import numpy as np
import math

#load plotting library
import matplotlib.pyplot as plt

def phi(t):
    return np.exp(np.sin(-0.3*t**2+0.2))

def a(t):
    return (10.*t)*np.cos(5*t)

def b(t):
    return (5.*(10-t)-1.5)*np.sin(5*t)

tmin = 2.
tmax = 8.
Nt   = 128

tgrid = np.linspace(tmin,tmax,Nt)
phiv  = phi(tgrid)
psiv  = np.zeros(Nt)
av    = a(tgrid)
bv    = b(tgrid)

plt.plot(tgrid,av,label="a(t)")
plt.plot(tgrid,bv,label="b(t)")
plt.xlabel("t")
plt.legend()
plt.show()

for it in range(Nt):
    psiv[it] = phiv[it] + math.atan2(bv[it],av[it])

plt.plot(tgrid,phiv,label="$phi(t)$")
plt.plot(tgrid,psiv,label="$psi(t)$")
plt.xlabel("t")
plt.legend()
plt.show()