L=1024;
S=64;

w1[k_,c_]:=Sign[k]Sqrt[Abs[c k+(1-c)k^3]]

w2[k_]:=k

Plot[{w1[k,.25],w2[k]},{k,0,1}]

f[t_]:=Module[{q},RotateRight[Chop[Fourier[RotateLeft[PadRight[Table[q=k/S; (q^2Cos[w1[q,c]t]+.2q Sin[w1[q,c]t])Exp[-5 q^2],{k,-S,S}],L],S]]],L/2]
]

c=.25;

splash=Table[ListPlot[-f[t],Joined->True,PlotRange->{-.3,.3},AspectRatio->.2,Filling->Bottom,FillingStyle->Blue,Axes->False,PlotStyle->Blue,ImageSize->600],{t,0,120,1.}];

ListAnimate[splash,20.]