Non-harmonic standing waves in rooms I've thought about this a bit more and decided to rework my question. Same basic idea, but I'll put it completely differently.  
The theory of acoustic standing waves in rooms is straightforward, and, assuming only that two waves of equal magnitude are traveling in opposite directions, leads to a simple equation where the amplitude of vibration depends on position in the room. Considering particle displacement (not pressure) gives:
$$y=2A\sin(2\pi x/\lambda)cos(\omega t)     $$
That's a general relationship describing amplitude versus position for two waves traveling in opposite direction. We haven't said anything about walls yet.
So now let's introduce walls at x=0 and =L, keeping this to one dimension. If you assume perfectly reflective walls (assumed implicitly in the equation above), the condition that the particle displacement be zero at the far wall constrains solutions to $\lambda=2L/n$, where L is a room dimension. Particles can't move longitudinally at the room boundary, and that constrains wavelengths to those given by the relationship above. It would appear then, according to this math, that you cannot have sound in your room at other wavelengths. 
But anyone who has ever set up a stereo or home theater system knows that although "room modes" are a PITA, they do NOT totally remove all other frequencies. Rooms >>do<< support sound at wavelengths that do not satisfy the above relationship. If they didn't, we'd have serious problems with our stereo systems--far beyond the odd standing wave. 
So what's going on? How should I think of this? Do rooms support non-eigen frequencies only because room boundaries aren't perfectly reflective? Is there something else going on? Conceptually or mathematically, how should I think of sound existing in a room at frequencies that don't correspond to room modes? Is there a way to relate amplitude to position and wavelength? 
Thanks. 
 A: It is a very fundamental good question and it is not limited by acoustical waves but is also valid for all harmonic wave propagations bounded by a reflecting surface. Here it is the room, ie, standing waves are observed also for electromagnetic waves very similarly.
I find it helpful to clarify some points first, especially for other readers following this question. What is interference and what are the requirements for interference? I apologize in advance since it will be a bit long. I wanted to explain this fundamental phenomenon using no math, as simple as possible. As Feynman said, 'if you can't explain it in a very simple and clear manner, then, most probably you don't really understand it yourself.' So, I believe math should come at last to verify it.
Directions of contributing waves must be different: 
Standing waves are created by the interference of waves travelling in different directions. Waves travelling in the same direction with same frequency and polarization etc. add up or cancel everywhere and they should not be treated as different waves.
The wavelength of interfering waves must be the same: 
To observe interference it is required that the waves moving in different directions to have the same frequency in order to interfere, add up and build its amplitude in time. Otherwise, it is similar to chaos and all waves add up incoherently. Interference only happens if they have the same wavelength (I can think of selecting two specific frequencies (harmonics) and to find a finite number of specific interference points, but this is out of this discussion.)
The interference pattern: 
It means that the maximum possible value for the total wave amplitude (observed at points in time) is a function of space. That is really interesting when this pattern becomes spatially stationary. 
Standing wave ratio: 
The interference pattern is the amplitude variation in space (not as a function of time and we are talking about the maximum possible value of the total wave amplitude in space.) The standing wave ratio is the ratio of the maximum/minimum value.
Observations: 
1. No reflection case:
If the room were to be covered with fully/ideally absorbing material, there would be no reflections and so, there would be no reflected waves to interfere. That means the standing wave ratio would be 1/1 = 1. This ratio is called Voltage Standing Wave Ratio (VSWR) for the electromagnetic wave case, simply SWR.
2. Single reflection from a single surface: 
Let us assume one of the wall surfaces is covered with a fully reflective material (independent of hard/soft surface boundary properties or E/H polarizations etc.), that is when the power of the incident wave (to the surface) is equal to the power of the reflected wave, we would have equal-power reflected wave. So, taking only those two waves into account, the standing wave ratio becomes infinity since there will be some points in space to add up negatively to have a minimum of zero value where there will be zero total-wave at that point! (we also need enough distance from the wall, actually should be larger than lambda/4 to be clear). 
3. The value of the standing wave ratio (SWR) as a function of time - single reflection case: 
If we assume a wave travelling towards a wall and when there is no reflected wave yet, there will be no standing waves, just the incident. Whenever time passes enough to let the reflected wave travel to distances where waves meet each other, we start to see the standing wave pattern at that point. It is true to say that wherever reflected wave travels, it brings the standing wave effect with it.
So, what are the effects of other walls?
