Accurately distort graphics

I would like to accurately distort this plot of $\sin(x^{1/2})$ (and others like it) so that the wavelengths are evened out (ie - "inverse-square root" it).

I should like to do the same to "de-log" plots too.

I have no idea where to start though.

Answer

plt1 = Plot[Sin[Sqrt[x]], {x, 0, 1000}, ImageSize -> 400];

ticks = Ticks /. AbsoluteOptions[plt1, Ticks];
ticks = MapAt[#^(1/2) &, ticks, {{1, All, 1}}];
plotrange = PlotRange[plt1];
plotrange = MapAt[#^(1/2) &, plotrange, {1}]; 

plt2 = Graphics[plt1[[1]] /. Line[x_] :> Line[{#[[1]]^(1/2), #[[2]]} & /@ x], 
                PlotRange -> plotrange, Ticks -> ticks, plt1[[2]]];
Row[{plt1, plt2}]

Update: The approach above works without issue in Version 9.0.1.0 (Windows 8 64bit). Unfortunately it does not work in Version 10 because AbsoluteOptions stopped working as expected in Version 10. A work-around until the AbsoluteOptions glitch is fixed (hopefully in Version 10.0.1.0) is to specify the ticks directly:

ticksb = {{#^(1/2),#}&/@FindDivisions[{0,1000},{5}][[1]],Automatic};
plt2b = Graphics[plt1[[1]] /. Line[x_] :> Line[{#[[1]]^(1/2), #[[2]]} & /@ x], 
            PlotRange -> plotrange, Ticks -> ticksb, plt1[[2]]];
Row[{plt1, plt2b}]