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.

Acoustic data from the Canada Basin Acoustic Propagation Experiment are discussed. These recordings were obtained under seasonally varying sea ice to the north of Alaska during a period of 154 days. They contain signals from sources that were deployed at ranges of 17.5, 29.6, and 237.8 km and ambient sounds from marine mammals and ice-related events. After the area was covered with ice, the amplitude of receptions from the most distant source gradually decreased as scattering features on the underside of the ice developed during fracturing, drifting, ridging, and rafting events. Improvements are presented for an Arctic acoustic model that is based on the parabolic equation method, and the approach is applied to a problem in which variable ice thickness acts as a loss mechanism by scattering energy out of the waveguide. Some of the recordings have a harmonic signature that is believed to be associated with the resonances of ice floes rubbing together, but variations in the harmonics over short time scales cannot be explained in terms of the resonances of an isolated floe. This behavior may be related to the coupling of vibrations at contact points that vary during the relative motions of floes.

The parabolic equation method is extended to handle problems involving ice cover and other thin elastic layers. Parabolic equation solutions are based on rational approximations that are designed using accuracy constraints to ensure that the propagating modes are handled properly and stability constrains to ensure that the non-propagating modes are annihilated. The non-propagating modes are especially problematic for problems involving thin elastic layers. It is demonstrated that stable results may be obtained for such problems by using rotated rational approximations [Milinazzo, Zala, and Brooke, J. Acoust. Soc. Am. 101, 760–766 (1997)] and generalizations of these approximations. The approach is applied to problems involving ice cover with variable thickness and sediment layers that taper to zero thickness.

A complete energy conservation correction is derived to improve the accuracy of the elastic parabolic equation for range-dependent problems. The correction is complete in the sense that it is valid for problems involving a broad spectrum of horizontal wave numbers. It is a linear condition that associates the incident and transmitted fields on a vertical interface with arrays of point sources having the appropriate energy flux densities. It is a generalization of a complete energy conservation correction for the acoustic parabolic equation [J. Acoust. Soc. Am. 94, 975–982 (1993)].

Despite the publication of three independent papers that reported sightings of the Ivory-billed Woodpecker (Campephilus principalis), its preservation became a controversial topic because nobody has been able to obtain a distinct photo, which is considered the standard form of evidence for documenting birds. An analysis based on a combination of factors related to habitat and behavior suggests that the expected waiting time for obtaining a distinct photo of an Ivory-billed Woodpecker is several orders of magnitude greater than it would be for a more typical baseline species of comparable rarity. Given these difficulties and the time pressure involved to ensure the conservation of this species, this paper discusses the need to use a different approach for an exceptional case. Presented here are three videos that show birds with flights, behaviors, field marks, and other characteristics that are consistent with the Ivory-billed Woodpecker but no other species inhabiting the region.

After the parabolic equation method was initially applied to scalar wave propagation problems in ocean acoustics and seismology, it took more than a decade before there was any substantial progress in extending this approach to problems involving solid layers. Some of the key steps in the development of the elastic parabolic equation are discussed here. The first breakthrough came in 1985 with the discovery that changing to an unconventional set of dependent variables makes it possible to factor the operator in the elastic wave equation into a product of outgoing and incoming operators. This innovation, which included an approach for handling fluid-solid interfaces, was utilized in the first successful implementations of the elastic parabolic equation less than five years later. A series of papers during that period addressed the issues of accuracy and stability, which require special attention relative to the scalar case. During the 1990s, the self-starter made it possible to handle all types of waves, rotated rational approximations of the operator square root made it possible to handle relatively thin solid layers, and there was some progress in the accurate treatment of sloping interfaces. During the next decade, an improved formulation and approach for handling interfaces facilitated the treatment of piecewise continuous depth dependence and sloping interfaces. During the last 10 years, the accuracy of the elastic parabolic equation was improved and tested for problems involving sloping interfaces and boundaries, and this approach was applied to Arctic acoustics and other problems involving thin layers. After decades of development, the elastic parabolic equation has become a useful tool for a wide range of problems in seismology, seismo-acoustics, and Arctic acoustics, but possible directions for further work are discussed.