Sommerfeld-Watson Transformation Research Articles

Backscattering from an elastic sphere: Sommerfeld–Watson transformation and experimental confirmation

The Sommerfeld–Watson transformation (SWT) of the partial wave series for the acoustical scattering from a fluid-loaded elastic sphere is examined. This research specifically focuses on the specular reflection and Rayleigh wave contribution to scattering at small backscattering angles. In a previous paper the angular dependence of the Rayleigh contribution to near backward scattering was measured and modeled [Williams and Marston, J. Acoust. Soc. Am. 78, 722–728 (1985)]. The SWT confirms the physical picture used and, for the first time, predicts the absolute Rayleigh contribution associated with one or more circumnavigations of the sphere. To test the SWT, tungsten carbide spheres in water were ensonified by tone bursts having central frequencies in the range 24<ka<80. Measurements were made of the first and second Rayleigh contributions to the backscattered pulse train. Plots of these measured distinct Rayleigh amplitudes as a function of ka confirm the results of the SWT and illustrate the significance of radiation damping and axial focusing.

Rayleigh‐mode backscattering from an elastic sphere: Sommerfeld‐Watson transformation and experimental confirmation

The acoustical scattering form function of fluid‐loaded bodies such as cylinders and spheres may be written as a partial wave series. The Sommerfeld‐Watson transformation (SWT) has been used previously to rewrite the form function of elastic cylinders with ka⩾1. The SWT allows one to understand the form function in terms of reflected, transmitted, and surface waves. In the present discussion we carry out the SWT on a fluid‐loaded elastic sphere. We concentrate on the specular reflection and Rayleigh wave contribution to scattering at small backscattering angles. We previously measured and modeled the angular dependence of the Rayleigh contribution to near backward scattering [K. L. Williams and P. L. Marston, J. Acoust. Soc. Am. Suppl. 1 76, S8 (1984)]. The SWT confirms the physical picture used and, for the first time, predicts the absolute Rayleigh contribution associated with one or more circumnavigations of the sphere. To test the SWT, tungsten carbide spheres in water were ensonified by tone bursts having central frequencies in the range 30 < ka < 80. Measurements were made of the first and second Rayleigh contributions to the backscattered pulse train. Plots of these measured distinct Rayleigh amplitudes as a function of ka confirm the results of the SWT and illustrate the significance of radiation damping and axial focusing. [Work supported by ONR.]

Acoustic high‐frequency scattering by elastic cylindrical shells

Acoustical scattering resulting from a high‐frequency plane‐wave incident upon an infinite aluminum circular cylindrical shell immersed in and filled with water is determined by applying the Sommerfeld–Watson transformation to the classical Rayleigh normal‐mode series solution. The resulting contour integrals are computed by both the saddle point method and by summing residues over poles which correspond to the zeros of a 6 ×6 determinant resulting from satisfying the necessary boundary conditions. The Bessel functions and their derivatives which appear in the 6 × 6 determinant are evaluated using asymptotic representations of both the Airy and Debye type depending upon the region of the complex plane. Emphasis is placed on the transitions in the scattering characteristics as the shell goes from a solid cylinder to a thin‐walled shell. This dual behavior will be discussed in terms of the number and trajectories of allowed modes of the scatterer, the dispersion curves of various modes, and the contributions of the various modes to the scattered pressure field.

High-frequency asymptotic forms of refracted and transmitted waves when sound is scattered by a hollow elastic sphere filled with liquid

The Sommerfeld-Watson transformation, the reflection method and the theory of graphs are used to investigate the components of the echo-signal from a hollow elastic sphere with a concentric liquid filler, formed by reflection and rereflection at the media interfaces.

Acoustic high-frequency scattering by elastic cylinders

Acoustic scattering from an infinite elastic cylinder in a fluid is described by using the Sommerfeld–Watson transformation. The wavenumbers for the surface and transmitted waves are determined by poles or saddle points, respectively, of the scattering amplitude in the complex wavenumber plane. Dispersion and attenuation curves for the dominant surface wave modes have been obtained from the pole positions and given in our previous work. The present study gives the amplitudes of all the known contributions to the scattered field by evaluating the corresponding residues and the saddle-point amplitudes. The results indicate that the individual surface wave modes (i.e., Franz, Rayleigh, Stoneley, and Whispering Gallery), remain comparable to the transmitted-wave contributions to the scattered field (all these being plotted versus scattering angle at several frequencies) through frequencies at least as high as ka=200. This indicates that sound scattering from an elastic body does not tend to become predominantly geometrical-diffractive in the short-wavelength limit. Finally, some surface wave modes are plotted versus frequency, and are shown to contain a series of narrow resonances corresponding to the excitation of eigenvibrations of the elastic cylinder.

