${y_2} - a\,\cos \left( {\omega t + kx + {\phi _0}} \right)$

$\therefore y = {y_1} + {y_2} = a\,\cos \left( {\omega t - kx + \frac{\pi }{3}} \right)$

$ + a\,\cos \left( {\omega t + kx + {\phi _0}} \right)$

$ = 2a\,\cos \left[ {\omega t + \frac{{\frac{\pi }{3} + {\phi _0}}}{2}} \right] \times \cos \left[ {kx + \frac{{{\phi _0} - \frac{\pi }{3}}}{2}} \right]$

$\because $ $y = 0$ at $x = 0$ for any $t$

$ \Rightarrow kx + \frac{{{\phi _0} - \frac{\pi }{3}}}{2} = \frac{\pi }{2}$

$\therefore {\phi _0} = \frac{{4\pi }}{3}.$ Hence ${y_2} = a\,\cos \left( {\omega t + kx + \frac{{4\pi }}{3}} \right)$