What is the meaning of the imaginary part of the plane wave function
Most likely ##\phi## is going to be used to align the phase globally. Likewise with ##A_0## for the amplitude. To model a realistic wave, each frequency component ##\omega_n## will have a unique phase offset that is accommodated by the combined amplitudes of the real and imaginary components for that frequency.
The wave function with only a single valued ##\omega## is a monochromatic wave. Most waves encountered experimentally are more complex.
