This is another simple way. The idea is to plot a $cos$ function, by changing its frequency and amplitude, you can adjust the twisting needed

 f1[x_]:=Sqrt[x^2+1]
 data=Table[z0= 1/10 Cos[50 x];z1=f1[x];{x,z0+z1},{x,-2,2,.02}];
 Show[Plot[f1[x],{x,-2,2},AxesOrigin->{0,.7}],
 ListLinePlot[data,PlotStyle->Red]]

![Mathematica graphics](https://i.sstatic.net/dxaK3.png)

This one has smaller frequency

 data=Table[z0=1/10 Cos[20 x];z1=f1[x];{x,z0+z1},{x,-2,2,.02}];

![Mathematica graphics](https://i.sstatic.net/qvheS.png)

[![enter image description here][1]][1]

 Manipulate[
 data = Table[z0 = h Cos[w x]; 
 z1 = f1[x]; {x, z0 + z1}, {x, -2, 2, .02}];
 Show[Plot[f1[x], {x, -2, 2}, AxesOrigin -> {0, .7}], 
 ListLinePlot[data, PlotStyle -> Red]],
 {{h, .1, "amplitude"}, 0, .5, .01, Appearance -> "Labeled"},
 {{w, 20, "frequency"}, 1, 100, 1, Appearance -> "Labeled"},
 TrackedSymbols :> {h, w},
 Initialization :>
 (
 f1[x_] := Sqrt[x^2 + 1];
 )
 ]

 [1]: https://i.sstatic.net/TYj5K.gif