Intuitively d'Alembert's solution to 1D wave equation

 P: 99 D'Alembert's solution to the wave equation is $$u(x,t) = \frac{1}{2}(\phi(x+ct) + \phi(x-ct)) + \frac{1}{2c}\int_{x-ct}^{x+ct} \psi(\xi)d\xi$$ where $\phi(x) = u(x,0)$ and $\psi(x) = u_t (x,0)$. I'm trying to understand this intuitively. The first term I get: a function like f = 0 (x/=0), = a (x=0) will "break into two functions" and become f = a/2 (x = +/- ct), = 0 (x /= +/- ct). But I can't see how the integral term comes about. Does anyone here have a good physical intuition about this? Thanks.