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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.

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At the risk of giving an answer which is to complete, I'm going to offer interpretations for the indicated terms. Let me start by pointing out that the mathematical expression you are using describe the rotation of vectors in the complex number plane. This is convenient for dealing with phase differences, but we normally assume that only the real components give a displacement of the wave at a point (z) and time (t). With both waves (with the same polarization) moving in the same direction, the resultant can have a maximum of (A + αA) when the phase difference is zero (or some multiple of (2π)) or a minimum of ǀA – αAǀ when they are “out of phase”. (An observer might hear this variation as a “beat".) With two (polarized) waves (with equal amplitudes and frequencies) moving in opposite directions, the resultant is a “standing wave”. The maximum amplitude at any point is modulated by the cosine function of position. There are “nodes” which show no apparent motion and “anti-nodes” in between where the maximum amplitude is (2A).

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