A phase space approach to wave propagation with dispersion

Area of Science:

• Physics
• Applied Mathematics
• Wave Propagation

Background:

• Linear dispersive wave propagation is crucial in many scientific fields.
• Existing methods often focus on single modes, limiting analysis of complex wave phenomena.
• Accurate modeling of arbitrary initial conditions remains a challenge.

Purpose of the Study:

• To develop a novel phase space approximation for linear dispersive wave propagation.
• To extend Wigner distribution methods to analyze the full wave, not just single modes.
• To provide a framework applicable to general linear wave equations.

Main Methods:

• Developed a phase space approximation method for linear dispersive wave equations.
• Transformed wave equations into phase space for analysis.
• Utilized initial modal functions and cross-Wigner distributions.
• Each modal function was shown to satisfy a Schrödinger-type equation.

Main Results:

• The approximation is valid for the full wave, encompassing multiple modes.
• The method requires initial modal functions and their cross-Wigner distributions.
• Modal functions evolve according to Schrödinger-type equations with a mode-specific dispersion relation as the Hamiltonian.

Conclusions:

• The phase space approximation offers a powerful tool for analyzing complex wave propagation.
• This method provides a more comprehensive understanding compared to single-mode approximations.
• The approach is demonstrated effectively using the beam equation as an example.