# Does sound propagate further in freezing weather?

A few days ago I went for a walk in the evening. We're having winter with a little snow and freezing temperatures. We're in a quiet, shallow valley with a train station about 1km from us. I heard a train coming so I wanted to wait for it to watch it arriving to the station. To my surprise, although I was hearing the sound coming from behind a hill, the train wasn't appearing. After several minutes, I gave up, and went back, and finally I saw the train arriving after another a few minutes. The train must have been several km away when I first heard it.

I watched this phenomenon later and I realized that also I could hear and understand people talking on much larger distances than usual.

This has not happened before, and my only idea is that it's because of the cold weather. I have two ideas how to explain it:

1. Cold air propagates sound better for some reason.
2. We have a few cm of snow covered by ice crust, as we had freezing rain a few days ago. I guess this can mean sounds aren't absorbed by earth and are reflected instead, which makes them propagate further. (I'd say this is more probable than 1.)

Is any of this reasonable, or is there another explanation?

(I'm not a native speaker so please feel free to correct any language errors.)

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.

• Very interesting. So if I'd be standing on a hill, above an inversion layer, it could be that I wouldn't hear the train at all? – Petr Pudlák Jan 27 '13 at 9:49
• Yes. It can work the other way around also. Occasionally people to see flashes of lightening from a thunderstorm that appears to be only miles away, yet they cannot hear any thunder from the flashes. (I've had it happen; it's eerie.) In that case, the cooler, slow-sound air is in a layer above the warmer air near the ground, channeling the sound away so effectively that not even a rumble gets through. That's also an energetically unstable layering configuration, unlike the stable, cold-below one you mentioned, so it is typically found in association with stormy conditions (the thunderstorm). – Terry Bollinger Jan 27 '13 at 18:52

Cold air decreases the velocity of sound, but doesn't affect it's intensity. However, the snow covered, windless environment would likely reduce the ambient noise environment, making it possible for more distant sounds to be heard than usual. Also, your valley location might magnify distance sounds in unexpected ways.

The are few possible reason or combinations of them:

1. There are wind blowing toward you. As sound are mediated by particles, the intensity of sound can be higher when it arrive your location. You are not necessary feel any wind since it might be blowing above your head.

2. Refraction of sound by different sound speed. This is similar to the effect of total internal reflection in the mirage, but now the direction is reversed. It occurs when the temperature near the ground is lower than the atmosphere so that the speed of sound is slower at the bottom but faster at upper layer. Therefore, the sound will blend toward the ground or even total reflected. This effect is usually observed at evening and night time when sun was set. From your first sentence and description 2, I think it would be the main reason.

3. The absorption of sound by snow. This results in a quiet environment so that you can easily hear the sound far away. Particularly, it means that the sound toward the ground will be absorbed and no reflection, so the sound you hear are likely the sound propagating directly toward you or upward with with the effect #1 and #2 above.

The train is behind a hill so the sound you hear is either reflected by other hills (should be weak in snow day) or refracted by different air layer. You have also mentioned that it easier for you to hear other's talking, so I would think that all three effects take place here, particular the seecond rason.