Ytukyg

Why Vertical axis changes name when we talk about waves? is there any differences between Range and Amplitude?

also i didn't knew what tag should i use for this, sorry for inappropriate tags.

EDIT:
I always see ppl using fourier transform and stuff, they write Amplitude as the label of their frequency domain plot.

This very much depends on how you represent a wave and what it means.

In time domain, generally one plots an $x$-axis representing time and the $y$-axis represents a physical quantity, such as pressure or voltage or displacement. The amplitude, while not being an axis of this graph, is a quantity that one can find from the graph - it is the distance between an extreme point and the average point ("equilibrium" in a physical context). In particular, the wave $Asin(x)$ has amplitude $A$ because its extreme points are at $pm A$ and its equilibrium would like at $0$. The "range" (or "peak-to-peak amplitude") sometimes refers to the difference of largest and smallest values in a wave - so $2A$ for $sin(x)$.

It's harder to give a single value for amplitude for more complex waveforms - there's still a notion that making a wave twice as tall doubles its amplitude, but there are many definitions that try to capture various desirable quantities of a wave.

For the most part, amplitude would not be an appropriate label for the $y$-axis of a plot of a waveform - the possible exception would be if you're plotting a standing wave where the $x$-axis is distance and the $y$-axis is the amplitude of some time-varying quantity when observed at the prescribed point in space.

In frequency domain, a complicated wave is shown to be decomposed into a sum of sine waves. For instance, if you started with a wave such as
$$F(t)=sin(2pi tcdot 1000 text{ Hz})$$
where $t$ is some duration, the Fourier transform would show* a single bar showing amplitude $1$ at $1000text{ Hz}$, meaning that the wave is just a sine wave with that amplitude at that frequency. A more complicated wave such as
$$F(t)=1/3cdot sin(2pi tcdot 1000 text{ Hz})+ 1/4 cdot sin(2pi tcdot 2000 text{ Hz})$$
would show an amplitude of $1/3$ at $1000text{ Hz}$ and $1/4$ at $2000text{ Hz}$, showing that the bigger wave is a sum of simpler waves with the prescribed amplitudes. In general, any complex wave can be written as a sum of sine waves with various phases, and the Fourier transform displays, at each frequency, what amplitude of wave would be necessary.

(*There are many caveats here: if you're, for instance, working with digital audio, you won't see a sharp peak because those programs take an arbitrary chunk of your sound, pretend it repeats forever, then decompose that wave; if the chunk doesn't line up with the periods of your wave, the transform won't represent a sine wave. There are also issues with the fact that Fourier transforms most naturally work on continuous signals whereas they are most commonly applied to discrete signals obtained via sampling at some rate. Also, in this context, usually both axes are using logarithmic scales, which makes them look a bit odd)

As mentioned in a comment, if you have a sine wave in the time domain $y=Asinomega t$, $A$ is the amplitude. If you add multiple such functions you get something like $$y=sum_i A_isinomega_it$$
When you do the Fourier transform, this will show up as a set of points $(pmomega_i, A_i)$ (up to some normalization constant). The same thing is occurring if you have a continuous distributions of $omega$ angular frequencies. So in the case of the Fourier transform the thing that you plot is the amplitude of each contributing sine wave.

The above answer is the same if you add sine and cosine contributions.

Important point: $y=sin(ωt)+1$ has exactly the same amplitude as $y=sin(ωt)$. But the first has $[0,2]$ as its range and the second, $[-1,1]$.

The amplitude measures the size of the waveform, not the set of values it assumes.