Diffraction of non-relativistic electron waves by a cylindrical capacitor

The diffraction of non-relativistic electron waves by a cylindrical capacitor is considered for an electric voltage at the capacitor small compared with the energy of the incident electrons. On the basis of the exact general solution of the Schrodinger equation for an electron in an attractive logarithmic potential, asymptotic solutions inside the capacitor which are similar to JWKB-type solutions, but with a significant modification, are derived. Application of appropriate boundary conditions to the asymptotic solutions yields an angular momentum expansion of the scattered wave which is further evaluated by means of the Sommerfeld-Watson transformation. The change of various well known diffraction phenomena with increasing electric voltage at the capacitor is calculated explicitly and discussed in detail; in particular, the convergence of electron interference fringes towards the optical axis is one of the main results of the investigation.

The Lorentz group in the oscillator realization. I. The group SO(2,1) and the transformation matrices connecting the SO(2) and SO(1,1) bases

The unitary transformation connecting the SO(2) and SO(1,1) bases for the principal and discrete series of representations of the three-dimensional Lorentz group is determined by using the oscillator representation technique. The Hilbert space and the SO(1,1) basis, in this realization, have a simple appearance while the compact basis appears as the solution of an ordinary differential equation reducible to the confluent hypergeometric equation by a simple substitution. The Taylor expansion of this solution obtained by the use of certain functional identities yields the continuous spectrum of the SO(1,1) representations and the unitary transformation from the compact to the noncompact basis after the Sommerfeld–Watson transformation.

Semiclassical scattering theory with complex trajectories. I. Elastic waves

The WKB approximation to the one-particle Schrödinger equation is used to obtain the wave function at a given point as a sum of semiclassical terms, each of them corresponding to a different classical trajectory ending up at the same point. Besides the usual, real trajectories, also possible complex solutions of the classical equations of motion are considered. The simplicity of the method makes its use easy in practical cases and allows realistic calculations. The general solution of the one-dimensional WKB equations for an arbitrary number of complex turning points is given, and the solution is applied to calculate the position of the Regge poles of the scattering amplitude. The solution of the WKB equations in three dimensions for a central analytical potential is also obtained in a way that can be easily generalized to N-dimensions, provided the problem is separable. A multiple reflection series is derived, leading to a separation of the scattering amplitude into a smooth “background” term (single reflection approximation) that can be treated using classical but complex trajectories and a second resonating term that can be treated using the Sommerfeld-Watson transformation. The physical interpretation of the complex solutions of the classical equations of motion is given: they describe diffractive effects such as Fresnel, Fraunhofer diffraction, or the penetration of the quantal wave into shadow regions of caustics. They arise also in the scattering by a complex potential in an absorptive medium. The comparison with exact quantal calculations shows an astonishingly good agreement, and establishes the complex semiclassical approximation as a quantitative tool even in cases where the potential varies rapidly within a fraction of a wavelength. An approximate property of classical paths is discussed. The general pattern of the trajectories depends only on the product ϵ = EΘ, and not on energy and angle separately. This property is confirmed by experiments and besides the signature it gives for the semiclassical behavior, it simplifies considerably the search for all trajectories scattering through the same angle. Finally, a general classification of the different types of elastic heavy ion cross sections is given.

The master analytic function and the Lorentz group. II. The Clebsch–Gordan problem for O(2,1)

The Clebsch–Gordan coefficients of the noncompact group O(2,1) representing Lorentz transformations in three-dimensional space–time are calculated in the compact O(2) basis. Considerable simplification is achieved by introducing a variable x and replacing all algebraic equations by differential equations. The coupled state appears in the theory as a solution of an ordinary differential equation reducible to the hypergeometric equation by a simple substitution. The coefficients in the Taylor–Laurent expansion of this solution in powers of x are shown to be identical with the Clebsch–Gordan coefficients. The inverse expansion, obtained by the use of certain identities for the hypergeometric function and the Sommerfeld–Watson transformation, yields the normalization factor and the values of j appearing in the reduction.

The master analytic function and the Lorentz group. I. Reduction of the representations of O(3,1) in O(2,1) basis

The reduction of the principal and supplementary series of representations of SL(2,C) in the SU(1,1) basis is carried out by using a basis function which formally resembles the coupled state of two angular momenta. The spectrum of the SU(1,1) representations contained in SL(2,C) and the transformation coefficients are obtained by expanding the SU(2) in terms of the SU(1,1) bases with the help of the Sommerfeld–Watson transformation. The orthogonality conditions for the principal and supplementary series are discussed. For the principal series this follows easily from the standard Sturm–Liouville theory of the second order differential equations. For the supplementary series the orthogonality condition is obtained from the fourth order differential equation satisfied by the Fourier transform of the basis function.

