Wave amplitude - Knowlpedia

In an oscillatory field

w(t,x)=a(t,x)e^{i\kappa\Phi(t,x)},

the coefficient $a$ multiplying the phase factor is the amplitude. It may be a complex scalar or vector. For real-valued fields one often takes $\operatorname{Re}(ae^{i\kappa\Phi})$ or the conjugate pair $ae^{i\kappa\Phi}+\bar a e^{-i\kappa\Phi}$ ; these conventions differ by a factor of two.

Separation assumptions

Calling $a$ an amplitude does not imply it varies slowly. Relative derivative bounds must be supplied. The decomposition also depends on convention: a phase factor can be moved between $a$ and the phase unless a choice has been fixed. An envelope is a bound on an amplitude, not necessarily the amplitude itself.