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On September 8, 2001 around 2 h UT, the largest uranian moon, Titania, occulted Hipparcos star 106829 (alias SAO 164538, a V = 7.2 , K0 III star). This was the first-ever observed occultation by this satellite, a rare event as Titania subtends only 0.11 arcsec on the sky. The star's unusual brightness allowed many observers, both amateurs or professionals, to monitor this unique event, providing fifty-seven occultations chords over three continents, all reported here. Selecting the best 27 occultation chords, and assuming a circular limb, we derive Titania's radius: R T = 788.4 ± 0.6 km ( 1 - σ error bar). This implies a density of ρ = 1.711 ± 0.005 g cm −3 using the value G M = ( 2.343 ± 0.006 ) × 10 11 m 3 s −2 derived by Taylor [Taylor, D.B., 1998. Astron. Astrophys. 330, 362–374]. We do not detect any significant difference between equatorial and polar radii, in the limit r eq − r po = − 1.3 ± 2.1 km , in agreement with Voyager limb image retrieval during the 1986 flyby. Titania's offset with respect to the DE405 + URA027 (based on GUST86 theory) ephemeris is derived: Δ α T cos ( δ T ) = − 108 ± 13 mas and Δ δ T = − 62 ± 7 mas (ICRF J2000.0 system). Most of this offset is attributable to a Uranus' barycentric offset with respect to DE405, that we estimate to be: Δ α U cos ( δ U ) = − 100 ± 25 mas and Δ δ U = − 85 ± 25 mas at the moment of occultation. This offset is confirmed by another Titania stellar occultation observed on August 1st, 2003, which provides an offset of Δ α T cos ( δ T ) = − 127 ± 20 mas and Δ δ T = − 97 ± 13 mas for the satellite. The combined ingress and egress data do not show any significant hint for atmospheric refraction, allowing us to set surface pressure limits at the level of 10–20 nbar. More specifically, we find an upper limit of 13 nbar ( 1 - σ level) at 70 K and 17 nbar at 80 K, for a putative isothermal CO2 atmosphere. We also provide an upper limit of 8 nbar for a possible CH4 atmosphere, and 22 nbar for pure N2, again at the 1 - σ level. We finally constrain the stellar size using the time-resolved star disappearance and reappearance at ingress and egress. We find an angular diameter of 0.54 ± 0.03 mas (corresponding to 7.5 ± 0.4 km projected at Titania). With a distance of 170 ± 25 parsecs, this corresponds to a radius of 9.8 ± 0.2 solar radii for HIP 106829, typical of a K0 III giant.

A parabolic wave equation is etablished with a particular approximation by rational fraction of degree N. An analytical solution is given for this approximate equation, in the case of a homogeneous medium of constant depth with a free surface and a rigid bottom. An analytical representation is also obtained for the numerical solution of this equation by the alternating direction implicit (ADI) method. The advantages of this method over the usual ones are indicated

In order to realize a complete geoacoustical model of the sea we have created a vertical variation in the sound velocity in a liquid layer presenting a minimum like in real sea by a diffusion technique in a tank. This technique involves the superimposing of several layers of miscible liquids with different densities without mixing. To simulate the sea bottom (with or without a 5% slope) we have used two types of material-Polyurethane resin to represent a sedimentary ground and marble to represent an elastic rock. To account for the scale, the measures were made at a frequency of 1 MHz corresponding to 50 Hz in a real situation. The measured values are compared with the results of a numerical integration of the Helmholtz equation. We use a variant of the parabolic approximation