On the calculation of normal mode group velocity and attenuation

Abstract

The group velocity for a normal mode can be calculated without invoking a finite difference approximation requiring a second eigenmode calculation. The reciprocity relation is employed in a derivation of the normal mode group velocity and attenuation coefficient. The group velocity thus calculated is more accurate than a comparable finite difference approximation. Arbitrarily arranged layers of solid and fluid media are considered. PACS numbers: 43.30. Bp, 43.30.Jx, 43.20.Hq, 43.20. Bi INTRODUCTIONFor low-frequency long range underwater acoustic propagation problems, a normal mode description has the advantage of being a direct wave theory capable of dealing with complicated depth dependent media. Various physical quantities may be computed once the modal eigenvalues and eigenfunctions are obtained. Two quantities typically of interest are the modal attenuation coefficients and the group velocities. For fluid media the attenuation of a given mode can be calculated using a perturbation approach through integrations involving the depth dependence of the absorption coefficients of the medium and of the modal eigenfunction at the frequency of interest. In calculating the group velocity, the usual approach I involves a second eigenvalue calculation at a nearby frequency followed by dividing the difference between the two eigenvalues into the corresponding frequency difference. The additional eigenvalue calculation required doubles the computational burden. This paper describes an exact group velocity calculation, similar to that used for calculating the mode attenuation coefficient, not requiring an additional eigenvalue calculation. The derivation of this approach is based on the reciprocity principle 2 and is valid for mixed fluid and solid layers. A general discussion 3 shows the relationship between group velocity and energy flux. The calculation of normal mode group velocity as described in our paper has also been proposed and applied to the layered fluid. 4 The present paper considers the problem from another viewpoint for a geometry frequently assumed in underwater acoustic propagation. We begin by describing in Sec. I the horizontally stratified model and our notation. In Sec. II the attenuation and group velocity formulas, for an arbitrary arrangement of horizontal fluid or solid layers, are derived from the reciprocity relation. Section III gives an example of numerical results. Section IV summarizes our results. The Appendix contains a derivation of the reciprocity relation in the form used here. //•. is the depth ofthejth layer from the surface. The last layer N is a homogeneous fluid or solid half-space. The formulation developed below can also be applied to an adiabatic description 5 of problems having range variable parameters. Under these circumstances the shear potential •p and the compressional potential 4 of an acoustic wave are well defined and satisfy 6 the simple wave equations C} 0t 2' V2•/'= C• 0t 2' (l) More complicated equations would be required if additional terms related to density, shear ...

Sounds in the Ocean at 1–100 Hz

“…Although many authors favor a ray-based approach to modeling range dependence within the water column, the elastic properties of anisotropic sediments require properly simulating polarization effects. In sedimentary layers, compressional (P) and vertically and horizontally polarized shear (SV and SH) waves combine to form propagating modes (Koch et al 1983, Stoll et al 1994, Godin 1998. Anisotropy induces these seismic modes to mix the polarizations of P, SV, and SH motions, even in the absence of range dependence in the media (Park 1996, Soukup et al 2013).…”

Acoustic-gravity waves in atmospheric and oceanic waveguides

“…As in Eq. (21), q m and q n can be positive or negative. Power flux density 24,26 in AGWs in inhomogeneous, moving fluids is I ¼p @ŵ @t þ qu @ŵ @t Á dŵ dt ;…”

“…(43) improves accuracy and decreases computation time in numerical simulations of guided propagation. 21 When the waveguide's boundary z ¼ 0 is either free or rigid, the second term in the denominator in Eq. (43) vanishes, and the latter reduces to results obtained earlier for acoustic waveguides in moving fluids 22,24 and for AGWs in atmospheric waveguides with smooth stratification.…”

“…[12][13][14][15][16][17][18] Any detailed description of AGW fields in the atmosphere and ocean undoubtedly requires numerical modeling for specific environmental conditions. However, theoretical investigations [19][20][21][22][23][24] of general properties and the resulting qualitative understanding of normal modes have proven very useful in acoustics, and are likely to play a similar role for AGWs. While earlier AGW studies typically assumed an ideal gas half-space overlying a rigid boundary, 1,4,[9][10][11]25 investigations of AGW propagation above the ocean surface and, more generally, of coupling of physical processes in the ocean and atmosphere, require an environmental model that allows for a general equation of state of the fluid as well as the presence of fluid-fluid interfaces and compliant boundaries.…”

Acoustic Modeling Using a Three-Dimensional Coupled-Mode Model

“…Solutions to range-dependent problems will be calculated using the stepwise coupled-mode approach described above with boundary conditions defined by the momentum conservation equations [Koch et al (1983)]…”