Abstract. In this paper, the asymptotic expansion of integrals of the form JÖ F(kr)f(k) dkis considered, as r tends to infinity, and where F(kr) are Bessel functions of the first and second kind, or functions closely related to these. Asymptotic expansions for several functions of this type are presented under suitable restrictions on f(k). The expansion given by Willis for Hankel transforms is seen to be valid under conditions of f(k) less restrictive than those imposed by that author. Introduction.The evaluation of the Hankel transform /" J,(kr)f(k) dk is a problem often arising in mathematical physics. So-called far-field approximations are of major interest and correspond to large values of the parameter r. A method for obtaining an asymptotic representation of the transform as a series in reciprocal powers of r was given by Willis [1], who obtained as results the following:
SUMMARYPressure fluctuations due to an explosive point-source in an atmosphere containing steady winds may be described in terms of a fundamental mode of long period followed by modes of the acoustic and gravity type, with relative amplitudes, group speeds and periods dependent upon winds. Far-field ground-pressure contributions of various modes are presented for a particular model atmosphere. Inverse dispersion and the Airy phase are found to be important features of the dispersion. A study is made of barograms in the light of new results obtained. Some previously unexplained features of barograms are seen to be attributable to winds.
A disk moves edgewise in a vertical plane with a sufficiently small constant velocity in a uniformly stratified inviscid fluid under gravity. The resultant hydrodynamic thrust on the disk is estimated. Configurations of some associated phase surfaces are shown. A transient Green's function is presented for the case when the Boussinesq approximation is not made, and also for an analogous case of inertial waves in a rotating fluid.
A summary is presented of previously unpublished microbarographic data associated with atmosphere thermonuclear bomb tests in the Marshall Islands and in the Soviet Union from 1954 to 1962. Some effects of winds upon atmospheric pressure waves are indicated through study of wind conditions over the paths of propagation and through comparison with reported Japanese data. It is shown that, as well as the speed of the waves, the maximum amplitudes of wave-trains depend upon wind conditions so that estimates of the size of explosions must take into account prevailing winds. The possible usefulness of microbarograms in the study of upper atmospheric winds is indicated. 1, INTRODUCTIONSince 1954 it has been known that atmospheric pressure waves moving with the speed of sound are generated by atmospheric nuclear explosions and that these waves are readily observable on microbarographs of moderate sensitivity (Yamamoto 1954(Yamamoto , 1955. Such waves had been observed previously in connection with the Krakatoa eruption of 1883 (Symonds 1888) and with the Siberian meteorite of 1908 (Whipple 1930). Interest in pressure waves from nuclear explosions was heightened through the nature of their generation. Microbarographic records were used to estimate location and size of explosions.It was seen that predictions of periods of abnormally great radioactive fall-out could be made in the absence of official announcements of bomb tests. In Great Britain a great amount of public interest was expressed in the Soviet explosion of 30 October 1961 because of its magnitude (60 megatons) and photographs of recordings of seismic and atmospheric waves were reproduced in London newspapers. Subsequent reports of a more scientific nature were made by Jones (1962) and by Carpenter, Harwood and Whitehead (1961) of the Atomic Weapons Research Establishment. It was noted that in addition to the direct (A,) waves, antipodal (A,) waves and (A3) waves which had made a full global passage were also detected from this explosion. Reports of British recordings for other explosions have been made by Hunt, Palmer and Penney (1960) and Jones and Forbes (1862). Many microbarograms showing pressure waves recorded in locations throughout the world have been published by : Yamamoto (1954Yamamoto ( , 1955Yamamoto ( , 1956aYamamoto ( , 1956bYamamoto ( , 1957aYamamoto ( , 1957b, Murayama (1962, 1963a, 1963b, 1965), Farkas (1962, Wexler and Hass (1962), Toperczer (1962), Donn and Ewing (1962a, 1962b), Donn, Pfeffer and Ewing (1963, Bhartendu and Currie (1964), among others. A useful summary of Japanese records has been given by Murayama (1965). There now exists a great amount of published information on these waves and it is probable that a great amount of unpublished and/or neglected data exists. In this regard it is unlikely that more useful recordings are available from the Atomic Weapons Establishment at Southend-on-Sea (private communication). Some records of the type reported by Jones (1962) may become available in future but at the time of writing ...
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