acousticstime
From this Stack Exchange Physics Post, I am certain that it is possible to view the past.
But then this interesting question came to me.
Is it possible to hear the past?
Ok, you might say, "Well, we are hearing the past, aren't we?" and go onto immense details about the speed of sound against the difference and stuff.
But what if I wanted to hear the sound of a place extremely close by days, weeks, months or even years back?
Would that be possible?
I've been doing experiments related to this back in 1994, so it's going to take a bit of recall.
The idea of a flute is that you create standing waves, which have a frequency that depends on the (variable) geometry. The reason they're standing waves is because you fix specific boundary conditions. In particular, p=0 at an open end.
Now, consider that you have a standing wave in a flute, with an wavelength that is a fraction of your flute length. That means that you have several nodes in the middle. If you would open a key at a node, there would be no effect. If you'd open one near a node, the pitch would change slightly.
The reason you are hearing the train farther away is more consequence of the the geometry of different spaces than anything else.
It starts with an inversion layer of cold air clinging close to the ground. Just as glass bends light by making light move more slowly through it, an inversion layer of cold air bends sound because sound moves more slowly through cold air (the molecules move more slowly is why). So, this inversion layer behaves like the audio equivalent of a large sheet of glass covering the ground and guiding the sound away from the air above it. This is the same principle that allows a fully transparent optical fiber to capture light moving through it and transmit that light for many kilometers with very few losses. (@hwlau already noticed the inversion in the second of his three possible answers.)
On the ground side of the inversion layer, a fresh fall of snow further helps confine and preserve sound because it looks smooth to the long wavelengths of sound. So, even though snow thoroughly jumbles up the much shorter wavelengths of light, which is why it looks white, it looks very different and much more mirror-like to sound.
Put those two together -- diffraction on the top and reflection on the bottom -- and you have an example of two-dimensional sound dispersion. By way of contrast, sound dispersion from a train on a summer day that lacks any inversion layer and has sound-absorbing grass on the ground is an example of largely three-dimensional sound dispersion.
So, why is the dimensionality of the sound dispersion important?
Because sound (or any other radiation) disperses at a rate that is dependent on the number of dimensions of the space into which into which it is dispersing. If $L_n$ is the perceived loudness of the sound, $s$ is the distance to the sound source, and $n$ is the number of space dimensions into which sound is dispersing, the general equation for how loud the train will sound is:
$L_n = 1/s^{n-1}$
Notice that the lower the number of dimensions, the more slowly the sound disperses. (I discussed this same issue from a slightly different perspective a few months ago in my answer to this question about why objects look smaller when they are farther away.)
This equation explains why optical fibers can transmit light many kilometers without loss any significant of intensity ("loudness"). The dispersion space $n$ for optical fibers is $n=1$, so $L_1 = \frac{1}{s^{1-1}} = \frac{1}{s^0} = 1$. That is, there is no diminution of intensity. The audio equivalent would be a long tube, like the ones they used to use as intercoms in old houses (and still use in some playgrounds).
Now for your inversion layer case, $n=2$ and:
$L_2 = 1/s^{n-1} = 1/s$
But because your perception of train distance was tuned to $n=3$ space, you expected the sounds of the train to diminish at the much faster rate of:
$L_3 = 1/s^{n-1} = 1/s^2$
The analysis of how much farther away the train really is turns out to be trickier than it might seem. That's because the model I just described assumes that the energy of a 3D sound source can be compressed into a mathematically precise 2D plane. The physical world just doesn't work that way, since the sound energy in a 3D volume cannot be forced into a true 2D plane without creating infinitely high energy densities in the plane. Why? Well, pretty much for the reason that you cannot compress a 3D volume of air into an infinitely thin 2D plane without creating infinite mass densities. Crossing dimensionalities is often done a bit casually in physics, but one need to be careful with it.
