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I got a little bit confused at my EM waves lectures when we were studying the topic of wave propagation. Totally because of my lack of understanding of the wave function graphs. In the chapter, we proved that the solution to the $(\frac{\partial^2 E_{X}}{\partial z^2}={\mu}{\epsilon}\frac{\partial^2 E_{X}}{\partial t^2})$ is in the form of $f(z-vt)$ and by assuming $z=z_{1}$ and $t=t_{1}$ the function has a value of $A$ which can be shown as;$$f_{1}(z_{1}-vt_{1})=A$$ Then we said for $z=z_{2}$ it will have the same value $A$ at a later time $t_{2}$, then we drawed the following graphs which is what confused me:

enter image description here

In these graphs, it is really hard for me to understand which axis is which? For example, is the vertical line referred to as time or magnitude? the horizontal is the space variable I can see but if the vertical is the magnitude, then where is the time variable in the wave graphs, and finally, what does the space variable mean exactly, is it like the location of the wave? I understand it is somewhat like traditional function graphs that it moves horizontally when you sum the variable with a constant and it contracts or expands if you multiply it but in the waves, it is like we have two variables in one function, I have never seen any graphs drawn with two independent variables, how does that work in wave graphs? I know I am asking too many questions, forgive me but I could really use some clear explanations, much thanks.

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What you are writting is a way to show that a wave that does not change its shape as it progresses through space and that has the general form of the one-dimensional wave function ($\psi(x, t)=f(x-v t)$) has velocity $v$ when is forward travelling and $-v$ reverse travelling.

You only have two axis: $\phi$ and $x$ and you are representing $\phi(x)$ . Indeed you are representing $\phi(x,t)$ but in two different times $\phi(x,t_{1})$ and $\phi(x,t_{2})$, therefore $\phi(x,t_{1})=\phi(x)_{t_{1}}$ and $\phi(x,t_{2})=\phi(x)_{t_{2}}$ do not depend on time.

If your wave $\phi(x,t)$ is, for example:

enter image description here

You are representing, two snaps of the wave at two different times: $t_{1}$ and $t_{2}$.

The quantity: $\Delta z = z_{2}-z_{1}$ is the distance the wave moved during $\Delta t = t_{2}-t_{1}$.

You can see more information in this amazing optics textbook:

E. Hecht, Optics. Pearson, 2012 - 6th edition.

In particular in section 2.1.- One-dimensional Waves.

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