How to plot different dependent varaibles against the same independent variable

Question

Let's say I have a dispersion relation $\omega=\exp{k}$, how should I write a mathematica code to ask for plots of $\omega$ against $k$ as well as the phase speed $\omega/k$ against $k$ for a range of $k\in(-10,10)$ and $\omega/k\in(-5,5)$?

I tried to write my code as follows:

ContourPlot[Dis==0,{k,-10,10},{\[Omega],-5k,5k}]

And this worked, a graph is plotted successfully. Here Dis is defined as Dis = $\omega-\exp{k}$.

However, when I try plotting phase speed against wavenumber, a graph cannot be plotted. My code is as follows

a=\[Omega]/k
ContourPlot[Dis==0,{k,-10,10},{a,-5,5}]

Could someone please let me know where went wrong please?

Answer 1

Clear[dis,reg];
dis = ω - E^k;
reg = ParametricRegion[{{k, a}, 
    dis == 0 && a == ω/k}, {{k, -10, 10}, ω, {a, -5, 
     5}}];
RegionPlot[reg, PlotRange -> {{-10, 10}, {-5, 5}}, 
 FrameLabel -> {"k", "a"}, LabelStyle -> Directive[Blue, Bold, 12]]
(*Region[reg]*)

We can also test another cases, for example:

dis = k*(ω^2 + ω) - E^k;