Physical intuition on the integral contained in D'Alembert's Formula for the wave equation If $\phi(t,x)$ is a solution to the one dimensional wave equation and if the initial conditions $\phi(0,x)$ and $\phi_t(0,x)$ are given, then D'Alembert's Formula gives  
$$\phi(t,x)= \frac 12[ \phi(0,x-ct)+ \phi(0,x+ct) ]+ \frac1{2c} \int_{x-ct}^{x+ct} \phi_t(0,y)dy . \tag{1}$$ 
Letting $g(x)=\phi(0,x)$ and $h(x)= \phi_t(0,x)$ (with $c=1$, so $ct=t$) this is commonly written
$$\phi(t,x)= \frac 12[ g(x-t)+ g(x+t) ]+ \frac12 \int_{x-t}^{x+t} h(y)dy . \tag{2}$$ 
My question:
What is the physically intuitive meaning of the integral term?
For example, why does  $\phi_t$ show up in the integral whereas $\phi$ shows up in the forward and backward waves? Why does the integral have those specific limits of integration (and region of integration) for $h$ whereas $g$ uses only the end points, $x+ct,x-ct$? Are there examples with specific functions for $\phi$ that would help to understand this? 
References:
http://mathworld.wolfram.com/dAlembertsSolution.html
https://en.wikipedia.org/wiki/D%27Alembert%27s_formula
http://www.jirka.org/diffyqs/htmlver/diffyqsse32.html
http://people.uncw.edu/hermanr/pde1/dAlembert/dAlembert.htm 
math.ualberta.ca 337week0405.pdf
after equation 180.
stanford univ waveequation3.pdf page 4 Lemma 3 and page 5.
math.nist.gov evolution.pdf page 537
math.usask.ca lamoureax_michael.pdf page 19.
univ. of penn. m425-dalembert-2.pdf first three pages.
univ. of ill. at urbana  286-dalemb.pdf
"Generalized Functions, vol 1", I. M. Gelfand, G. E. Shilov, page 114
"Mathematics for the Physical Sciences", L. Schwartz, pages 253-257
"The Mathematical Theory of Wave Motion", G. R. Baldock, T. Bridgeman, pages 40-45
From these, for example, I know that for a string $\phi_t(0,y)$ represents the velocity at time zero, but why physically (and intuitively) does it end up in the integral. Or I know $(x+ct),(x-ct)$ define the edges of a cone with vertex at $(x,t)$ which form the boundary of the region of the argument of $h$ where $h$ can effect $\phi(t,x)$, but why does the integral have those specific limits of integration for $h$ whereas $g$ uses only the end points, $x+ct,x-ct$?
 A: The D'Alembert solution has a simple interpretation. Using your notation, it reads
$$\phi(x, t) = \frac{g(x-t) + g(x+t)}{2} + \frac12 \int_{x-t}^{x+t} h(x') dx'.$$
where $g(x)$ is the initial position, $h(x)$ is the initial velocity, and $v = 1$.
Mathematically, the first term above solves the wave equation with initial position $g(x)$ but zero initial velocity, while the second term does the same for zero initial position and initial velocity $h(x)$. By the superposition principle, their sum has initial position $g(x)$ and initial velocity $h(x)$, and is hence equal to $\phi(x, t)$ for all times. 
We can understand each of these individual terms by physical intuition.
First consider the term for initial position $g(x)$. We know all solutions of the wave equation are a linear combination of functions of the form $f(x\pm t)$, so the only things we can use are $g(x+t)$ and $g(x-t)$, which both have the right initial position. Finally, we notice that averaging them produces zero initial velocity by the chain rule. Intuitively, if you hold a string and let go, you will make equal-sized waves going in both directions.
Next consider the term for initial velocity $h(x)$. To understand it, consider the following simpler question: suppose $h(x)$ is zero everywhere except for a sharp spike at $x = 0$, corresponding to us hitting the string there at time $t = 0$. What is the resulting shape?
Physically, we expect a 'shockwave' to propagate outward from this event. Solving the wave equation using a similar technique to the one in the previous paragraph, we find
$$y(x, t) = \frac12\left( \theta(x+t) - \theta(x-t) \right).$$
That is, everything inside the "light cone" of $(x, t) = (0, 0)$ has been raised up by $1/2$. 
For a general $h(t)$, then, the solution is to integrate this $1/2$ over all light cones that could have affected the point $(x, t)$. The limits of this light cone are $x-t$ and $x+t$, yielding the term
$$\frac12 \int_{x-t}^{x+t} h(x') dx'$$
in agreement with the formula.
