Today we look at how the sound field in a (non-physical) room can be calculated analytically. The application we look at is again the subwoofer, and we seek insight as to where to place our subwoofer(s) to get a desired bass response.
The sound pressure in a room is highly dependent on where in the room you place your sources and where you place yourself. Also, the dimensions of the room are important, as the so-called modes are defined by the dimensions and their boundary conditions. For a rectangular room with low damping on the walls an analytical expression The analytical function may not reveal much about an actually room with higher degrees of damping and/or non-rectangular, but it is very good way to learn about the general behavior of the sound field, and also to investigate the different proposed methods for optimum subwoofer placement.
The function gives the pressure phasor p in a point for a volume velocity source phasor Q as
The details of the function can be explored e.g. in , , and , since the blog format does not lend itself well to all of these Greek letters, but note that you sum over the room modes, and that essentially all modes will affect any frequencies, although at varying degrees. Note that the complete sound field is contained in the above expression, and so an additional 'direct sound' term should not be added, even though this is suggested in several papers. If you have some hours to spare, go down the rabbit hole and read this thread on diyaudio.com. Big names like Geddes (gedlee), John Kreskovski (john k...), and Todd Welti (cap'n todd) enter the stage and have some heated discussions, for example about the analytical function itself. Enjoy the mudslinging.
Since the superposition principle has to hold, you can expand the expression to include multiple sources. These sources need not have the same amplitude or phase, and can be positioned arbitrarily within the boundaries of the room. This way, you can investigate e.g. the CABS  approach (I am pretty sure the concept was known before, I just don't remember the abbreviation), where you place subwoofers near your front speakers, and at the rear wall you have more subwoofers with a phase shift related to the lenght of the room, so that you get a situation where the rear wall seems to have been removed, effectively getting rid of some room modes.
I wrote a MATLAB program, where I can input two sources, since I do not imagine anyone having much more than that in a normal living situation. I also modified the sources, so that they better emulate real subwoofers, e.g. they have low frequency roll-off, and their phases and amplitudes can be adjusted individually. Now the fun challenge lies in placing the subwoofers, so that you obtain e.g. the smoothest response in a single point, or the least variation across several points.
I highly recommend writing this program yourself, and start fiddling with different parameters, if you are interested in subwoofers and their sound reproduction in rooms. You can also investigate more elusive ideas, such as 'room gain', to see if it really exists or not ;-).
If you want to expand on the above, so that you can have more complex boundary conditions on the wall or similar modifications, feel free to take on that task. If you come up with some interesting results, feel free to let us all know.
: "Low-Frequency Room Responses: Part 2 -Calculation methods and experimental results", R.Walker, BBC Research Department Report
: "Low-Frequency Optimization Using Multiple Subwoofers", T.Welti, A.Devantier, J. Audio Eng. Soc., Vol. 54, No. 5, 2006 May
: "Low frequency sound field enhancement system for rectangular rooms using multiple loudspeakers", Ph.D. thesis, Adrian Celestinos