y=F(x+at) + f(x-at)
y=F(w) + f(v) (w=x+at, v=x-at)
don't see how to get from here
to partial derivative y with
respect to t
= F"(w)a^2 + f"(v)a^2
ignoring all the stuff with F and f, I assume you meant
y = f(x+at) + f(x-at)
∂y/∂t= f'(w)∂w/dt + f'(v)∂v/dt
= a(f'(w)-f'(v))
∂2y/∂t2 = a(f"(w)∂w/∂t - f"(v)∂v/∂t)
= a^2(f"(w)+f"(v))