On three-dimensional propagation and dispersion of wave packets described by spatial moments


  • B. Dong,

  • K. C. Yeh


In this paper we study the three-dimensional propagation of a vector wave packet in a lossless, homogeneous but anisotropic medium. By using the superposition principle and Fourier transform theory a general formulation of wave packet propagation theory is developed. We prove that the square magnitude of the complex amplitude of the wave packet over the whole space integrates to a constant. The conservation of this integral merely reflects the lossless nature of the medium. This integral can be considered as the zeroth spatial moment. Upon this foundation the first and second spatial moments are introduced. The first moment determines the position vector of the centroid of a wave packet. It is a linear function of time, showing the motion of the centroid at a spectrum weighted average of the gradient of frequency in wave number space. Thus the concept of group velocity is extended to the wide banded wave packets. The second moment of the wave packet describes the property of dispersion. It reflects the anisotropic nature of the propagating medium and reveals the fact that the length of the wave packet expands with time in the form of time squared.