So, in this case, instead of assuming a simple 2D plane, what you have to do is model the problem by using a "pancake" that more realistically represents the thickness of the inversion layer confining the sound. That allows sound intensities that "look" 3D in the immediate vicinity of the train, but then fade off more according to the dimensionality diffusion rules as distances increase to many times the thickness of the inversion layer.
So, everything from this point is obviously guesswork about what happened in your case, but a nice ballpark height for your inversion layer might be 10 meters. Approximating again, that 10 meters also becomes the "unit of equality" for distance from the train at which the sound of the train is perceived as the same in both cases. This approximation should work reasonably well for any more-or-less point source of sound coming from the train, in particular its whistle. So, call this unit of distance $s_u$ for hearing a similar loudness for the whistle $s_u = s_w = 10 m = 0.01$ km.
Alas, it gets messier. The sound of the train itself is anything but a point source, since you may be able to hear wheel-on-rail sounds for very long lengths, such as a kilometer for a long train. That also messes up the model and adds even more complexity in the form of orientation and sound delays. So, I'm going to wrap all of that complexity up into a single huge approximation and say that for a long train, the sound of the all the train wheels on all the track sounds "about the same" for anyone within a kilometer of the train as it passes by, inversion layer or not. So, the length unit for assessing how train track noises changes over distance becomes $s_u = s_t = 1$ km.
The equation now has to be altered slightly so that these "sounds the same" units $s_u$ are factored in:
Actual: $L_2 = s_u/s^{n-1} = s_u/s$
Perceived: $L_3 = s_u/s^{n-1} = s_u/s^2$
Solving for the $s$ distances in terms of loudness:
Actual: $s_2 = s_u/L$
Perceived: $s_3 = \sqrt{s_u/L}$
The error factor $e$ for how far off your distance estimate was then is:
$e = \text{(actual)}/\text{(perceived)} = s_2/s_3 = \frac{s_u/L}{\sqrt{s_u/L}} = \sqrt{s_u/L}$
For the train whistle, $s_u = s_w = 0.01 km$. With $L$ in km:
$e_w = \sqrt{s_u/L} = \sqrt{0.01/L} = 0.1/\sqrt{L}$
For the track noise from the entire train, $s_u = s_t = 1 km$. With $L$ in km:
$e_t = \sqrt{s_u/L} = 1/\sqrt{L}$
So, finally, a couple of very rough estimate examples are possible.
Assume the train is actually about $L = 16$ km away. In that case, the whistle sounds like it is $e_wL$ km away, or:
$e_wL = 16e_w = 16(0.1/\sqrt{16}) = 0.4$ km away.
In the same case, the train track sound will appear to be $e_tL$ km away, or:
$e_tL = 16e_t = 16/\sqrt{16} = 4$ km away.
So, not only are sounds moving through a winter inversion layer highly deceptive for estimating distances, they can be deceptive in different ways at the same time! A point source such as the train whistle may well sound like it is even closer than the train as a whole -- and both perceptions will sound way, way closer than the actual distance.
Best Answer
Luboš' comment really answers your question, but to more specifically address your comment:
The big difference between sound and light is that sound requires a medium to travel in while light does not. In fact light travels best in a vacuum where by definition there is no medium. The reason we can see back 13.5 billion years is because the light has been travelling through an almost perfect vacuum so there's nothing to impede it.
The problem with anything travelling through a medium is that no medium is perfectly elastic and there are always energy losses due to viscous damping. This is why when you strike a bell it may ring for a while but won't ring for anything like 13.5 billion years. You could ring the bell and wait for the sound wave to travel right round the Earth. You'd hear the sound about 120,000 seconds later, so this would allow you to hear a day and a bit into the past. However, unless it was an extraordinarily loud bell the sound would have decayed to below thermal noise in that time so you wouldn't be able to hear it.
This isn't really an answer to your question, but I mention it because it's cool: cosmic events like supernovae form shock waves when they hit ares of concentrated interstellar gas, and this is arguably a form of sound. In that case you can hear sound from many years ago, though you'd need an awfully big ear!