Power Series, Analyticity, and Current-Hadron Scattering Amplitudes

We propose to expand the current-hadron scattering amplitude as a doublc power series in the mass of the current and its laboratory energy. The series is summed by a double Sommerfeld-Watson transformation. Scaling is seen to provide a powerful constraint on the location of singularities in a complex parameter plane associates with the mass variable. (auth)

A nonperturbative solution of a nonlinear field equation of motion

In a previous paper an infinite-component generalized free fieldϕ(x) was constructed which transforms locally under the Poincare group and has a propagator which is invariant under aTP transformation.ϕ(x) carries an infinite tower of unstable (excepting the lowest member) self-compounds which is defined by a spin trajectory. The propagator ofϕ(x) can be written as an infinite partial series (in spinj) which can be summed by a Sommerfeld-Watson transformation with a particular constraint on the spin trajectory. The present paper investigates in what sense the fundamental fieldϕ(x) fixed in such a way in formal detail can provide a nonperturbative solution of the nonlinear field equation of motion (in aTCP- and Lorentz-invariant manner) $$\begin{gathered} \square _x \left\langle {0\left| {\left\{ {\varphi (y),\varphi ^ + (x) + \varphi ( - y),\varphi ^ + ( - x)} \right\}} \right|0} \right\rangle = \hfill \\ g\left\langle {0\left| {\left\{ {\varphi (y),\varphi ^ + (x) \cdot \varphi ^ + (x) \cdot \varphi ^ + (x) + \varphi ( - y),\varphi ^ + ( - x) \cdot \varphi ^ + ( - x)} \right\}} \right|0} \right\rangle \hfill \\ \end{gathered} $$ withx0 =y0. It is found necessary in terms of a dynamical hypothesis to consider how the reaction of the two-particle states corresponding to the terms nonlinear inϕ†(x) leads to compounds as single-particle states. The consequently required covariant reduction of the nonlinear term, where it involves the product representations of discrete and continuum mass distributions, may be generalized from known theory with due recognition of the scale factor problem. The latter is solved in a way which appears to be of importance for the concept of a fundamental field. One then finds by a particular way of taking the momentum integrals involved off the mass shells, and considering the equation in the particular form above, that all the extra momentum integrals for all spin components are completely convergent. One ends up with purely dynamical equations involving the mass spectral functions nonlinearly, and giving the conditions forϕ(x) to be a solution of the nonlinear equation in this particular sense. These spectral-function equations are examined for convergence of the integrals overr and sums overj involved, and for the asymptotic behaviour inr andj. The examination shows that in principle a solution may be obtained which is consistent with the independent determination ofϕ(x) in the previous paper, provided that the scale factor function has a suitable dependence onr andj.

An infinite-component free field carrying compounds lying on a Regge trajectory

One herein constructs an infinite-component generalized free field ϕ(x) which carries an infinite tower of unstable (excepting the lowest-mass member) self-compounds which is defined by a spin trajectory. ϕ(x) transforms locally under the Poincare group. The norm or propagator of ϕ(x) can be written as an infinite partial (in spinj) series of contributions of positive-definite metric, which, with the suggested constraint on the mass2 (squared) functionmj2 as |j|→∞, permits summation by the Sommerfeld-Watson transformation. In close analogy to the Gupta-Bleuler form of quantum electrodynamics, the consequent Reggeized propagator then involves contributions of indefinite metric completely equivalent to the positive-metric partial series. The indefinite metric is here directly associated with the instability of the higher Regge poles. The constraint onmj2 together with the effect of invariance underTP for the ϕ propagator leads to a particular dispersion relation for our mass2 functionmΛZ2. The latter involves Λ=(j+1/2)2,Z=P2 the fourmomentum squared, on a completely reciprocal basis. It also, necessarily, introduces two mass2 continua (r>4m02,r<0). The latter, tachyon component suggests that ϕ has to be interpreted as a fundamental field which is not directly observed. The phenomenological field having no tachyon component and which is directly observed will be discussed on the basis of unitarity in a subsequent paper. The propagator of the free field ϕ clearly obeys unitarity in a trivial way in terms of the free states defined by itself.

