Why is it a bad approximation summing accelerations of small loudspeaker-cones for determining sound pressure caused by a vibrating plate? In a previous answer, it was explained that a loudspeaker creates sound pressure which is proportional to its cone acceleration.
Based on this, I got the idea that calculating sound pressure from a vibrating plate can be approximated well by using a lot of little loudspeakers as the plate. For example, I subdivide the plate into 1000 small elements, measure the acceleration of each of them: $a_i(t)$, where $i$ is the index of the little element, then I do a sum: $$p(t) = \frac{\rho S_D}{2 \pi r} \, \sum_{i=1}^{1000} a_i(t)$$
to determine $p(t)$, the sound pressure at a distant point.
But jcandy told me (the one who answered my previous question) this formula is not suitable for doing such a thing like this.
Why is it?
How can I approximate a the sound pressure from a vibrating plate at a distant point?
 A: Actually, the idea of summing acceleration as you suggest is on the right track, but its not directly suitable.  Let me outline a general procedure that is conceptually simple and similar to what you propose; namely, the Method of Fundamental Solutions (MFS).  The text below is actually a summary of the method presented in this AES article.
The pressure for a radiating system satisfies the Helmholtz equation (assuming a uniform, non-viscous medium at rest in the absence of sound).  In the region exterior to all sources, the pressure satisfies the time-dependent wave equation 
\begin{equation}
\left( \nabla^2 - \frac{1}{c_s^2} 
\frac{\partial^2}{\partial t^2} \right) p({\bf x},t) = 0 \; ,
\end{equation}
where the velocity of sound, $c_s$, is given by $c_s = 1/\sqrt{\rho \kappa}$, 
with $\rho$ the density, and $\kappa$ the adiabatic compressibility of air (see Morse and Ingard).  For oscillations at an angular frequency 
$\omega= 2 \pi f$, the pressure is
\begin{equation}
p({\bf x},t) = e^{-i\omega t} p_\omega({\bf x}) \; .
\end{equation}
The function $p_\omega$ is then determined by the solution of 
\begin{equation}
\left( \nabla^2 + k^2 \right) p_\omega({\bf x}) = 0 \; ,
\label{eq.helmholtz}
\end{equation}
where $k = \omega/c_s$ is the wavenumber.  In the case of a loudspeaker enclosure, for example, the velocity on each speaker panel is zero, except at the location of radiating elements (woofer, tweeter).  On a radiating 
element, the boundary condition is
\begin{equation}
-\rho \frac{\partial \mathbf{u}}{\partial t} = \nabla p 
\quad\leftrightarrow\quad
i \omega v_n({\bf x}) = \frac{\partial p_\omega}{\partial n} \; ,
\end{equation}
with $v_n$ the velocity dependence of the radiator.  More specifically, 
if the radiator is a rigid flat piston (circular, square, oval, whatever), then the velocity profile 
is
\begin{equation}
v_n = \left\{
 \begin{array}{cc} 
  V & \mbox{on piston surface} \; ,  \\
  0 & \mbox{off piston surface} \; .  
 \end{array} \right. \; 
\end{equation}
To solve the Helmholtz equation subject to the boundary conditions, we construct a solution by summing Green's functions for the exterior 
Helmholtz problem
\begin{equation}
G_j({\bf x}) = \frac{e^{i k R_j({\bf x})}}{4 \pi R_j({\bf x})} \; ,
\quad\mbox{with}\quad  
R_j = \left| {\bf x}-{\bf x}_j \right| \; .
\label{eq.greenj}
\end{equation}
$G_j$ is equivalently an acoustic point source, and solves the Helmholtz equation everywhere except at ${\bf x} = {\bf x}_j$, where $G_j$ is singular.  But as long as we keep the point sources outside the region of interest there is no problem.  Now expand the pressure due to $n_s$ point sources as
\begin{equation}
\frac{p_\omega({\bf x})}{p_0} = a \sum_{j=1}^{n_s} c_j G_j({\bf x}) \; ,
\end{equation}
where $p_0$ is the normalizing pressure
\begin{equation}
p_0 \doteq i k a \rho c_s V \; .
\end{equation}
Above, $a$ is a normalizing length that is taken to be the radius of the radiating piston, and $V$ is a normalizing velocity that is taken to be the uniform piston velocity, $V$.  Different normalization can be taken to suit the specifics of the problem. 
Thus, instead of summing accelerations, we sum pressures due to point sources.  The coefficients $c_j$ in the summation are chosen to satisfy the boundary conditions on all surfaces.
A: One reason the summation method might not be valid in this application is as follows:
If it is important for you to know the effect of the driving frequency on the radiated sound pressure, then you cannot approximate (for example) the behavior of a single 18" diameter loudspeaker by considering the behavior of a close-packed array of 1" diameter loudspeakers. This is because their frequency responses are completely different; the 18" will have a peak in its radiation resistance somewhere around 500Hz while the 1" speaker will have a radiation resistance peak at about 12kHz. Their mechanical resonances will also be completely different as well.
There may be other reasons why this approximation technique isn't strictly valid and I invite experts in the field to weigh in. 
