# Some interpretations of interfering waves

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.