Think about the definition of pressure:
$$P=\frac{F}{A}$$
Now, let's consider the definition of a force.
$$F=\frac{dp}{dt}=m\frac{dv}{dt}$$
Hence, for a given area and particle mass, the pressure is a function of the velocity:
$$P=\frac{m}{A}\frac{dv}{dt} $$
(Aero-)Acoustics (among other parts of fluid-dynamics) loves the velocity potential
It is something that has nothing to do with Acoustic or any other area where potentials come into play. Rather, it is a general property of fields provided certain assumptions hold.
For (usually) simply connected regions whenever you have an irrotational field, that is $\textrm{curl}\,\mathbf {v}=0$, then it exists a function $f$ such that $\mathbf{v} = \textrm{grad}\,f$ in every point $x$ where the domains of definition make sense. The function $f$ is then said to be a potential for the field $\mathbf{v}$. Likewise, knowledge of the potential function in any point of the space allows to derive back the field, equivalently (again, provided the correct assumptions on connectivity of the regions to hold).
This pretty much summarise the entire fairy tale about the potential in any area of physics where they come into play. It is always easier to solve equations involving scalar quantities rather than vectors, therefore whenever potentials may be defined, equations are usually rewritten in terms of the latter for the sake of simplicity. The original vector quantities can eventually be derived back taking derivatives thereof.
Notice on close that, nevertheless, there are some very special cases where, in physics, potentials and field are not completely equivalent (in the sense of invertible one from the other) in every point of space and time; this is mostly due to some very special topologies one deals with: the most remarkable example is the Aharonov-Bohm effect. But that is a corner case and other than that no special features appear when passing from vectors to their potentials except moving the complexity of the equations onto scalar ones.
Best Answer
The Neumann boundary condition is jsut a condition/constraint placed on the gradients of some parameter, $Q$, normal to the boundary surface, or: $$ \mathbf{n} \cdot \nabla Q = f\left(\mathbf{r},t\right) \tag{1} $$ where $\mathbf{n}$ is the outward unit normal vector to the surface boundary and $f\left(\mathbf{r},t\right)$ is some known/given scalar function of position and/or time.
In the specific example you show, there is no pressure gradient along the outward unit normal vector. From the Euler equations, we know that: $$ \rho \left( \partial_{t} \ \mathbf{u} + \mathbf{u} \cdot \nabla \mathbf{u} \right) = - \nabla \ P + \mathbf{F}_{ext} \tag{2} $$ where $\rho$ is the mass density of the fluid, $\mathbf{u}$ is the fluid element velocity, $P$ is the scalar pressure, $\mathbf{F}_{ext}$ is some external force (usually assumed to be gravity), and $\partial_{j}$ is just the partial derivative with respect to parameter $j$. In the absence of gravity or an external force and spherical symmetry, then Equations 1 and 2 show that: $$ \left( \partial_{t} \ \mathbf{u} + \mathbf{u} \cdot \nabla \mathbf{u} \right) = 0 \tag{3} $$
We can further reduce this using the continuity equation which is given by: $$ \partial_{t} \ \rho + \nabla \cdot \left( \rho \ \mathbf{u} \right) = 0 \tag{4} $$ and a steady state assumption to find that $\nabla \cdot \mathbf{u} = 0$. Generally a divergenceless velocity is interpreted as an incompressible flow but in a 1D spherically symmetric system (i.e., only radial direction matters) this also corresponds to no flow.
In a generic dimensional analysis, a flux is just a density multiplied by a velocity. This is often shown in various forms of the continuity equation (e.g., see Equation 4 above for fluid flow), where the first term is the time rate of change of a density and the second is the divergence of a flux. Pressure is a type of momentum flux. Thus, the condition that $\mathbf{n} \cdot \nabla P = 0$ means there is no change in momentum flux along the outward unit normal of the boundary. The general form of pressure is a rank-2 tensor, not a scalar. It reduces to a scalar when the system is symmetric and one-dimensional.