Dispersive Lamb Systems

The original Lamb system was introduced by Horace Lamb in 1900, [13], as a simple model for the phenomenon of radiation damping experienced by a vibrating body in an energy conducting medium. Examples include vibrations of an elastic sphere in a gaseous medium, electrical oscillations of a spherical conductor, a dielectric sphere with large inductance, relativistic radiation of energy from a concentrated mass via gravity waves, and quantum resonance of nuclei. To understand this phenomenon, Lamb proposed a simpler one-dimensional model consisting of an oscillatory point mass–spring system directly coupled to an infinite string modeled by the usual second order wave equation. The oscillator transfers energy to the string, generating waves that propagate outwards at the intrinsic wave speed. Meanwhile, the string induces a damping force on the oscillator, with the effect that the propagating waves are of progressively smaller and smaller amplitudes. It is worth pointing out a potential source of confusion: Lamb refers to the point mass as a “nucleus”, although he clearly does not have atomic nuclei in mind as these were not discovered by Rutherford until 1911. More recently, Hagerty, Bloch, and Weinstein, [11], constructed gyroscopically coupled Lamb systems, also known as Chetaev systems, including the spherical pendulum and rigid bodies with internal rotors, that can induce instabilities through the effects of Rayleigh dissipation. They also investigate the effects of dissipation as modeled by the Klein–Gordon equation, although as we will see, this does not cover the range of phenomena exhibited by dispersive Lamb models. This paper analyzes the both the full line and periodic Lamb problems when the mass is coupled to more general one-dimensional dispersive media. Our motivation is to see whether the remarkable phenomenon of dispersive quantization, [15], also known as the