w1=Sqrt[1-b^2]
begin=Integrate[tp/T Exp[-b (t-tp)]Sin[w1(t-tp)],{tp,0,t}]/w1
first=Integrate[tp/T Exp[-b (t-tp)]Sin[w1(t-tp)],{tp,0,T}]/w1
end=Integrate[ Exp[-b (t-tp)]Sin[w1(t-tp)],{tp,T,t}]/w1
x[t_]=If[t<T,Evaluate[begin],Evaluate[first+end]]
b=.01
T=8 Pi
sloww=Plot[x[t],{t,0,25 Pi},PlotRange->All]
T=7 Pi
sloww0=Plot[x[t],{t,0,25 Pi},PlotRange->All]
T=4 Pi
slow=Plot[x[t],{t,0,25 Pi},PlotRange->All]
T=3Pi
slow0=Plot[x[t],{t,0,25 Pi},PlotRange->All]
T=Pi
fast=Plot[x[t],{t,0,25 Pi},PlotRange->All]
T=Pi/10
faster=Plot[x[t],{t,0,25 Pi},PlotRange->All]
GraphicsGrid[{{sloww,sloww0},{slow,slow0},{fast,faster}}]

a=.02
f[t_]=Integrate[Exp[-a tp^2]Cos[tp]Exp[-b(t-tp)]Sin[w1(t-tp)]/w1,{tp,-Infinity,t}]
gauss=Plot[Re[f[t]],{t,-15,70}] 
GraphicsGrid[{{sloww,sloww0},{slow,slow0},{fast,faster},{gauss}}]
