# Why wave graphs sometimes use space and sometimes time?

On some graphs representing waves, the horizontal axis is marked as space (x), on other graphs the horizontal axis is marked as time (t).

This is the example for space:

This is the example for time:

Personally, I first started to represent waves by means of phasors: the projection of point travelling along the circumference perfectly represents some physical quantity oscillating around one specific point in space. For me it became natural to think of the circumference representing time.

Later I saw wave graphs where time was a straight line. Although this representation seemed odd to me at first, I started to realize that it's just another "system of coordinates" representing the same thing.

But then I started to read about wavelength. They used graphs with straight line and marked the wavelength on them. I was a bit puzzled how time axis may be used to show distance, until I noticed that the straight line was marked not with t, but with x.

I was a little astonished. Can anybody explain why the wave graph is the same no matter if we use time or space on horizontal line. And how to show wavelength on phasor and on graph with straight line marked with t.

It just depends on which way of representing the wave is most convenient, given the objective. Note that in any case, as long as you know the wave propagation speed, you can work back and forth between time and distance and not lose the big picture.

Assuming that the wave is sinusoidal, we can write the following (ignoring a phase for the moment):

$$y(x,t)=A\sin(kx-\omega t),$$

where $$k=2\pi/\lambda$$ with $$\lambda$$ the wavelength, $$\omega$$ is the angular frequency of the wave (rounds per unit time), $$A$$ is the fixed maximal amplitude (so maximal value of $$y(x,t)$$).

Plotting time versus amplitude means that you fix $$x$$ (so a fixed position, think about a rod and you look at a fixed point how the amplitude changes as function of time) and plot $$y(t)$$, note that $$x$$ is now a constant.

However you could also look at an instantaneous picture of the rod. Then of course the time $$t$$ is fixed and you can plot $$y(x)$$ since $$t$$ is constant now.

Note that in both cases: time or position fixed, $$y(x)$$ resp. $$y(t)$$ will still be described by a sine wave as you can see in the plots.

• Note that the wave does not need to be periodic--in general it could be any function of the form f(x-ct). It could be a pulse. – user45664 Mar 1 at 16:59
• I know, therefore I assumed it is periodic since the plot the OP showed is a sinusoid. In general this is not the case. – Dani Mar 1 at 17:24
• Do you know where to find a video animation or a drawing for your explanation? It's hard for me to understand your answer without a representation. Anyway, I'm interested to find any animations for waves for better understanding of the subject. – Igor Liferenko Mar 5 at 9:41