symmetry of some wave field quantity with respect to interchange of the source and receiver positions and the simultaneous reversal of the ambient flow. A simple but rather general proof of the FRT becomes possible due to a particular choice of a mixed Eulerian-Lagrangian description of fluid motion. Relation between the FRT proved and a number of known FRTs established earlier for various specific cases is analyzed. Wave quasi-energy and wave-action conservation laws, related to the FRT, are proved for linear waves in an inhomogeneous moving steady compressible fluid. Validity domains of the FRT and the conservation laws are discused. Some possible applications of the FRT and the conservation laws in the investigation of sound generation and propagation in a moving medium are considered.

RECIPROCITY AND ENERGY CONSERVATION WITHIN THE PARABOLIC APPROXIMATION

Abstract Acoustic reciprocity in stationary media at rest and energy conservation are widely recognized as fundamental properties of the sound field to be inherited by any successful paraxial approximation. Rigorous validity of the reciprocity principle for a parabolic equation (PE) solution in motionless media is crucial for the PE to be capable of modeling subtle nonreciprocal acoustic effects due to ocean currents. The energy-conserving property of a PE is of particular importance for correct modeling of the field amplitude in shallow water with significant bathymetric variations and/or considerable range-dependence of geoacoustic parameters. In this paper, reciprocity and energy conservation are considered for two-dimensional (2D) PEs for fluids with piecewise-continuous range-dependence. In the case of lossless media, equivalence of the reciprocity and the energy conservation is demonstrated. A new class of wide-angle PEs is derived such that its members ensure energy conservation and reciprocity, and possess higher asymptotic accuracy in range-dependent environments than previously known PEs. Energy-conserving boundary conditions consistent with the parabolic approximation are obtained for stair-case and sloping interfaces.

Plane Waves in Discretely Layered Fluids

A discretely layered medium is a set of homogeneous layers in contact. Such a model is widely used because it is a good approximation of many real geophysical and technical systems. The theory of sound-wave propagation in discretely layered media has been elaborated rather thoroughly. Media with continuously varying parameters can be approximated by discrete media by assuming an increasing number of layers with decreasing thickness.

Plane-Wave Reflection from the Boundaries of Solids

In this chapter we study the elastic waves behavior in discretely layered solid media. The basic equations and boundary conditions for this problem were discussed in Sect. 1.3. Because the propagation of shear waves of horizontal polarization in a layered solid is independent of that of waves of vertical polarization and is formally analogous to the propagation of sound waves in liquid, in this chapter we shall consider only the case of vertical polarization.

Basic Equations for Wave Processes in Fluids and Solids

In this chapter the equations and boundary conditions for elastic waves in liquids and solids, with particular attention to layered media, are obtained. Special consideration is given to harmonic waves since superpositions of such waves can be generally applied to many fields in acoustics.

Energy Conservation and Reciprocity for Waves in Three-Dimensionally Inhomogeneous Moving Media

In this chapter, we present an approach [8.1–3] which is quite different from those used in previous chapters and turns out to be particularly efficient to treat waves in a general three-dimensionally (3-D) inhomogeneous fluid in the presence of a mean flow. The approach is based on an introduction of a new vectorial acoustic quantity, the oscillatory displacement of fluid particles, as a dependent variable to describe the wave field. Mathematically, the approach may be viewed as a particular case of a mixed Eulerian—Lagrangian representation of fluid motion due to Eckart [8.4]. As it is demonstrated below, the use of the concept of oscillatory displacement leads to a drastic simplification of the linearized equations of motion and the respective boundary conditions. The effect of a nonuniform gravity field on the wave propagation can be taken into consideration without making the analysis more complicated. The approach will be applied to establish the energy-conservation law and to derive reciprocity-type relations for sound and acoustic-gravity waves in general inhomogeneous moving media with time-independent parameters. Our presentation is based mainly on [8.5].

High Frequency Sound Fields

In this chapter we shall discuss the fundamentals of the ray method, in particular, its origination from the wave theory of the sound field. Monochromatic waves in a stationary, inhomogeneous in three dimensions, moving medium are considered. Some relevant questions were considered in [Ref. 5.1, Chaps. 8,10] but only in the case when the field dependence on the horizontal coordinates is harmonic. It will be shown below that many results obtained in [5.1] are also valid in the general case. A rather full description of the geometrical acoustics of inhomogeneous (including nonstationary) media at rest as well as its numerous applications can be found in [5.2,3]. Special aspects of ray theory for elastic waves in solids have been discussed in [5.4].

The Field at and near a Caustic

In this chapter we shall continue the study of the high frequency sound field in continuously layered media. We start with the integral representation of a field of the type:

Universal Properties of the Plane-Wave Reflection and Transmission Coefficients

\begin{gathered} p\left( {r,z,{z_0}} \right) = {\left( {\frac{{{k_0}}} {r}} \right)^{1/2}}{e^{ - i\pi /4}}\int_{ - \infty }^{ + \infty } {dq\;F} \left( {z,{z_0},q} \right){e^{i{k_0}\varphi }}, \hfill \\ \varphi = qr + \omega \left( {z,{z_0},q} \right) \hfill \\ \end{gathered}