Polynomial expansions are used to approximate the equations of the eigenvalues of the Schrödinger equation for a finite square potential well. The technique results in discrete, approximate eigenvalues which, it is shown, are identical to the corresponding eigenvalues of a wider, infinite well. The width of this infinite well is easy to calculate; indeed, the increase in width over that of the finite well is simply the original width divided by the well strength. The eigenfunctions of this wider, infinite well, which to first order has the same width for the ground state and all excited states, are also good approximations to the exact eigenfunctions of the finite well. These approximate eigenfunctions and eigenvalues are compared to accurate numeric calculations and to other approximations from the literature.

Long-term monitoring of endangered species abundance based on acoustic recordings has not yet been pursued. This paper reports the first attempt to use multi-year passive acoustic data to study the impact of the Deepwater Horizon oil spill on the population of endangered sperm whales. Prior to the spill the Littoral Acoustic Demonstration Center (LADC) collected acoustic recordings near the spill site in 2007. These baseline data now provide a unique opportunity to better understand how the oil spill affected marine mammals in the Gulf of Mexico. In September 2010, LADC redeployed recording buoys at previously used locations 9, 25, and 50 miles away from the incident site. A statistical methodology that provides point and interval estimates of the abundance of the sperm whale population at the two nearest sites is presented. A comparison of the 2007 and the 2010 recordings shows a decrease in acoustic activity and abundance of sperm whales at the 9-mile site by a factor of 2, whereas acoustic activity and abundance at the 25-mile site has clearly increased. This indicates that some sperm whales may have relocated farther away from the spill. Follow-up experiments will be important for understanding long-term impact.

In the summer of 2003, the Littoral Acoustic Demonstration Center conducted an acoustic characterization experiment for a 21-element marine seismic exploration airgun array of total volume of 0.0588 m 3 ͑3590 in. 3 ͒. Two Environmental Acoustic Recording System buoys, one with a desensitized hydrophone, were deployed at a depth of 758 m in a water depth of 990 m, near Green's Canyon in the Gulf of Mexico. Shots over a grid were recorded and calibrated to produce absolute broadband ͑up to 25 kHz͒ pressure-time dependencies for a wide range of offsets and arrival angles in the water column. Experimental data are analyzed to obtain maximum received zero-to-peak pressure levels, maximum received sound exposure levels, and pressure levels in 1 / 3-octave frequency bands for each shot. Experimental data are quantitatively modeled by using an upgraded version of an underwater acoustic propagation model and seismic source modeling packages for a variety of ranges and arrival angles. Experimental and modeled data show good agreement in absolute pressure amplitudes and frequency interference patterns for frequencies up to 1000 Hz. The analysis is important for investigating the potential impact on marine mammals and fish and predicting the exposure levels for newly planned seismic surveys in other geographic areas.

This paper presents a multilayer generalization of an algebraic method of inverting frequency-domain airborne active electromagnetic (AEM) data in terms of 1-D layered earth models. The processing of the AEM data, which includes a recalibration procedure, is also outlined. The inversion is applied to synthetic fields generated from a multilayer model which is intended to approximate a measured conductivity profile of the water column in the Gulf of Mexico and to measured AEM data from a survey of the Barataria Bay estuary region of the Louisiana Gulf of Mexico coast. The inversion results from the synthetic data are in good agreement with the forward model. The conductivities calculated from the inversions of measured AEM data are compared to ground-and water-based measurements. The depth variations of the calculated electrical conductivities in the nearshore Gulf waters are in good agreement with measurements of conductivity versus depth by conductivitytemperature-depth (CTD) casts at several points on the over-the-water portion of two flight lines.

Higher-order correlations and spectra may be used for detection, time delay estimation, classification, and discrimination of signals. For these applications, a detailed knowledge of their attributes can be highly useful. In this paper, the properties of the bicorrelation and tricorrelation of bandlimited deterministic transients, i.e., energy signals, and their spectra, the bispectrum and trispectrum are studied. Bandlimited transients that contain frequencies down to and including zero and those that have a nonzero lower cutoff frequency are both considered. Using symmetries inherent in the bispectrum of a signal, the entire bispectrum can be mapped from bispectral elements defined in two polygons, one for the unaliased and one for the aliased domain, each of which is one-twelfth the area of its total domain. The nonredundant unaliased region of the trispectrum is contained in two principal unaliased polyhedra, each replicated 48 times to reproduce the full trispectrum. If there is aliasing in the trispectrum, then the total aliased domain can similarly be mapped from two principal aliased polyhedra. It is shown that the critical sampling interval for the bicorrelation, i.e., the sampling interval sufficient to avoid aliasing, is 2/3 the critical sampling interval for the ordinary correlation, and the critical sampling interval for the tricorrelation is 1/2 the critical sampling interval for the ordinary correlation. If the lowest frequency of the bandlimited transient fb is greater than half the highest frequency ft, then there is no unaliased bispectrum. The unaliased trispectrum is made up of two different replicated domains, one of which disappears for 3fb>ft.

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Higher-order cross and ordinary correlation detectors are applied to four deterministic transients contaminated by uncorrelated Gaussian noise only. Histograms and moments are used to examine the properties of the signals and their effect on detector performance. Receiver operating characteristic (ROC) curve analysis and limiting signal-to-noise ratios for ‘‘good’’ detection provide comparative measures for different detectors. Probability density functions of detection ordinate values of signal-present and noise-only correlations are used to explain ROC curve behavior. Using a known source, the cross-correlation detector performs better than the higher-order correlation detectors for each transient studied. However, for an unknown narrow pulse source signal, the bicorrelation and tricorrelation detectors outperform the cross-correlation detector. In contrast, the bicorrelation detector performs very poorly for low-frequency narrow-band signals with a small third moment embedded in uncorrelated Gaussian noise. Rectification as part of the detection process improves the performance of the bicorrelation detector and also places the peak of maximum magnitude at the origin. This eliminates the problem in detection or time delay estimation that the existence of multiple peaks due to symmetries in the auto-bicorrelation or the bicorrelation of repeated signals may cause. The tricorrelation detector also performs better with rectification than without. For an unknown source, the bicorrelation and tricorrelation detectors with rectification perform on a level comparable to the cross-correlation detector for certain signals. Comparisons are made between repeating a known source and repeating noisy received signals in the bicorrelation.