Initial conditions for wave equation One of the common initial conditions given for the wave equation,
$$\frac{\partial^2 p}{\partial t^2} - \nabla^2 p = 0,$$
is $p(\overline{x},t=0) =0$ and $p^\prime (\overline{x},t=0) =0$. What is the physical interpretation of the initial condition $p^\prime (\overline{x}, t=0) =0$?
Edit 1: I messed up the initial conditions. They are now fixed.
Edit 2: Since someone asked, it is an acoustic wave.
 A: Just like in describing the trajectory of an object under a constant force (like an apple falling under gravity), we have a differential equation describing how it moves $F=ma=-mg$ but the full motion is given by $x(t) = x_0 + x'_0 t -\frac{1}{2}gt^2$. The fact this is a second order differential equation means that you need 2 initial parameters, the initial position and the initial velocity. 
Now for the string each point on the string has its own second order differential equation so each point needs an initial position and an initial velocity to describe fully determine the motion.
As a side note I think you should check he initial conditions that you give, because if the string is flat in the beginning, and it is not moving, then it will not move. A far more common initial condition would be something like $p_1(x,t=0) = \text{sin}(x),$ and $p_1'(x,t=0) = 0$. This would describe a string in an initial sin wave shape, with no initial velocity.
Similarly you could have $p_2(x,t=0) = 0,$ and $p_2'(x,t=0) = \text{sin}(x)$. This would be a flat string with initial velocity distributed like a sine wave. These two problems are actually related to each other by a time  shift of $\Delta t = -\frac{\pi}{2}$, with corresponding solutions of $p_1(x,t) = \text{sin}(x)\text{cos}(t)$, and $p_2(x,t) = \text{sin}(x)\text{sin}(t)$
Thanks to @MichaelSeifert for the insightful addition in the comments (which I add here). My solutions are specifically for a string that has both ends tied down at $p(0,t) = p(\pi,t) = 0$. Therefore no disturbances are entering from the outside, and the string will not start moving. If instead we had one side was waving up and down in time, $p(0,t) = f(t)$ this would no longer be the appropriate motion. A good example of this kind of phenomena would be a speaker on one side of an open tube, it is driving the oscillations at $x = 0$ which then disperse over the medium. In this case even if the string was not moving and was flat it would start to move because of the disturbances that are entering into the region of interest.
I say string because that is the simplest way for me to visualize 1 dimensional oscillations, but if this were in 3-dimensions it would just as well explain sound waves in air.
A: Consider a string of length $L$ and mass $M$, and homogeneous density $\lambda = M / L$ and tension $F$, obeying the wave equation
\begin{equation}\tag{1}
\frac{1}{v^2} \, \frac{\partial^2 y}{\partial t^2} - \frac{\partial^2 y}{\partial x^2} = 0,
\end{equation}
where $v = \sqrt{F/\lambda}$ is the wave velocity on the string.  The general solution to (1) is this:
\begin{equation}\tag{2}
y(x, t) = \frac{1}{2} \big(\mathcal{Y}(x - v t) + \mathcal{Y}(x + v t) \big) + \frac{1}{2 v} \int_{x - v t}^{x + v t} \mathcal{V}(u) \, du,
\end{equation}
where $\mathcal{Y}(x) \equiv y(x, 0)$ is the initial displacement of the string (for each $x$) and $\mathcal{V}(x) \equiv \dot{y}(x, 0)$ is the initial velocity of each of its elements.  $\mathcal{Y}(x)$ and $\mathcal{V}(x)$ are arbitrary continuous and derivable functions.
If the string is fixed at $x = 0$: $y(0, t) = 0$, then $\mathcal{Y}(x)$ and $\mathcal{V}(x)$ must be odd functions: $\mathcal{Y}(-v t) = - \mathcal{Y}(v t)$.  For example: $\mathcal{Y}(x) = A \sin{(k \, x)}$ and $\mathcal{V}(x) = 0$ (no velocity at time $t = 0$).  If the string is also fixed at the other side: $y(L, t) = 0$, then $k = n \pi / L$ where $n = 1, 2, 3, \ldots, \infty$.
You may also have a flat string at $t = 0$: $\mathcal{Y}(x) = 0$, but give an initial velocity to each of its elements: $\mathcal{V}(x) = B \sin{(k \, x)}$ (with $k = n \pi / L$ so the string stay fixed at both of its ends: $\mathcal{V}(0) = \mathcal{V}(L) = 0$).
A: To answer your question "What is the physical interpretation of the initial condition $p^\prime (\overline{x}, t=0) =0$ ". 
$p^\prime (\overline{x}, t=0) $ is the initial velocity at time zero as a function of $\overline{x}$ . 
The first initial condition, $p (\overline{x}, t=0) $, is the initial displacement at time zero as a function of $\overline{x}$ .   Note that the initial conditions can be given independently of each other. The initial displacement can be zero with a non zero initial velocity.  If both are zero there will be no subsequent wave (if there is no driving force). 
