Search

A COMMITTEE of Section A of the British Association for the Advancement of Science, appointed to undertake the further tabulation of Bessel functions, is at present considering the advisability of unifying and completing the existing tables with the view of the publication of a complete table of Bessel functions.

Lommel's function s,,v(x) is a particular solution of the differential equation X2y" + Xy' + (x2 _ v2)y = X# +1. It is shown here that sg,v(x)> 0 for x >0, if IL=2 and IvI ' and IvI

Calculations of Bessel Functions of real order and argument for large values of the argument can be greatly facilitated by the use of the so called phase-amplitude method [1]. In this method two auxiliary functions, the amplitude and phase functions, are defined in terms of the regular and irregular solution of Bessel's equation. These auxiliary functions have the great advantage that they are monotonic functions of the argument; moreover, for large arguments these functions are slowly varying and, hence, easily amenable to computation and interpolation. The Bessel Functions of the third kind, so-called Hankel Functions, are defined as follows: (1.1) H,(')(x) = J,(x) + iY,(x) and (1.2) H,(2)(x) = J,(x) iY,(x). These definitions suggest the alternative definition of the Bessel Function in terms of an amplitude and phase function, viz., (2.1) AV ,1(x) I--II(2x(x) I or (2.2) Hp(') = Av+ifv (2.3) = A so that (2.4) J,(x) =A, cos 4, and (2.5) Y,(x) = A, sin 4, The phase function P,(x) obeys the first order differential equation

Let n be a positive integer. Then we know that, if m>– 1,Consider the integralwhich is equal to

A number of authors have obtained diffraction corrections for a circular piston source by numerical or graphical integration of an approximate expression for the piston field attributable to E. Lommel [Abhandl. Bayer. Akad. Wiss. 15, 233 (1886)]. Lommel's expression gives the piston field in terms of trigonometric functions and Lommel functions of two variables, the latter being defined as infinite sums involving cylindrical Bessel functions. It is shown here that the required integral of Lommel's expression can be evaluated analytically to obtain a simple closed form expression for the diffraction correction. The extrema of this expression are obtained as roots of simple transcendental equations. It is also shown that the same expression can be obtained by taking the limit as ka → ∞ (where k is the wavenumber and a is the piston radius) of Williams's exact integral expression for the diffraction correction [J. Acoust. Soc. Am. 23, 1 (1951)]. Finally, it is shown both analytically and by numerical integra...

Among the many formulae which show special relations existing between the circular functions and the Bessel-Function J n ( x ), when n is half an odd integer, there is one due to Lommel

The Rayleigh formula for acoustic radiation from an ideal piston source reduces, under suitable approximation given below, to the ordinary Fresnel integral for diffraction through a circular opening. The Fresnel integral is usually expressed in a series of Bessel functions (Lommel function of two arguments) and reduces in the region of validity, to the correct solution along the axis, to the Fraunhofer solution at large distances, and agrees with R. B. Lindsay's solution on a cylinder based on the piston and coaxial with it. In this expansion, the first term represents the Fraunhofer solution but all the terms of the series must be included to give the solution along the axis. An alternative expansion can be given in which the first term represents both the Fraunhofer solution at large distances and the solution along the axis. This expansion is much easier to handle for regions near the axis than the first. For a piston of radius a, these expansions are valid for points at a distance d from the center of the piston within a cone of half‐angle θ coaxial with the piston such that (a/d) sin2θ≪λ/a, where λ is the wave‐length, and where (a/z)2≪1, z being the distance along the axis from the center of the piston.