William L. Siegmann

Examination of three‐dimensional effects using a propagation model with azimuth‐coupling capability (FOR3D)

A three‐dimensional wave propagation model of parabolic approximation type (FOR3D) is used to examine 3‐D ocean environmental variations. The background theory and characteristics of the model are reviewed briefly. Propagation situations that are classified as 3‐D, N×2‐D, and 2‐D are described in connection with FOR3D and are interpreted in several ways. An analytic exact solution is used to demonstrate the model’s accuracy and its capability for treating fully 3‐D propagation, when coupling exists between solutions in adjacent vertical planes of constant azimuth. It is also employed to illustrate a procedure for using approximate conditions at vertical side boundaries in a 3‐D calculation. An application is made to an Atlantic Ocean shelf‐slope environment with realistic bottom topographic variations and sound‐speed profiles. The occurrence of significant azimuthal coupling is demonstrated in propagation loss versus range curves. It follows that, while the N×2‐D approximation of no azimuthal coupling is ...

A wide‐angle three‐dimensional parabolic wave equation

A third‐order partial differential equation with wide‐angle capability is formulated to predict three‐dimensional underwater sound propagation. The development is based on physical acoustic characteristics and mathematical theory. Both operator and asymptotic analyses are given to thoroughly discuss the validity of the formulation. Physical conditions are indicated when a three‐dimensional approach is needed.

Internal waves in a rotating stratified fluid in an arbitrary gravitational field

Abstract Small amplitude oscillations of a uniformly rotating, density stratified, Boussinesq, non-dissipative fluid are examined. A mathematical model is constructed to describe timedependent motions which are small deviations from an initial state that is motionless with respect to the rotating frame of reference. The basic stable density distribution is allowed to be an arbitrary prescribed function of the gravitational potential. The problem is considered for a wide class of gravitational fields. General properties of the eigenvalues and eigenfunctions of square integrable oscillations are demonstrated, and a bound is obtained for the magnitude of the frequencies. The modal solutions are classified as to type. The eigenfunctions for the pressure field are shown to satisfy a second-order partial differential equation of mixed type, and the equation is obtained for the critical surfaces which delineate the elliptic and hyperbolic regions. The nature of the problem is examined in detail for certain speci...

Modeling Rayleigh and Stoneley waves and other interface and boundary effects with the parabolic equation

An improved approach for handling boundaries, interfaces, and continuous depth dependence with the elastic parabolic equation is derived and benchmarked. The approach is applied to model the propagation of Rayleigh and Stoneley waves. Depending on the choice of dependent variables, the operator in the elastic wave equation may not factor or the treatment of interfaces may be difficult. These problems are resolved by using a formulation in terms of the vertical displacement and the range derivative of the horizontal displacement. These quantities are continuous across horizontal interfaces, which permits the use of Galerkins method to discretize in depth. This implementation extends the capability of the elastic parabolic equation to handle arbitrary depth dependence and should lead to improvements for range-dependent problems.

A wide angle and high Mach number parabolic equation

Various parabolic equations for advected acoustic waves have been derived based on the assumptions of small Mach number and narrow propagation angles, which are of limited validity in atmospheric acoustics. A parabolic equation solution that does not require these assumptions is derived in the weak shear limit, which is appropriate for frequencies of about 0.1 Hz and above for atmospheric acoustics. When the variables are scaled appropriately in this limit, terms involving derivatives of the sound speed, density, and wind speed are small but can have significant cumulative effects. To obtain a solution that is valid at large distances from the source, it is necessary to account for linear terms in the first derivatives of these quantities [A. D. Pierce, J. Acoust. Soc. Am. 87, 2292-2299 (1990)]. This approach is used to obtain a scalar wave equation for advected waves. Since this equation contains two depth operators that do not commute with each other, it does not readily factor into outgoing and incoming solutions. An approximate factorization is obtained that is correct to first order in the commutator of the depth operators.

