set polar
set xrange[-2.9:2.9]
set yrange[-2.9:2.9]
set size square
set zeroaxis lt -1 lw 1

unset tics

set label "v_A = a" at -2,-2 font "Arial,15"
set label "B_0" at -0.55,2.2 font "Arial,12"

set style arrow 1 size character 2,20 filled linewidth 2
set arrow 1 from -0.2,1.7 to -0.2,2.4 arrowstyle 1

set label "(v_A^2+a^2)^{1/2}" at 1.5,-0.2 font "Arial,10"
set label "v_A, a" at -0.5,1.3 font "Arial,10"

a=1
VA=a
p=3.14

plot   VA*cos(p/2-t) title"v_p (shear Alfven wave)" lc rgb "red" linewidth 5, -VA*cos(p/2-t) notitle lc rgb "red" linewidth 5
replot sqrt(0.5*(VA**2+a**2+sqrt((VA**2+a**2)**2-4*VA**2*a**2*cos(p/2-t)**2))) title "v_{pf} (fast mode)" lc rgb "dark-green" linewidth 5
replot sqrt(0.5*(VA**2+a**2-sqrt((VA**2+a**2)**2-4*VA**2*a**2*cos(p/2-t)**2))) title "v_{ps} (slow mode)" lc rgb "blue" linewidth 5


set terminal png
set output "V3.png"
replot