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

[ralqs]

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.

 

[lurflurf]

The integral comes from the initial conditions. In particular the fact that we are given a derivative of our solution so our solution must be an integral of what we are given. Think of it in two steps.
We have the operator

(D_t)²-(c D_x)²=(D_t+c D_x)(D_t-c D_x)

In the factored form it is easy to see our solution is the sum of two parts. In one x+ct is held constant and in the other c-ct is constant.

f(x+ct)+g(x-ct)

Then apply the Cauchy boundary condition and preform some elementary algebra to express the solution in terms of the Cauchy data.