The concentrated toroidal wave

TL;DR

This paper introduces a method to generate toroidally concentrated waves by superposing bandpass-filtered spherical Bessel functions, enabling wave amplitude focus around a ring without long oscillatory tails, with applications in optics and acoustics.

Contribution

The study presents a novel approach to produce toroidal wave concentration through bandpass superpositions of spherical Bessel functions, expanding classical solutions for practical wave focusing.

Findings

Superpositions of bandpass-filtered spherical Bessel functions concentrate wave amplitude around a ring.

The method avoids the long oscillatory tails of classical spherical Bessel solutions.

Numerical simulations confirm effective wave concentration with practical source shapes.

Abstract

The classical solution to the Helmholtz wave equation in spherical coordinates is well known and has found many important applications in wave propagation, scattering, and imaging in optics and acoustics. The separable solution is comprised of spherical Bessel functions in the radial direction and spherical harmonics in the angular directions. The nature of the spherical Bessel functions includes a long asymptotic oscillatory tail at large radii, not conducive to applications where a tight concentration of wave amplitude around a ring is desired, for example in toroidal configurations. However, we have found that certain practical bandpass spectral shapes, centered around a peak frequency, can create a superposition of spherical Bessel functions that effectively concentrate the wave amplitude around a defined ring at the time instant of coherent addition, avoiding the long tail asymptotic oscillations of the single frequency solution. Theoretical solutions are shown for different bandpass spectra applied to the spherical Bessel functions, along with numerical solutions of transient wave propagation using practical hemispherical source shapes. These findings introduce a framework by which ring or toroidal concentrated waves can be produced with a simple bandpass superposition applied to hemispherical source shapes and with reference to the classical solutions in spherical coordinates.