4. Multi reflections from two parallel reflecting surfaces:
Let us move one step forward to assume that our waves are bounded by two surfaces which are perpendicular to both opposite directions of propagation. Then, we would have the reflected wave to reach the other surface and reflect once again. Let us also assume that reflections are perfect - reflected wave power is equal to the incident. This is a very special and interesting case since we can see the power building up between the two walls. In this imaginary ideal case, we expect the total power coming from wherever the source is to built-up inside the room since it is being kept. This ideal case suggests that the energy inside the space diverge to infinity, equally the average power at any point in space to grow in time steadily. Here, the summation of multiple reflected waves is very similar to the sum of a diverging series of infinite length.
In this case, to be able to discuss any standing wave in space, we need to be clear about what we are talking about. Initially, we can observe transient standing waves close to the walls just after the first reflections. However, in the case of a large energy build-up due to a long series of reflected waves, the first two waves' standing-waves would be overweighed by others if the two walls are placed randomly. This is because the standing wave ratio (maximum/minimum ratio) will converge to 1. 
5. Room environment (finite space) - Standing waves at steady state in time:
In order to really talk about a standing wave inside a room, we should be talking about the steady-state case where the reflected waves having enough time to bounce back and forth, and we are observing this finalized pattern (due to loss it will converge.) 
Assuming some loss on the surfaces, the energy build-up will not diverge but will eventually converge to a point where the power generated by the source becomes equal to the total loss inside the room. If the surface loss is small, then the steady-state amplitudes are expected to be very large (think of singing in the shower room or inside a microwave oven). Otherwise, it will be smaller, but we will have some steady-state value.
6. Standing waves inside a room:
In order to have some minimum amplitude at some point(s) inside the room, we need to have interference between some reflected waves (rays) which would repeat their travelling path over and over again. Further, along with this specific path, we need the reflected waves to be coherent to each other, meaning that a full length of such a path should be related to wavelength/2. Given the room geometry and surface reflection properties (coefficients), we might have some ray paths which would provide coherent addition of waves at some specific frequencies.
Case 1: there is no such a path and no such frequency:
This means all waves will be incoherently added yielding a standing wave ratio of 1. This means we can't observe spatial interference.
Case 2: there could be such a (closed-loop) ray path trajectory that would allow a certain set of wavelengths/frequencies (principal critical frequency and its harmonics) to allow such a coherent summation of waves at some part of the room defined by this trajectory. To observe coherence between the series of waves, the closed-loop length needs to be some multiple of wavelength/2.  If that happens we might be able to observe some standing waves. 
Standing wave ratio for Case 2: Generalizing the definition of the standing wave ratio, we would need to observe the 'total' wave which will be composed of coherently interfering waves and the remaining parts. Taking the 'total wave amplitude' into account, if we do have a standing wave ratio sufficiently smaller than 1, we could say that we do have a standing wave in the room. 
Example: These points are generally in the neighbourhood of the far corners of a concert hall.
Now, after all that, I am ready to answer your question.
As you say, the surface reflection coefficient can be close to 1 or -1. It only matters to us if we are talking about the total observed wave amplitude at the surface. The wave might sum up to zero, that is fine. But, when you are observing the total wave inside the room, the phase of the reflection point only moves the standing waves in space, that's all. The magnitude of the reflection coefficient matters most which decides how much energy is stored in space. 
'Particles can't move longitudinally at the room boundary, and that constrains wavelengths to those given by the relationship above.' 
=> Since they can't move longitudinally, they bounce right back with a reflection coefficient of magnitude 1 or somewhat smaller!
'It would appear then, according to this math, that you cannot have sound in your room at other wavelengths.'
For the full reflection case where the surface boundary condition forces the wave to be zero 'only' at the surface boundary does not mean the reflected power or the energy inside the room is zero. Actually, strong reflection helps standing waves to build up.
Do rooms support non-eigenfrequencies only because room boundaries aren't perfectly reflective?  Conceptually or mathematically, how should I think of sound existing in a room at frequencies that don't correspond to room modes? Is there a way to relate amplitude to position and wavelength?
Eigen means 'certain', 'specific', 'characteristic.' In our case, given room geometry and wall properties, there could be such a path and such wavelength(s) that we can have closed-loop path length = some multiple of wavelength/2. This happens independent of the magnitude of the reflection coefficients, only their phases affect the closed-loop phase length (actually it is called the optical path length.) This is often explained mathematically using the definition of 'eigenfrequencies'. Any other frequency means chaos, and incoherence. But, local weaker interferences can create local standing waves which could be observed, and that is why we need to agree on the definition of SWR.
I hope that you find this helpful in understanding the basics of standing waves and interference. GokhunT