Analysis of the Scattering Process of Two Majorana Fields

The s-channel scattering process in which all external fields are Majorana ones are considered. We discuss here the elastic scattering of a hybrid-type field (the direct product of Dirac and Majorana field) and Majorana field. A rule for calculating the three-Majorana vertex is given. Because of the effect of this vertex, besides the unequal mass kinematics, the partial wave amplitude has an unphysical left-hand cut in the s-plane which is different from the ordinary theory. It is also shown that a partial wave amplitude satisfies the condition for performing the Sommerfeld-Watson transformation, and a scattering amplitude has a Regge pole term.

On the zero-mass superpropagator

A rigorous construction for the zero-mass exponential superpropagator is given. It is based on an explicit regularization of the propagator in coordinate space and a suitable choice of the coupling constant is taken in order to damp down the regularized superpropagator to an L 2 function. By applying a Sommerfeld-Watson transformation and a Fourier transformation we are able to show that the regularized superpropagator tends, in an appropriate sense, both in momentum and coordinate space, to some localizable generalized function. Analytic continuation in the coupling constant shows the appearance of a logarithmic cut. By means of the regularization of the propagator and the special choice of the coupling constant we have neither to cut divergent integrals nor to pass to Euclidean region. Extensions to superpropagators which are formally written as entire functions of the propagator are given.

Complex helicity and the Sommerfeld-Watson transformation of group-theoretic expansions

The connection of group-theoretic expansions (in both continuous and discrete bases) of a function defined onSO3 andSO2,1 and analytic on their complexification (except for certain singularities), is discussed. This is achieved by making various transformations of the Sommerfeld-Watson type. The analyticity assumptions needed for these transformations are considered and found to depend on the basis chosen for the representation functions. The applications of this work to multi-Reggeon expansions and, in particular, to the behaviour of Reggeon vertex functions are discussed. The connections between the various Sommerfeld-Watson and group-theoretic signatures are also given. As a preliminary, transforms are used to relate the Fourier series to the Fourier integral.

Nonmodal Solution of Spherical Shells With Cutouts Excited by High-Frequency Axisymmetric Forces

A closed-form solution is obtained for the high-frequency response of a thin spherical shell embodying a circular cutout and excited axisymmetrically by a concentrated radial force. The solution is constructed by combining the shell response to the radial exciting force with its response to radial, tangential, and moment line loads applied along the cutout boundary, these line loads being selected to match the boundary conditions. Concise expressions for the shell response are obtained by applying the Sommerfeld-Watson transformation to the slowly converging high-frequency modal series which is thereby reduced to only two terms, viz., an exponentially decaying near-field and a standing or propagating-wave field. These two terms are in the nature of the creeping waves commonly used to formulate electromagnetic or acoustic diffracted wave fields in the short-wavelength limit. The method is illustrated for the simple case of a circular cutout with a clamped boundary, but lends itself to more complicated boundary conditions, viz., intersecting shells or wave guides. The natural frequencies and mode shapes are found from a single, characteristic equation involving trigonometric functions.

Two-Variable Expansion of the Scattering Amplitude: An Application of Appell's Generalized Hypergeometric Functions

Appell's polynomials in two variables orthogonal in a triangle are described and some of their properties and those of related generalized hypergeometric functions are given. An application to the expansion of the scattering amplitude is suggested, the equal-mass case being discussed in some detail. A simple crossing matrix is derived. Difficulties introduced by inequality of the particle masses are explained. A Neumann formula is presented which permits an analytic continuation in the parameters to be made of the expansion coefficients for parts of the amplitude: This is in analogy with the Froissart-Gribov continuation. A conjectured analog of the Sommerfeld-Watson transformation then suggests the existence of fixed cuts in the partial-wave scattering amplitude.

Use of Nonsignatured Partial-Wave Amplitudes in Regge-Pole Theory

It is shown that a Froissart-Gribov-type partial-wave amplitude without signature can be defined for both two-body and multiparticle processes. Such an amplitude permits a Sommerfeld-Watson transformation, and its poles govern asymptotic behavior in the usual way, although unitarity properties of the amplitude are complicated. The relation to Toller amplitudes is discussed.

A crossing-symmetric Sommerfeld-Watson transformation of the two-variable expansion of the scattering amplitude

We consider the scattering of two particles of equal mass and show how a Sommerfeld-Watson transformation may be applied to the expansion of the amplitude given by Balachandran and Nuyts, if it is assumed that the amplitude is continuable in the complex plane of a crossing-symmetric angular-momentum variable. We show that analyticity in this variable can be derived from the assumption of analyticity in the usual complex angular-momentum planes.