Wave Interactions and X-Ray Crystallography

Abstract

The Fourier representation of plane waves in fluid motions may be expressed in the form

$$ u(r,t) = \sum\limits_{n} {{a_{n}}\exp [i2\pi n({k_{n}} \cdot r - {\omega _{n}}t)] + c.c} $$

. The coefficients α _n , the wave numbers k _n , and the wave frequencies ω _n are determined by the conservation laws of mass and momentum as well as boundary initial conditions. Each term in the Fourier representation is called a mode. In a linear system, the modes do not interact with each other. In a nonlinear system, the modes do interact with each other and these interactions generate new modes. So, strictly speaking, the above Fourier representation of a plane wave for a nonlinear system is valid only for a specified moment of time or a specified short time interval. The new modes are generated only when resonance conditions

$$ \begin{array}{l} {{\rm{k}}_{1}}\pm {k_{2}}\pm {k_{3}} = 0 for three waves, \\ {\omega _{1}}\pm {\omega _{2}}\pm {\omega _{3}} = 0, \\ \end{array} $$

or

$$ \begin{array}{l} {{\rm{k}}_{1}}\pm {k_{2}}\pm {k_{3}}\pm {k_{4}} = 0 for four waves, \\ {\omega _{1}}\pm {\omega _{2}}\pm {\omega _{3}}\pm {\omega _{4}} = 0, \\ \end{array} $$

are satisfied. The third mode is generated by the first two modes in a three-wave interaction process and similarly the fourth mode is generated by the first three modes in a four-wave interaction process.