A n analytic formula for the three-electron generating integral I(a &, a2, o. 3, o. », a», a 3) ) ( r 1 r2 r3 r» r23 r3'j ) exp(a l r l -a2r& -a3r3e»r»n»r»a» r» )d r l d r2d r3 is given -1 3 3 3 which is valid for all values of a&, a~, a3, cx», +23,+31 for which this integral converges. A large class of integrals can be evaluated analytically by taking derivatives of I with respect to the a' s. More general integrals whose integrands contain products of spherical harmonics can also be evaluated analytically. In particular, all of the matrix element integrals which arise in a variational calculation on the lithium atom with Hylleraas-type basis functions can be evaluated in closed form. Certain dif5cult two-electron integrals can be obtained as a limiting case. Two-center two-electron molecular integrals can be obtained via an inverse Laplace transform; this observation is used to discuss the computation and convergence of series expansions for these molecular integrals.
This paper presents a derivation of the time reversal operator decomposition ͑DORT͒ using the sonar equation. DORT is inherently a frequency-domain technique, but the derivation is shown in the time-frequency domain to preserve range resolution. The magnitude of the singular values is related to sonar equation parameters. The time spreading of the time-domain back-propagation image is also related to the sonar equation. Noise-free, noise-only, and signal-plus-noise data are considered theoretically. Contamination of the echo singular component by noise is shown quantitatively to be very small at a signal-to-noise ratio of 0 dB. Results are shown from the TREX-04 experiment during April 22 to May 4, 2004 in 94 m deep, shallow water southwest of the Hudson Canyon. Rapid transmission of short, 500 Hz wide linear frequency modulated beams with center frequencies of 750, 1250, 1750, 2250, 2750, and 3250 Hz are used. Degradation caused by a lack of time invariance is found to be small at 750 Hz and nearly complete at 3250 Hz. A back-propagation image at 750 Hz shows focusing on the echo repeater. These results are discussed with comments about further research.
At long-range, shallow-water reverberation can be driven by sub-critical-angle scattering, i.e. by rough interface scattering. The Naval Research Laboratory has recently developed a small-slope model for elastic seafloors that provides physics-based estimates of the dependence of scattering on the incident and scattered angles, and physical descriptors of the environment. In this paper, this incoherent model is used as kernels in reverberation models, which in turn are used to assess the sensitivity at 3.5 kHz of long-range monostatic reverberation to the roughness of the water-sediment interface. It is shown that when sub-critical-angle scattering dominates, the acoustic field could be quite sensitive to the parameter values of the roughness, thus arguing for the need for regional in-situ methods for its estimation.
On 3 July 2004, an aggregation of ca. 150–200 melon-headed whales (Peponocephala electra) appeared in the shallow waters of Hanalei Bay, Kauai and congregated there for over 27 h. Preceding the whales’ appearance and partially coincident with their time in the Bay, midrange (3.5–5 kHz) tactical sonars were intermittently deployed during the Rim of the Pacific 2004 (RIMPAC) joint military exercises being conducted in waters near Kauai by the U.S., Japan, and Australia Navies. An NOAA report (Southall et al., 2006) attributed the active sonar usage as a plausible, if not likely, contributing factor. A detailed timeline and reconstruction of the RIMPAC activities is presented showing the worst-case estimates of the sonar sound levels in the waters surrounding Kauai. A re-examination of available evidence combined with a new report of a simultaneous and similar aggregation in Sasanhaya Bay, Rota, Commonwealth of the Northern Mariana Islands, brings the plausibility conclusion into question. [This work was sponsored by multiple sources. D. Fromm and S. Martin conducted acoustic analyses with funds provided by the U.S. Pacific Fleet. J. Mobley received funding from the U.S. Geological Survey. P. Nachtigall is sponsored by the Office of Naval Research for marine mammal audiometric studies.]
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