How can I validate a numerical model using an analytical solution of the same scenario? In the course of my research project I have been tasked with validating an acoustic model, however I am having trouble conceptualising what I am supposed to do. I have found an analytical solution for a given scenario:
A planar acoustic wave is incident on an infinitely thin sensor at some arbitrary angle.

Analytical solution
I am able to determine the average pressure across the sensor for the given angle (this is time independent). 
Numerical model
In my numerical model I have constructed the numerical analogue of the above situation, however my incident acoustic field is time varying. 
What I would like to do 
I would like to compare the average pressure across the sensor in the analytical and numerical solutions. I'm just not sure how I can compare a time varying spatially averaged pressure (numerical model) to the analytical solution. I'm pretty certain I'm missing something trivial here - but I cannot for the life of me seem to wrap my head around this. 
[edit]
Specific information about the scenario


*

*The amplitude of the plane, harmonic wave is unity and it is incident in the xz plane

*The sensor is of finite diameter

*The domain in the numerical solution is bounded by perfectly matched layers, however I have made the grid large enough (for now) that the acoustic wave does not interact with these yet. All boundaries perpendicular to the plane of the wave are rigid so as to (temporarily) avoid diffraction.


To be more clear, as I understand it the analytical solution is computing the acoustic pressure across the sensor independently of time i.e.
$$
P(x,y,z) = \int_{-\infty}^\infty ...
$$
Whereas the numerical approximation computes it over some finite time i.e.
$$
P(x,y,z,t) = \int_0^t ...
$$
Given this difference, how can I compare one with the other?
 A: Presumably, the analytical solution is using
\begin{equation}
P(x,y,z) = \lim_{T\to \infty} \frac{1}{T} \int_{-T/2}^{T/2} P(x,y,z,t)\, dT
\end{equation}
Note the limit that takes $T$ to infinity.
If the solution is periodic with period $T$, then this is precisely equivalent to writing
\begin{equation}
P(x,y,z) = \frac{1}{T} \int_{-T/2}^{T/2} P(x,y,z,t)\, dT
\end{equation}
which is the same thing over just one cycle.  The average over infinite time averages over infinitely many cycles, but in the truly periodic case, these cycles are all identical, so they don't change the average at all.
Now, your numerical simulation will naturally not be precisely periodic, in the sense that it lasts for only a limited amount of time.  However, that finite amount of time is just a model for the infinite time of the analytical solution.  So if some portion of the numerical solution is periodic, you can just use this last version of the equation over one cycle -- or several cycles to get better numerical behavior.  If it's not periodic over even a single cycle, it sounds like the numerical and analytical models just don't agree very well.
[Note, of course, that you may have to shift your time axis, since you probably just start your simulation with $t=0$ like most of us.  Just shift it so that the integration starts at -- let's say -- the peak of the pressure, and stops at another peak.  Then divide by the time between those peaks to get the average.  Of course, using troughs or any other identifiable feature would work just as well.]