Parabolic equations for gravity and acousto-gravity waves

Parabolic equations for gravity and acousto-gravity waves are derived and implemented. The wave equations for these problems contain singularities at depths at which the buoyancy frequency equals the forcing frequency. One of the advantages of the parabolic equation solution is that it is easy to avoid numerical problems associated with the singularities. Some problems involve an infinite number of propagating modes. This artifact of neglecting viscosity is handled by including stability constraints in the rational approximations used in the implementation of the parabolic equation. The parabolic equation is tested for idealized problems involving surface, internal, and interface gravity waves. Parabolic equation solutions are also presented for range-dependent problems involving internal waves in the ocean and acousto-gravity waves in the atmosphere.

Internal waves in a contained rotating stratified fluid

Small-amplitude time-dependent motions of a uniformly rotating, density-stratified, Boussinesq non-dissipative fluid in a rigid container are examined for the case of the rotation axis parallel to gravity. We consider a variety of container shapes, along with arbitrary values for the (constant) Brunt-Vaisala and rotation frequencies. We demonstrate a number of properties of the eigenvalues and eigenfunctions of square-integrable oscillatory motions. Some of these properties hold generally, while others are shown for specific classes of containers (such as with symmetry about the container axis). A full solution is presented for the response of fluid in a cylindrical container to an arbitrary initial disturbance. Features of this solution which are different from the cases of no stratification or no rotation are emphasized. For the situation when Brunt-Vaisala and rotation frequencies are equal, characteristics of the oscillation frequencies and modal structures are found for containers of quite general shape. This situation illustrates, in particular, effects which are possible when rotation and stratification act together and which have been overlooked in previous investigations that assume that the vertical length scale is much smaller than the horizontal scales.

Parabolic equation solution of seismo-acoustics problems involving variations in bathymetry and sediment thickness

Recent improvements in the parabolic equation method are combined to extend this approach to a larger class of seismo-acoustics problems. The variable rotated parabolic equation [J. Acoust. Soc. Am. 120, 3534-3538 (2006)] handles a sloping fluid-solid interface at the ocean bottom. The single-scattering solution [J. Acoust. Soc. Am. 121, 808-813 (2007)] handles range dependence within elastic sediment layers. When these methods are implemented together, the parabolic equation method can be applied to problems involving variations in bathymetry and the thickness of sediment layers. The accuracy of the approach is demonstrated by comparing with finite-element solutions. The approach is applied to a complex scenario in a realistic environment.

Modeling and analysis of sound transmission in the Strait of Korea

The experiment called the Acoustic Characterization Test III was performed in the oceanographically complex Strait of Korea. It was designed to provide accurate measurements of sound transmission and array signal gain under known environmental conditions. Bottom sampling and sub-bottom surveys coupled with archival geophysical information provided the basis for the geoacoustic depth profiles of sound speed, density and attenuation. The bottom was a sand-silt sediment for which shear wave propagation at experimental frequencies was determined to be unimportant. Using the compressional wave profiles, the measured bathymetry and the water sound speed as input parameters, good agreement was obtained between measured and calculated narrowband transmission loss when an attenuation profile with a frequency dependence to the 1.8 power in the near sediment-water interface layer was used. This power law was determined by using an effective attenuation coefficient and a least squares comparison of calculated and measured sound transmission of five narrowband tones between 47 and 604 Hz. The resulting geoacoustic model was used to compare measured and calculated broadband sound transmission and signal spread and excellent agreement was found. These results are consistent with measurements in other sand-silt areas where site specific frequency dependent characteristics have been observed.

Current and current shear effects in the parabolic approximation for underwater sound channels

The effect of currents on the acoustic pressure field in an underwater sound channel is investigated. Based on fundamental fluid equations, model equations are formulated for sound pressure while including nonuniform currents in the source–receiver plane. Application of parabolic‐type approximations yields a collection of parabolic equations. Each of these is valid in a different domain determined by the magnitudes of current speed, current shear, and depth variation of sound speed. Under certain conditions, it is possible to interpret current effects in terms of an effective sound speed. Using this effective sound speed in an existing numerical code, we examine sound speed in a shallow water isospeed channel with a simple shear flow and a lossy bottom. It is found that even small currents can induce very substantial variations in relative intensity. The degree of variation depends upon current speed, source and receiver geometry, and acoustic frequency. Particular emphasis is placed on intensity‐differen...