f1[u_]=Cos[gamma*Cos[Pi*u/sigma1]]

f2[u_]=Sin[gamma*Cos[Pi*u/sigma1]]

gamma=Pi*Sqrt[2]/2

a=1

sigma1=a/BesselJ[0,gamma]

N[sigma1]

Plot[NIntegrate[f1[x],{x,0,u}],{u,0,8Pi}]

Plot[NIntegrate[f2[x],{x,0,u}],{u,0,8Pi}]

t=0

0

c=1

ParametricPlot[{1/2(NIntegrate[f1[x],{x,0,c*t+sigma}]-NIntegrate[f1[x],{x,0,c*t-sigma}]),1/2(NIntegrate[f2[x],{x,0,c*t+sigma}]-NIntegrate[f2[x],{x,0,c*t-sigma}])},{sigma,0,sigma1}]

t=sigma1/4

ParametricPlot[{1/2(NIntegrate[f1[x],{x,0,c*t+sigma}]-NIntegrate[f1[x],{x,0,c*t-sigma}]),1/2(NIntegrate[f2[x],{x,0,c*t+sigma}]-NIntegrate[f2[x],{x,0,c*t-sigma}])},{sigma,0,sigma1}]

t=sigma1/3

ParametricPlot[{1/2(NIntegrate[f1[x],{x,0,c*t+sigma}]-NIntegrate[f1[x],{x,0,c*t-sigma}]),1/2(NIntegrate[f2[x],{x,0,c*t+sigma}]-NIntegrate[f2[x],{x,0,c*t-sigma}])},{sigma,0,sigma1}]

Animate[ParametricPlot[{1/2(NIntegrate[f1[x],{x,0,c*t+sigma}]-NIntegrate[f1[x],{x,0,c*t-sigma}]),1/2(NIntegrate[f2[x],{x,0,c*t+sigma}]-NIntegrate[f2[x],{x,0,c*t-sigma}])},{sigma,0,sigma1}],{t,0,sigma1},AnimationRunning->False]