I'm studying propagation of waves. We usually have the most simple cases, when the amplitude, direction of propagation, frequency are equal, but I'm studying some other different cases, that I want to give some physic interpretation to the results. Because as we could expect we want to end up with an expression of other wave, analogous as the general form, $$ u(z,t)=Ae^{-i(\omega t-kz)} e^{i\phi} $$

$\bullet$ First I wanted to study waves with same frequency and direction of propagation but different amplitudes $A_2=\alpha A_1$, and a initial phase in each wave,

$$ \left\{\begin{matrix} u_1(z,t)=Ae^{-i(\omega t-kz)} e^{i\phi_1}\\ u_2(z,t)=(\alpha A)e^{-i(\omega t-kz)} e^{i\phi_2} \end{matrix}\right. \quad \Rightarrow u(z,t)=A \underbrace{\left( \alpha+e^{i\Delta \phi} \right)}_{\text{Unknown interpretation}} e^{-i(\omega t-kz)}e^{i\phi_1} $$

I dont know if the term of the equation is part of the amplitude or if it could be simplifyed, or others.

$\bullet$ Other case that I studied, is the superposition of waves with same frequency, polarization, amplitude, but opposite propagation direction $\vec{k}_1=-\vec{k}_2$, (\vec{P} is the polarization vector, but in this case it doesn't affect to the result)

$$ \left\{\begin{matrix} u_1(z,t)=Ae^{-i(\omega t-kz)} \vec{P}\\ u_2(z,t)= Ae^{-i(\omega t+kz)} \vec{P} \end{matrix}\right. \quad \Rightarrow u(z,t)=2A \underbrace{\cos kz}_{\text{Unknown interpretation}} e^{-i\omega t}\vec{P} $$

Again i could supose that this term is inside the amplitude, and now the amplitude depends on the position, it could be a kind of beat interferences.